Physics (042) · Class XII · Chapter-wise Board Questions

A particle of mass m and charge q starts from rest and moves in an electric field E = E₀ i. After travelling a distance x in the field along x-axis, the kinetic energy of the particle will be :

1. q E₀ x²
2. q E₀ x
3. q² E₀ x
4. q² E₀² x

Two small identical metallic balls having charges q and – 2q are kept far at a separation r. They are brought in contact and then separated at distance r2. Compared to the initial force F, they will now :

1. attract with a force F2
2. repel with a force F2
3. repel with a force F
4. attract with a force F

A conducting sphere of radius R has a charge Q on it. Which of the following graphs correctly represents the value of electric field E in the range 0 ≤ x < ∞, where x is the distance of the point from the centre of the sphere ?

Figure shows variation of Coulomb force (F) acting between two point charges with 1r², r being the separation between the two charges (q₁, q₂) and (q₂, q₃). If q₂ is positive and least in magnitude, then the magnitudes of q₁, q₂ and q₃ are such that

1. q₂ < q₃ < q₁
2. q₃ < q₁ < q₂
3. q₁ < q₂ < q₃
4. q₂ < q₁ < q₃

Two charges –q each are placed at the vertices A and B of an equilateral triangle ABC. If M is the mid-point of AB, the net electric field at C will point along

1. CA
2. CB
3. MC
4. CM

Which one of the following statements is correct ?
Electric field due to static charges is

1. conservative and field lines do not form closed loops.
2. conservative and field lines form closed loops.
3. non-conservative and field lines do not form closed loops.
4. non-conservative and field lines form closed loops.

A body acquires charge 8·0 × 10⁻¹² C. The mass of the body :

1. increases by 4·5 × 10⁻⁷ kg
2. decreases by 1·0 × 10⁻⁶ kg
3. decreases by 4·55 × 10⁻²³ kg
4. increases by 9·1 × 10⁻²³ kg

A small metallic sphere S of charge +q is placed exactly at a point midway between two-point charges A (+Q) and B (+Q) where Q >> q. If the sphere is slightly displaced towards charge A and released, then :

1. it will move further towards A
2. it will move towards B
3. it will oscillate about its original position
4. it will not move at all

A charge Q is enclosed by a spherical Gaussian surface of radius R. If the radius is doubled, then the total electric flux through the surface :

1. becomes four times
2. remains the same
3. becomes half
4. becomes twice

A thin plastic rod is bent into a circular ring of radius R. It is uniformly charged with charge density λ. The magnitude of the electric field at its centre is :

1. λ2ε₀R
2. Zero
3. λ4πε₀R
4. λ4ε₀R

Two charged particles P and Q, having the same charge but different masses m_P and m_Q, start from rest and travel equal distances in a uniform electric field E in time t_P and t_Q respectively. Neglecting the effect of gravity, the ratio t_Pt_Q is :

1. m_Pm_Q
2. m_Qm_P
3. m_Pm_Q
4. m_Qm_P

Two identical small conducting balls B₁ and B₂ are given –7 pC and + 4 pC charges respectively. They are brought in contact with a third identical ball B₃ and then separated. If the final charge on each ball is –2 pC, the initial charge on B₃ was

1. –2 pC
2. –3 pC
3. –5 pC
4. –15 pC

Assertion (A) : Equal amount of positive and negative charges are distributed uniformly on two halves of a thin circular ring as shown in figure. The resultant electric field at the centre O of the ring is along OC.

Reason (R) : It is so because the net potential at O is not zero.

1. If both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. If both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. If Assertion (A) is true but Reason (R) is false.
4. If both Assertion (A) and Reason (R) are false.

A point charge situated at a distance ‘r’ from a short electric dipole on its axis, experiences a force F. If the distance of the charge is ‘2r’, the force on the charge will be :

1. F/16
2. F/8
3. F/4
4. F/2

The magnitude of the electric field due to a point charge object at a distance of 4.0 m is 9 N/C. From the same charged object the electric field of magnitude, 16 N/C will be at a distance of

1. 1 m
2. 2 m
3. 3 m
4. 6 m

An electron experiences a force (1·6 × 10^-16 N) i in an electric field E. The electric field E is :

1. (1·0 × 10³ NC) i
2. – (1·0 × 10³ NC) i
3. (1·0 × 10^-3 NC) i
4. – (1·0 × 10^-3 NC) i

Which one of the following is not a scalar quantity ?

1. Electric field
2. Voltage
3. Resistivity
4. Power

Two charges q₁ and q₂ are placed at the centres of two spherical conducting shells of radius r₁ and r₂ respectively. The shells are arranged such that their centres are d [ > (r₁ + r₂)] distance apart. The force on q₂ due to q₁ is :

1. 14π ε₀ q₁q₂d²
2. 14π ε₀ q₁q₂(d − r₁)²
3. Zero
4. 14π ε₀ q₁q₂[d − (r₁ + r₂)]²

An electric dipole of length 2 cm is placed at an angle of 30° with an electric field 2 × 10⁵ N/C. If the dipole experiences a torque of 8 × 10^-3 Nm, the magnitude of either charge of the dipole, is

1. 4 μC
2. 7 μC
3. 8 mC
4. 2 mC

Assertion (A): For three point charges to be in equilibrium, they must be collinear.

Reason (R): One of the three charges must have different polarity than rest of the two.

1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Both Assertion and Reason are false.

A point charge +Q is placed at the centre of a spherical Gaussian surface of radius R. If the radius of the sphere is doubled, the electric flux through the surface will

1. become half
2. become double
3. remain the same
4. become four times

An electric dipole consists of two point charges +1 μC and −1 μC, held 10 cm apart. It is subjected to a uniform electric field of 100 N/C. Calculate the amount of work done in turning the dipole from its position of stable equilibrium to the position of unstable equilibrium, in the field.

Four point charges of 1 μC, –2 μC, 1 μC and –2 μC are placed at the corners A, B, C and D respectively, of a square of side 30 cm. Find the net force acting on a charge of 4 μC placed at the centre of the square.

Three point charges, 1 pC each, are kept at the vertices of an equilateral triangle of side 10 cm. Find the net electric field at the centroid of triangle.

Two point charges of 3 µC and 4 µC are kept in air at (0·3 m, 0) and (0, 0.3 m) in x-y plane. Find the magnitude and direction of the net electric field produced at the origin (0, 0).

Two identical dipoles are arranged in x-y plane as shown in the figure. Find the magnitude and the direction of net electric field at the origin O.

Depict the orientation of an electric dipole in (a) stable and (b) unstable equilibrium in an external uniform electric field.

Write the potential energy of the dipole in each case.

The inward and the outward electric flux through a Gaussian surface are 2ϕ and ϕ respectively.

(a) What is the net charge enclosed by the surface ?

(b) If the net outward flux through the surface were zero, can it be concluded that there were no charges inside the surface ? Justify your answer.

A charge q is placed inside a sphere of radius ‘a’ filled with water and another charge 2q is placed inside cube of side ‘2a’ which is vacuumed inside. Find the ratio of the flux linked with the sphere to that linked with the cube. (Take relative permittivity of water as 80)

(a) Using Gauss’s law, deduce an expression for electric field at a point due to a uniformly charged infinite plane thin sheet.

(b) Two large thin plane sheets, each having surface charge density σ are held close and parallel to each other in air. What is the net electric field at a point (i) inside and (ii) outside, the sheets ?

Figure shows a narrow beam of electrons entering with a velocity of 3 × 10⁷ m/s, symmetrically through the space between two parallel horizontal plates P₁P₁′ and P₂P₂′ kept 2 cm apart.

If each plate is 3 cm long, calculate the potential difference V applied between the plates so that the beam just strikes the end P₂′.

Two infinitely long thin straight wires AB and CD with charge densities – 12 μCm and + 12 μCm are lying parallel to y-axis at points (–1 m, 0, 0) and (1 m, 0, 0) respectively, as shown in the figure. Find the net electric field E at points P₁ (– 2m, 0, 0), O (0, 0, 0) and P₂ (2m, 0, 0).

Two infinitely long straight wires ‘1’ and ‘2’ are placed d distance apart, parallel to each other, as shown in the figure. They are uniformly charged having charge densities λ and – λ2 respectively. Locate the position of the point from wire ‘1’ at which the net electric field is zero and identify the region in which it lies.

Define electric flux. Is it a scalar or a vector quantity ?

A point charge q is kept at a distance of d2 directly above the centre of a square of side d. Use Gauss’ law to obtain the expression for the electric flux through the square.

If the point charge is now moved to a point at a distance ‘d’ from the centre of the square and the side of the square is doubled, explain how the electric flux through the square will be affected.

A cube of side 0·1 m is placed, as shown in the figure, in a region where electric field E = 500 x i exists. Here x is in meters and E in NC⁻¹. Calculate :

(a) the flux passing through the cube, and
(b) the charge within the cube.

1. Define the term ‘electric flux’ and write its dimensions.
2. A plane surface, in shape of a square of side 1 cm is placed in an electric field E = (100 NC) i such that the unit vector normal to the surface is given by n = 0.8i + 0.6k. Find the electric flux through the surface.

Two large plane sheets P₁ and P₂ having charge densities + σ and – 3 σ respectively are arranged parallel to each other as shown in the figure. Find the net electric field (E) at points A, B and C.

Dipoles, whether electric or magnetic, are characterised by their dipole moments, which are vector quantities. Two equal and opposite charges separated by a small distance constitute an electric dipole, while a current carrying loop behaves as a magnetic dipole. Electric dipoles create electric fields around them. Electric dipoles experience a torque when placed in an external electric field.

(i) Two identical electric dipoles, each consisting of charges – q and + q separated by distance d, are arranged in x-y plane such that their negative charges lie at the origin O and positive charges lie at points (d, 0) and (0, d) respectively. The net dipole moment of the system is : [1]

1. – q d (i + j)
2. q d (i + j)
3. q d (i – j)
4. q d (j – i)

(ii) E₁ and E₂ are magnitudes of electric field due to a dipole, consisting of charges –q and +q separated by distance 2a, at points r (>> a) (1) on its axis, and (2) on equatorial plane, respectively. Then (E₁E₂) is : [1]

1. 14
2. 12
3. 2
4. 4

(iii) An electric dipole of dipole moment 5.0 × 10^–8 Cm is placed in a region where an electric field of magnitude 1.0 × 10³ N/C acts at a given instant. At that instant the electric field E is inclined at an angle of 30° to dipole moment P. The magnitude of torque acting on the dipole, at that instant is : [1]

1. 2.5 × 10^–5 Nm
2. 5.0 × 10^–5 Nm
3. 1.0 × 10^–4 Nm
4. 2.0 × 10^–6 Nm

(iv) Answer either (a) or (b) :

(a) An electron is revolving with speed v around the proton in a hydrogen atom, in a circular orbit of radius r. The magnitude of magnetic dipole moment of the electron is : [1]

1. 4 evr
2. 2 evr
3. 12 evr
4. 14 evr

(b) A square loop of side 5.0 cm carries a current of 2.0 A. The magnitude of magnetic dipole moment associated with the loop is : [1]

1. 1.0 × 10^–3 Am²
2. 5.0 × 10^–3 Am²
3. 1.0 × 10^–2 Am²
4. 5.0 × 10^–2 Am²

Electrostatics deals with the study of forces, fields and potentials arising from static charges. Force and electric field, due to a point charge is basically determined by Coulomb’s law. For symmetric charge configurations, Gauss’s law, which is also based on Coulomb’s law, helps us to find the electric field. A charge/a system of charges like a dipole experience a force/torque in an electric field. Work is required to be done to provide a specific orientation to a dipole with respect to an electric field.

Answer the following questions based on the above :

(a) Consider a uniformly charged thin conducting shell of radius R. Plot a graph showing the variation of |E| with distance r from the centre, for points 0 ≤ r ≤ 3R.

(b) The figure shows the variation of potential V with 1r for two point charges Q₁ and Q₂, where V is the potential at a distance r due to a point charge. Find Q₁Q₂.

(c) An electric dipole of dipole moment of 6 × 10⁻⁷ C-m is kept in a uniform electric field of 10⁴ N/C such that the dipole moment and the electric field are parallel. Calculate the potential energy of the dipole.

Electrostatics deals with the study of forces, fields and potentials arising from static charges. Force and electric field, due to a point charge is basically determined by Coulomb’s law. For symmetric charge configurations, Gauss’s law, which is also based on Coulomb’s law, helps us to find the electric field. A charge/a system of charges like a dipole experience a force/torque in an electric field. Work is required to be done to provide a specific orientation to a dipole with respect to an electric field.

Answer the following questions based on the above :

(a) Consider a uniformly charged thin conducting shell of radius R. Plot a graph showing the variation of |E| with distance r from the centre, for points 0 ≤ r ≤ 3R.

(b) The figure shows the variation of potential V with 1r for two point charges Q₁ and Q₂, where V is the potential at a distance r due to a point charge. Find Q₁Q₂.

(c) An electric dipole of dipole moment p is initially kept in a uniform electric field E such that p is perpendicular to E. Find the amount of work done in rotating the dipole to a position at which p becomes antiparallel to E.

(a) An electric dipole consists of two point charges q and –q separated by a distance 2a. Derive an expression for the electric field E due to this dipole at a point distant r from the centre of the dipole on the equatorial plane. Write the expression for the electric field at a far off point, i.e. r >> a.

(b) A dipole is placed in x-y plane such that charges q and –q are located at x = a and x = b respectively. There exists an electric field E = 2 i N/C in the region. Calculate the force F and torque τ experienced by the dipole.

(i) Two infinitely long straight wires having linear charge densities − λ and 3λ are held vertically parallel to each other, distance r apart in free space. Find the nature and magnitude of the force/length exerted by one wire on the other.

(ii) A small hollow conducting sphere of radius r₁ is given a charge Q. It is surrounded by a concentric conducting spherical shell of inner radius r₂ and outer radius r₃, having charge − 3q.

If a point charge 2q were kept at the centre, find :

(I) the electric flux through a concentric spherical Gaussian surface of radius x for (1) x < r₁, and (2) r₁ < x < r₂.

(II) electric field at a point distant x from the centre for (1) x > r₃, and (2) r₁ < x < r₂.

(III) surface charge density on the inner surface of (1) sphere, and (2) shell.

1. What is difference between an open surface and a closed surface ? Draw elementary surface vector dS for a spherical surface S.
2. Define electric flux through a surface. Give the significance of a Gaussian surface. A charge outside a Gaussian surface does not contribute to total electric flux through the surface. Why ?
3. A small spherical shell S₁ has point charges q₁ = –3 μC, q₂ = –2 μC and q₃ = 9 μC inside it. This shell is enclosed by another big spherical shell S₂. A point charge Q is placed in between the two surfaces S₁ and S₂. If the electric flux through the surface S₂ is four times the flux through surface S₁, find charge Q.

1. Two point charges + q and − q are held at (a, 0) and (− a, 0) in x-y plane. Obtain an expression for the net electric field due to the charges at a point (0, y). Hence, find electric field at a far off point (y >> a).
2. Three point charges of − 2 nC, − 1 nC, and + 5 nC are kept at the vertices A, B and C of an equilateral triangle of side 0·2 m. Find the total amount of work done in shifting the charges from A to A₁, B to B₁ and C to C₁. Here A₁, B₁ and C₁ are the midpoints of sides AB, BC and CA, respectively.

1. Show that Gauss’s theorem is consistent with Coulomb’s law. Using it, derive an expression for the electric field due to a uniformly charged thin spherical shell of radius r at a point at a distance y from the centre of the shell such that (I) y > r, and (II) y < r.
2. A point charge of + 2 nC is kept at the origin of a three-dimensional coordinate system. Find the type and magnitude of the charge which should be kept at (0, 0, − 6m) so that the potential due to the system becomes zero at (0, 0, 2m).

1. Obtain an expression for the electric field E due to a dipole of dipole moment p at a point on its equatorial plane and specify its direction. Hence, find the value of electric field :
   (I) at the centre of the dipole (r = 0), and
   (II) at a point r >> a,
   where 2a is the length of the dipole.
2. An electric field E = (10x + 5) i N/C exists in a region in which a cube of side L is kept as shown in the figure. Here x and L are in metres. Calculate the net flux through the cube.

(i) Write Coulomb’s law of electrostatics in vector form. Apply it to determine the electric field at a point due to a system of point charges.

(ii) ABC is an equilateral triangle of side l. Two point charges +2 μC each, are located at points B and C. Find the sign and magnitude of the point charge q to be kept at the midpoint M of the side BC, so that the net electric field at point A becomes zero.

(i) Derive an expression for potential energy of an electric dipole p in an external uniform electric field E. When is the potential energy of the dipole (1) maximum, and (2) minimum ?

(ii) An electric dipole consists of point charges − 1·0 pC and + 1·0 pC located at (0, 0) and (3 mm, 4 mm) respectively in x − y plane. An electric field E = (1000 Vm) i is switched on in the region. Find the torque τ acting on the dipole.

(i) State Gauss’s Law in electrostatics. Apply this to obtain the electric field E at a point near a uniformly charged infinite plane sheet.

(ii) Two long straight wires 1 and 2 are kept as shown in the figure. The linear charge density of the two wires are λ₁ = 10 µC/m and λ₂ = − 20 µC/m. Find the net force F experienced by an electron held at point P.

(i) A charge + Q is placed on a thin conducting spherical shell of radius R. Use Gauss’s theorem to derive an expression for the electric field at a point lying (i) inside and (ii) outside the shell.

(ii) Show that the electric field for same charge density (σ) is twice in case of a conducting plate or surface than in a nonconducting sheet.

1. Using Gauss’s law, show that the electric field E at a point due to a uniformly charged infinite plane sheet is given by E = σ2ε₀ n where symbols have their usual meanings.
2. Electric field E in a region is given by
   E = (5x² + 2) i
   where E is in N/C and x is in meters.
   A cube of side 10 cm is placed in the region as shown in figure.

   

   Calculate (1) the electric flux through the cube, and (2) the net charge enclosed by the cube.

1. A thin spherical shell of radius R has a uniform surface charge density σ. Using Gauss’ law, deduce an expression for electric field (i) outside and (ii) inside the shell.
2. Two long straight thin wires AB and CD have linear charge densities 10 μC/m and –20 μC/m, respectively. They are kept parallel to each other at a distance 1 m. Find magnitude and direction of the net electric field at a point midway between them.

(i) State Coulomb’s law in electrostatics and write it in vector form, for two charges.

(ii) ‘Gauss’s law is based on the inverse-square dependence on distance contained in the Coulomb’s law.’ Explain.

(iii) Two charges A (charge q) and B (charge 2q) are located at points (0, 0) and (a, a) respectively. Let i and j be the unit vectors along x-axis and y-axis respectively. Find the force exerted by A on B, in terms of i and j.

(i) Derive an expression for the electric field at a point on the equatorial plane of an electric dipole consisting of charges q and −q separated by a distance 2a.

(ii) The distance of a far off point on the equatorial plane of an electric dipole is halved. How will the electric field be affected for the dipole ?

(iii) Two identical electric dipoles are placed along the diagonals of a square ABCD of side 2 m as shown in the figure. Obtain the magnitude and direction of the net electric field at the centre (O) of the square.

1. Use Gauss’ law to obtain an expression for the electric field due to an infinitely long thin straight wire with uniform linear charge density λ.
2. An infinitely long positively charged straight wire has a linear charge density λ. An electron is revolving in a circle with a constant speed v such that the wire passes through the centre, and is perpendicular to the plane, of the circle. Find the kinetic energy of the electron in terms of magnitudes of its charge and linear charge density λ on the wire.
3. Draw a graph of kinetic energy as a function of linear charge density λ.

(i) Define electric flux and write its SI unit.

(ii) Use Gauss’ law to obtain the expression for the electric field due to a uniformly charged infinite plane sheet.

(iii) A cube of side L is kept in space, as shown in the figure. An electric field E = (Ax + B) i NC exists in the region. Find the net charge enclosed by the cube.

In a region, the electric potential varies as V = 10 – 50x, where V is in volts and x in meters. The electric field in the region is :

1. 10 N/C along +x
2. 10 N/C along –x
3. 50 N/C along +x
4. 50 N/C along –x

A conducting wire connects two charged metallic spheres A and B of radii r₁ and r₂ respectively. The distance between the spheres is very large compared to their radii. The ratio of electric fields, (E_A/E_B) at the surfaces of spheres A and B will be

1. r₁r₂
2. r₂r₁
3. r₁²r₂²
4. r₂²r₁²

Three point charges 2q, –2q and q are kept at the vertices of an equilateral triangle of side l. The potential energy of the system is

1. zero
2. –2q²πε₀l
3. q²2πε₀l
4. – q²πε₀l

Two metal spheres of radii r₁ and r₂ (> r₁) having charges q₁ and q₂ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

1. The total charge of the two spheres is conserved.
2. Both spheres attain the same potential.
3. The final potential of the system equals 14πε₀ (q₁ + q₂)(r₁ + r₂)
4. The final potential of the system equals 14πε₀ (q₁ + q₂) (r₁ + r₂)r₁ r₂

The figure represents the variation of the electric potential V at a point in a region of space as a function of its position along the x-axis. A charged particle will experience the maximum force at :

1. P
2. Q
3. R
4. S

Assume that the surface charge density of the Earth is 1 electron charge/m². The potential of the Earth’s surface (radius 6400 km) will be :

1. – 0·12 V
2. – 0·21 V
3. + 2·1 V
4. + 1·2 V

A metal sheet is inserted between the plates of a parallel plate capacitor of capacitance C. If the sheet partly occupies the space between the plates, the capacitance :

1. remains C
2. becomes greater than C
3. becomes less than C
4. becomes zero

The electric field at a point in a region is given by E = α r|r|³, where α is a constant and r is the distance of the point from the origin. The magnitude of potential of the point is :

1. αr
2. α r²2
3. α2r²
4. – αr

Two horizontal plates, separated by 1 cm, are arranged one above the other. A particle of mass 5 mg and charge 2 nC is released in air between the plates. The potential difference that should be applied to the plates so that the particle remains suspended between them, is :

1. 250 V
2. 200 V
3. 100 V
4. 50 V

Questions number 13 to 16 are Assertion (A) and Reason (R) type questions. Two statements are given — one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.

Assertion (A) : Electrostatic force is a conservative force.

Reason (R) : In an electrostatic field, the work done between two points per unit positive charge depends on the path followed.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Both Assertion (A) and Reason (R) are false.

Ten capacitors, each of capacitance 1 µF, are connected in parallel to a source of 100 V. The total energy stored in the system is equal to :

1. 10⁻² J
2. 10⁻³ J
3. 0·5 × 10⁻³ J
4. 5·0 × 10⁻² J

Consider a group of charges q₁, q₂, q₃ ... such that Σq ≠ 0. Then equipotentials at a large distance, due to this group are approximately :

1. Plane
2. Spherical surface
3. Paraboloidal surface
4. Ellipsoidal surface

A proton is taken from point P₁ to point P₂, both located in an electric field. The potentials at points P₁ and P₂ are – 5 V and + 5 V respectively. Assuming that kinetic energies of the proton at points P₁ and P₂ are zero, the work done on the proton is :

1. – 1·6 × 10^–18 J
2. 1·6 × 10^–18 J
3. Zero
4. 0·8 × 10^–18 J

Two charges + q each are kept ‘2a’ distance apart. A third charge – 2q is placed midway between them. The potential energy of the system is –

1. q²8πε₀a
2. –6q²8πε₀a
3. – 7q²8πε₀a
4. 9q²8πε₀a

Two particles A and B of the same mass but having charges q and 4q respectively, are accelerated from rest through different potential differences V_A and V_B such that they attain same kinetic energies. The value of (V_AV_B) is :

1. 14
2. 12
3. 2
4. 4

A parallel plate capacitor is charged by a battery. The battery is then disconnected and the plates of the charged capacitor are then moved farther apart. In the process :

1. the charge on the capacitor increases.
2. the potential difference across the plates decreases.
3. the capacitance of the capacitor increases.
4. the electrostatic energy stored in the capacitor increases.

A point P lies at a distance x from the mid point of an electric dipole on its axis. The electric potential at point P is proportional to

1. 1x²
2. 1x³
3. 1x⁴
4. 1x^1/2

Assertion (A) : Work done in moving a charge around a closed path, in an electric field is always zero.

Reason (R) : Electrostatic force is a conservative force.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is NOT the correct explanation of Assertion (A).
3. Assertion (A) is true and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is also false.

The capacitors, each of 4 μF are to be connected in such a way that the effective capacitance of the combination is 6 μF. This can be achieved by connecting

1. All three in parallel
2. All three in series
3. Two of them connected in series and the combination in parallel to the third.
4. Two of them connected in parallel and the combination in series to the third.

If a charged hollow sphere and a solid sphere of aluminum and copper of equal radii are in electrostatic equilibrium, then which of the following statements is true?

1. Both the spheres are having equal charges.
2. The hollow sphere will have more charge than solid sphere at its surface.
3. The aluminum sphere will have more charge on its surface than copper sphere.
4. If hollow sphere is also made up of aluminum then it will have more charge.

Assertion (A): If a conductor is placed in an external electric field, the electric field inside the conductor becomes infinity in electrostatic equilibrium.

Reason (R): Free electrons inside the conductor rearrange themselves in such a way that the internal electric field is in the same direction as that of the external field.

1. both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. both Assertion and Reason are false.

Obtain an expression for electrostatic potential energy of a system of three charges q, 2q and −3q placed at the vertices of an equilateral triangle of side a.

Two small conducting balls A and B of radius r₁ and r₂ have charges q₁ and q₂ respectively. They are connected by a wire. Obtain the expression for charges on A and B, in equilibrium.

A uniform electric field E of 500 N/C is directed along +x axis. O, B and A are three points in the field having x and y coordinates (in cm) (0, 0), (4, 0) and (0, 3) respectively. Calculate the potential difference between the points (i) O and A, and (ii) O and B.

Three point charges 1 μC, −1 μC and 2 μC are kept at the vertices A, B and C respectively of an equilateral triangle of side 1 m. A₁, B₁ and C₁ are the midpoints of the sides AB, BC and CA respectively. Calculate the net amount of work done in displacing the charge from A to A₁, from B to B₁ and from C to C₁.

Two like charges +q and +q are placed at a distance d. What work must be done by an external agent to bring another charge +q from infinity to the midpoint of the line joining the two given charges?

A parallel plate capacitor of capacitance C has a dielectric slab between its plates. It is charged to a potential difference V by connecting it across a battery. The battery is then disconnected. If the dielectric slab is now withdrawn from the capacitor, how will the following be affected ?

1. Capacitance of the capacitor,
2. Energy stored in the capacitor, and
3. The potential difference between the plates of the capacitor.

Justify your answer in each case.

(a) An electric field E = E₀i exists in a region of space. Draw three equipotential surfaces in the region.

(b) Two point charges –q and +q are located at points (–a, 0, 0) and (a, 0, 0) respectively. Find the electrostatic potential at point (x, 0, 0) where x ≫ a.

Two parallel plate capacitors X and Y are connected in series to a 6 V battery. They have the same plate area and same plate separation but capacitor X has air between its plates, whereas capacitor Y contains a material of dielectric constant 4.

1. Calculate the capacitances of X and Y, if the equivalent capacitance of the combination of X and Y is 4 μF.
2. Calculate the potential difference across the plates of X and Y.

1. Briefly explain the process of charging a parallel plate capacitor, when it is connected across a dc source. [1]
2. A capacitor, of capacitance C, plate area A and plate separation d is charged to V volts by a battery. The energy stored in it is U. After some time, the battery is disconnected and a slab of dielectric constant K, is inserted to fill the space between the plates of the capacitor. Let V′ and U′ now be the potential difference across the plates of the capacitor and the energy stored in it respectively. Find (V′V) and (U′U). [2]

Two charges of 8 μC and – 4 μC are located at points (–18 cm, 0, 0) and (18 cm, 0, 0) respectively.

1. Calculate the electrostatic potential energy of the system of the two charges.
2. The same system of charges is placed in a region where electrostatic potential V(r) = Ar, where A = 9 × 10⁴ V.m. Calculate the potential energy of the system.

A parallel plate capacitor has plate area A and plate separation d. Half of the space between the plates is filled with a material of dielectric constant K in two ways as shown in the figure.

Find the values of the capacitance of the capacitors in the two cases.

Two small solid metal balls A and B of radii R and 2R having charge densities 2σ and 3σ respectively are kept far apart. Find the charge densities on A and B after they are connected by a conducting wire.

The electric field in a region is given by

E = (10x + 4) i

where x is in m and E is in N/C. Calculate the amount of work done in taking a unit charge from

1. (5 m, 0) to (10 m, 0)
2. (5 m, 0) to (5 m, 10 m)

Two point charges of 10 µC and 20 µC are located at points (– 4 cm, 0, 0) and (5 cm, 0, 0) respectively, in a region with electric field E = Ar², where A = 2 × 10⁶ NC^–1 m² and r is the position vector of the point under consideration. Calculate the electrostatic potential energy of the system.

(a) Twelve negative charges of same magnitude are equally spaced and fixed on the circumference of a circle of radius R as shown in Fig. (i). Relative to potential being zero at infinity, find the electric potential and electric field at the centre C of the circle.

(b) If the charges are unequally spaced and fixed on an arc of 120° of radius R as shown in Fig. (ii), find electric potential at the centre C.

Two charged conducting spheres of radii a and b are connected to each other by a wire. Find the ratio of the electric fields at their surfaces.

A parallel plate capacitor (A) of capacitance C is charged by a battery to voltage V. The battery is disconnected and an uncharged capacitor (B) of capacitance 2C is connected across A. Find the ratio of

1. final charges on A and B.
2. total electrostatic energy stored in A and B finally and that stored in A initially.

Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance U = 12CV², where symbols have their usual meanings.

Two capacitors, one of 3 μF and the other of 6 μF, are connected in series in the circuit as shown in the figure, for a long time.

(i) The total capacitance of the circuit is :

1. 6 μF
2. 3 μF
3. 9 μF
4. 2 μF

(ii) The current in the 10 Ω resistor is :

1. 0·3 A
2. 0·6 A
3. 0·2 A
4. 0

(iii) The potential difference between point A and B is :

1. 2 V
2. 0·3 V
3. 0·2 V
4. 3 V

(iv) Answer either (a) or (b) :

(a) The value of charge on the plates of the 6 μF capacitor is :

1. 6 μC
2. 4 μC
3. 12 μC
4. 8 μC

(b) The wire between two capacitors is cut at point P. The current in the circuit will :

1. increase
2. decrease
3. remain the same
4. first increase then become stable

The electric potential (V) and electric field (E) are closely related concepts in electrostatics. The electric field is a vector quantity that represents the force per unit charge at a given point in space, whereas electric potential is a scalar quantity that represents the potential energy per unit charge at a given point in space. Electric field and electric potential are related by the equations E_r = –dVdr and E = E_r r, i.e., electric field is the negative gradient of the electric potential. This means that electric field points in the direction of decreasing potential and its magnitude is the rate of change of potential with distance. The electric field is the force that drives a unit charge to move from higher potential region to lower potential region and electric potential difference between the two points determines the work done in moving a unit charge from one point to the other point.

A pair of square conducting plates having sides of length 0·05 m are arranged parallel to each other in x-y plane. They are 0·01 m apart along z-axis and are connected to a 200 V power supply as shown in the figure. An electron enters with a speed of 3 × 10⁷ ms^–1 horizontally and symmetrically in the space between the two plates. Neglect the effect of gravity on the electron.

(i) The electric field E in the region between the plates is :

1. (2 × 10² Vm) k
2. – (2 × 10² Vm) k
3. (2 × 10⁴ Vm) k
4. – (2 × 10⁴ Vm) k

(ii) In the region between the plates, the electron moves with an acceleration a given by :

1. – (3·5 × 10¹⁵ ms^–2) k
2. (3·5 × 10¹⁵ ms^–2) k
3. (3·5 × 10¹³ ms^–2) i
4. – (3·5 × 10¹³ ms^–2) i

(iii) (a) Time interval during which an electron moves through the region between the plates is :

1. 9·0 × 10^–9 s
2. 1·67 × 10^–8 s
3. 1·67 × 10^–9 s
4. 2·17 × 10^–9 s

OR

(b) The vertical displacement of the electron which travels through the region between the plates is :

1. 10 mm
2. 4·9 mm
3. 5·9 mm
4. 3·0 mm

(iv) Which one of the following is the path traced by the electron in between the two plates ?

1. a
2. b
3. c
4. d

A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit.

As soon as the circuit is completed by closing key S₁ (keeping S₂ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference V_c (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as q = Q[1 – e^–t/RC]

The charging current can be obtained by differentiating it and using ddx (e^mx) = me^mx.

Consider the case when R = 20 kΩ, C = 500 μF and V = 10 V.

(i) The final charge on the capacitor, when key S₁ is closed and S₂ is open, is

1. 5 μC
2. 5 mC
3. 25 mC
4. 0.1 C

(ii) For sufficient time the key S₁ is closed and S₂ is open. Now key S₂ is closed and S₁ is open. What is the final charge on the capacitor ?

1. Zero
2. 5 mC
3. 2.5 mC
4. 5 μC

(iii) The dimensional formula for RC is

1. [M L² T^–3 A^–2]
2. [M⁰ L⁰ T^–1 A⁰]
3. [M^–1 L^–2 T⁴ A²]
4. [M⁰ L⁰ T A⁰]

(iv) The key S₁ is closed and S₂ is open. The value of current in the resistor after 5 seconds, is

1. 12e mA
2. e mA
3. 1e mA
4. 12e mA

Alternative to (iv) : (iv) The key S₁ is closed and S₂ is open. The initial value of charging current in the resistor, is

1. 5 mA
2. 0.5 mA
3. 2 mA
4. 1 mA

A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.

1. The electric field between the plates of a parallel plate capacitor is E. Now the separation between the plates is doubled and simultaneously the applied potential difference between the plates is reduced to half of its initial value. The new value of the electric field between the plates will be :
   1. E
   2. 2E
   3. E4
   4. E2
2. A constant electric field is to be maintained between the two plates of a capacitor whose separation d changes with time. Which of the graphs correctly depict the potential difference (V) to be applied between the plates as a function of separation between the plates (d) to maintain the constant electric field ?

   1. 

   2. 

   3. 

   4. 

3. 

   In the above figure P, Q are the two parallel plates of a capacitor. Plate Q is at positive potential with respect to plate P. MN is an imaginary line drawn perpendicular to the plates. Which of the graphs shows correctly the variations of the magnitude of electric field strength E along the line MN ?

   1. 

   2. 

   3. 

   4. 

(iv) Alternative 1 –

Three parallel plates are placed above each other with equal displacement d between neighbouring plates. The electric field between the first pair of the plates is E₁ and the electric field between the second pair of the plates is E₂. The potential difference between the third and the first plate is –

1. (E₁ + E₂) · d
2. (E₁ – E₂) · d
3. (E₂ – E₁) · d
4. d(E₁ + E₂)2

(iv) Alternative 2 –

A material of dielectric constant K is filled in a parallel plate capacitor of capacitance C. The new value of its capacitance becomes

1. C
2. CK
3. CK
4. C(1 + 1K)

A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side ‘L’ held parallel and equidistant from each other. The space between P₁ and P₂ and P₂ and P₃ is completely filled with mica sheets of dielectric constant ‘K’. The plate P₂ is connected to point A and other plates P₁ and P₃ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.

1. The capacitance of the system between A and B will be :
   1. ε₀KL²d
   2. ε₀KL²2d
   3. 2ε₀KL²d
   4. 2ε₀KdL²
2. The charge on plate P₁ is :
   1. ε₀ VKL²2d
   2. ε₀ VKL²d
   3. 2ε₀ VKL²d
   4. ε₀ VKL²4d
3. The electric field in the region between P₁ and P₂ is :
   1. Vd
   2. 2Vd
   3. V2d
   4. dV
4. (a) The separation between the plates of same area (L²) of a parallel plate air capacitor having capacitance equal to that of this system, will be :
   1. dK
   2. 2dK
   3. d2K
   4. d4K

   OR

   (b) If the source of potential difference applied between A and B is removed, and then A and B are connected by a conducting wire, the net charge on the system will be :

   1. ε₀ VKL²4d
   2. ε₀ VKL²2d
   3. ε₀ VKL²d
   4. Zero

A capacitor is a system of two conductors separated by an insulator. In practice, the two conductors have charges Q and – Q with potential difference V = V₁ – V₂ between them. The ratio QV is a constant, denoted by C and is called the capacitance of the capacitor. It is independent of Q or V. It depends only on the geometrical configuration (shape, size, separation) of the two conductors and the medium separating the conductors. When a parallel plate capacitor is charged, the electric field E₀ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density σ), the dielectric is polarised by the field. Consequently opposite charges appear on the faces of the slab, near the plates, with surface charge density of magnitude σ_p. For a linear dielectric σ_p is proportional to E₀. Introduction of a dielectric changes the electric field, and hence, the capacitance of a capacitor, and hence, the energy stored in the capacitor.

Like resistors, capacitors can also be arranged in series or in parallel or in a combination of series and parallel.

(i) Consider a capacitor of capacitance C, with plate area A and plate separation d, filled with air [Fig. (a)]. The distance between the plates is increased to 2d and one of the plates is shifted as shown in Fig. (b). The capacitance of the new system now is :

1. C4
2. C2
3. 2C
4. 4C

(ii) A slab (area A and thickness d₁) of a linear dielectric of dielectric constant K is inserted between charged plates (charge density σ) of a parallel plate capacitor [plate area A and plate separation d (> d₁)] and opposite charges with charge density of magnitude σ_p appear on the faces of the slab. The dielectric constant K is given by :

1. σ + σ_pσ
2. σσ – σ_p
3. σ + σ_pσ_p
4. σσ_p

(iii) An electric field E is established between the plates of an air filled parallel plate capacitor, with charges Q and – Q. V is the volume of the space enclosed between the plates. The energy stored in the capacitor is :

1. 12 ε₀E²
2. ε₀Q²E
3. 12 ε₀E²V
4. ε₀EQV

(iv) (a) Three capacitors A, B and M, each of capacitance C are connected to a capacitor N of capacitance 2C and a battery as shown in the figure. If the charges on A and N are Q and Q′ respectively, then Q′Q is :

1. 16
2. 13
3. 3
4. 6

(iv) (b) (alternative to (iv)(a)) A slab (area A and thickness d2) of dielectric constant K is inserted in a parallel plate capacitor of plate area A and plate separation d. If C and C₀ are the capacitances of the capacitors with and without the dielectric, then CC₀ is :

1. K + 12K
2. 2KK + 1
3. KK – 1
4. K – 1K

The figure shows four pairs of parallel identical conducting plates, separated by the same distance 2·0 cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.

1. For which pair of the plates is the electric field E along i ?
   1. I
   2. II
   3. III
   4. IV
2. An electron is released midway between the plates of pair IV. It will :
   1. move along i at constant speed
   2. move along – i at constant speed
   3. accelerate along i
   4. accelerate along – i
3. Let V₀ be the potential at the left plate of any set, taken to be at x = 0 m. Then potential V at any point (0 ≤ x ≤ 2 cm) between the plates of that set can be expressed as :
   1. V = V₀ + αx
   2. V = V₀ + αx²
   3. V = V₀ + αx^1/2
   4. V = V₀ + αx^3/2
   where α is a constant, positive or negative.
4. (a) Let E₁, E₂, E₃ and E₄ be the magnitudes of the electric field between the pairs of plates, I, II, III and IV respectively. Then :
   1. E₁ > E₂ > E₃ > E₄
   2. E₃ > E₄ > E₁ > E₂
   3. E₄ > E₃ > E₂ > E₁
   4. E₂ > E₃ > E₄ > E₁

   OR

   (b) An electron is projected from the right plate of set I directly towards its left plate. It just comes to rest at the plate. The speed with which it was projected is about : (Take (e/m) = 1·76 × 10¹¹ C/kg)
   1. 1·3 × 10⁵ m/s
   2. 2·6 × 10⁶ m/s
   3. 6·5 × 10⁵ m/s
   4. 5·2 × 10⁷ m/s

Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric.

The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.

1. Which of the following is a polar molecule ?
   1. O₂
   2. H₂
   3. N₂
   4. HCl
2. Which of the following statements about dielectrics is correct ?
   1. A polar dielectric has a net dipole moment in absence of an external electric field which gets modified due to the induced dipoles.
   2. The net dipole moments of induced dipoles is along the direction of the applied electric field.
   3. Dielectrics contain free charges.
   4. The electric field produced due to induced surface charges inside a dielectric is along the external electric field.
3. When a dielectric slab is inserted between the plates of an isolated charged capacitor, the energy stored in it :
   1. increases and the electric field inside it also increases.
   2. decreases and the electric field also decreases.
   3. decreases and the electric field increases.
   4. increases and the electric field decreases.
4. An air-filled capacitor with plate area A and plate separation d has capacitance C₀. A slab of dielectric constant K, area A and thickness (d5) is inserted between the plates. The capacitance of the capacitor will become
   1. [4K5K + 1]C₀
   2. [K + 54]C₀
   3. [5K4K + 1]C₀
   4. [K + 45K]C₀

Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric.

The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.

1. Which of the following is a polar molecule ?
   1. O₂
   2. H₂
   3. N₂
   4. HCl
2. Which of the following statements about dielectrics is correct ?
   1. A polar dielectric has a net dipole moment in absence of an external electric field which gets modified due to the induced dipoles.
   2. The net dipole moments of induced dipoles is along the direction of the applied electric field.
   3. Dielectrics contain free charges.
   4. The electric field produced due to induced surface charges inside a dielectric is along the external electric field.
3. When a dielectric slab is inserted between the plates of an isolated charged capacitor, the energy stored in it :
   1. increases and the electric field inside it also increases.
   2. decreases and the electric field also decreases.
   3. decreases and the electric field increases.
   4. increases and the electric field decreases.
4. Two capacitors of capacitances 2 C₀ and 6 C₀ are first connected in series and then in parallel across the same battery. The ratio of energies stored in series combination to that in parallel is
   1. 14
   2. 16
   3. 215
   4. 316

A beam of electrons moving horizontally with a velocity of 3 × 10⁷ m/s enters a region between two plates as shown in the figure. A suitable potential difference is applied across the plates such that the electron beam just strikes the edge of the lower plate.

Answer the following questions based on the above :

(a) How long does an electron take to strike the edge ? [1]

(b) What is the shape of the path followed by the electron and why ? [1]

(c) Find the potential difference applied. [2]

Alternative to (c) :

(c) Find the magnitude and direction of the magnetic field which should be created in the space between the plates so that the electron beam goes straight undeviated. [2]

A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.

1. Find the equivalent capacitance between points A and B in the given diagram.

   

2. A dielectric slab is inserted between the plates of a parallel plate capacitor. The electric field between the plates decreases. Explain.
3. A capacitor A of capacitance C, having charge Q is connected across another uncharged capacitor B of capacitance 2C. Find an expression for (a) the potential difference across the combination and (b) the charge lost by capacitor A.

A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.

1. Find the equivalent capacitance between points A and B in the given diagram.

   

2. A dielectric slab is inserted between the plates of a parallel plate capacitor. The electric field between the plates decreases. Explain.
3. Two slabs of dielectric constants 2K and K fill the space between the plates of a parallel plate capacitor of plate area A and plate separation d as shown in figure. Find an expression for capacitance of the system.

   

A parallel plate capacitor is an arrangement of two identical metal plates kept parallel, a small distance apart. The capacitance of a capacitor depends on the size and separation of the two plates and also on the dielectric constant of the medium between the plates. Like resistors, capacitors can also be arranged in series or parallel or a combination of both. By virtue of electric field between the plates, charged capacitors store energy.

(a) The capacitance of a parallel plate capacitor increases from 10 μF to 80 μF on introducing a dielectric medium between the plates. Find the dielectric constant of the medium. [1]

(b) n capacitors, each of capacitance C, are connected in series. Find the equivalent capacitance of the combination. [1]

(c) A capacitor is charged to a potential (V) by connecting it to a battery. After some time, the battery is disconnected and a dielectric is introduced between the plates. How will the potential difference between the plates, and the energy stored in it be affected ? Justify your answer. [2]

A parallel plate capacitor is an arrangement of two identical metal plates kept parallel, a small distance apart. The capacitance of a capacitor depends on the size and separation of the two plates and also on the dielectric constant of the medium between the plates. Like resistors, capacitors can also be arranged in series or parallel or a combination of both. By virtue of electric field between the plates, charged capacitors store energy.

(a) The capacitance of a parallel plate capacitor increases from 10 μF to 80 μF on introducing a dielectric medium between the plates. Find the dielectric constant of the medium. [1]

(b) n capacitors, each of capacitance C, are connected in series. Find the equivalent capacitance of the combination. [1]

(c) Find the equivalent capacitance between points A and B, if capacitance of each capacitor is C. [2]

(a) Derive an expression for the capacitance of a parallel plate capacitor of plate area A and plate separation d with air present between the plates.

(b) Two air-filled capacitors of capacitances C₁ and C₂ are connected in parallel with a dc battery. After the capacitors are fully charged, a slab of dielectric constant K is inserted between the plates of each capacitor. How will the (i) charge on each capacitor and (ii) energy stored in the capacitor affected after the slab is introduced.

(i) In the figure, OA and OB show the variation of electric potential V at a point due to two point charges Q₁ and Q₂ with 1r respectively. Here r represents the distance of the point from the two point charges.

(I) Identify the nature of the two charges Q₁ and Q₂.

(II) What is the value of (Q₁Q₂) ? Justify your answer.

(ii) Two point charges − 2 μC and 5 μC are placed at (− 30 cm, 0) and (30 cm, 0) respectively in an external electric field E = Ax² i, where A = 9 × 10⁵ Nm² C⁻¹. Find the electrostatic potential energy of this configuration.

(a) (i) Explain the following statements giving reason :

1. An equipotential surface through a point is normal to the electric field at that point.
2. When a dielectric is placed in an external electric field, the electric field inside the dielectric is less than that outside it.
3. The potential difference between the plates of a charged parallel plate capacitor decreases when its plates are brought closer.

(ii) Obtain an expression for the work done to dissociate the system of three charges q, − 4q and 2q placed at the vertices A, B and C respectively of an equilateral triangle of side ‘a’.

(i) Two point charges 5 μC and –1 μC are placed at points (–3 cm, 0, 0) and (3 cm, 0, 0) respectively. An external electric field E = Ar² r where A = 3 × 10⁵ Vm is switched on in the region. Calculate the change in electrostatic energy of the system due to the electric field.

(ii) A system of two conductors is placed in air and they have net charge of +80μC and –80μC which causes a potential difference of 16 V between them.

(1) Find the capacitance of the system.

(2) If the air between the capacitor is replaced by a dielectric medium of dielectric constant 3, what will be the potential difference between the two conductors ?

(3) If the charges on two conductors are changed to +160 μC and –160 μC, will the capacitance of the system change ? Give reason for your answer.

(i) Consider three metal spherical shells A, B and C, each of radius R. Each shell is having a concentric metal ball of radius R/10. The spherical shells A, B and C are given charges +6q, –4q, and 14q respectively. Their inner metal balls are also given charges –2q, +8q and –10q respectively. Compare the magnitude of the electric fields due to shells A, B and C at a distance 3R from their centres.

(ii) A charge –6 μC is placed at the centre B of a semicircle of radius 5 cm, as shown in the figure. An equal and opposite charge is placed at point D at a distance of 10 cm from B. A charge +5 μC is moved from point ‘C’ to point ‘A’ along the circumference. Calculate the work done on the charge.

1. A small conducting sphere A of radius r charged to a potential V, is enclosed by a spherical conducting shell B of radius R. If A and B are connected by a thin wire, calculate the final potential on sphere A and shell B.
2. Write two characteristics of equipotential surfaces. A uniform electric field of 50 NC^-1 is set up in a region along +x axis. If the potential at the origin (0, 0) is 220 V, find the potential at a point (4m, 3m).

1. The electric field in a region is given by E = 40x i N/C. Find the amount of work done in taking a unit positive charge from a point (0, 3m) to the point (5m, 0).
2. A charge Q is distributed over two concentric hollow spheres of radii r and R (> r) such that their surface charge densities are equal. Find :
   (I) the electric field, and
   (II) the potential
   at their common centre.

(i) A parallel plate capacitor with plate area A and plate separation d has a capacitance C₀. A slab of dielectric constant K having area A and thickness (d4) is inserted in the capacitor, parallel to the plates. Find the new value of its capacitance.

(ii) You are provided with a large number of 1 μF identical capacitors and a power supply of 1200 V. The dielectric medium used in each capacitor can withstand up to 200 V only. Find the minimum number of capacitors and their arrangement, required to build a capacitor system of equivalent capacitance of 2 μF for use with this supply.

(i) An electric dipole of dipole moment p consists of point charges q and – q, separated by 2a. Derive an expression for electric potential in terms of its dipole moment at a point at a distance x (>> a) from its centre and lying (I) along its axis, and (II) along its bisector line.

(ii) An electric dipole of dipole moment p = (0·8 i + 0·6 j) 10^–29 Cm is placed in an electric field E = 1·0 × 10⁷ k Vm. Calculate the magnitude of the torque acting on it and the angle it makes with the x-axis, at this instant.

(i) A parallel plate capacitor A of capacitance C is charged by a battery to a potential ‘V’. The battery is disconnected and an uncharged identical capacitor B is connected across it. Calculate for the capacitor A the new value of the :

(I) charge

(II) potential difference

(III) energy stored

Justify your answers.

(ii) Draw the pattern of electric field lines due to :

(I) positively charged conducting sphere, and

(II) an electric dipole.

(i) An electric dipole (dipole moment p = p i), consisting of charges − q and q, separated by distance 2a, is placed along the x-axis, with its centre at the origin. Show that the potential V, due to this dipole, at a point x, (x >> a) is equal to 14πε₀ · p·ix².

(ii) Two isolated metallic spheres S₁ and S₂ of radii 1 cm and 3 cm respectively are charged such that both have the same charge density (2π × 10⁻⁹) C/m². They are placed far away from each other and connected by a thin wire. Calculate the new charge on sphere S₁.

(i) Obtain an expression for the electric potential due to a small dipole of dipole moment p, at a point r from its centre, for much larger distances compared to the size of the dipole.

(ii) Three point charges q, 2q and nq are placed at the vertices of an equilateral triangle. If the potential energy of the system is zero, find the value of n.

(i) Obtain the expression for the capacitance of a parallel plate capacitor with a dielectric medium between its plates.

(ii) A charge of 6 μC is given to a hollow metallic sphere of radius 0·2 m. Find the potential at (i) the surface and (ii) the centre of the sphere.

1. A dielectric slab of dielectric constant ‘K’ and thickness ‘t’ is inserted between plates of a parallel plate capacitor of plate separation d and plate area A. Obtain an expression for its capacitance.
2. Two capacitors of different capacitances are connected first (1) in series and then (2) in parallel across a dc source of 100 V. If the total energy stored in the combination in the two cases are 40 mJ and 250 mJ respectively, find the capacitance of the capacitors.

1. Draw equipotential surfaces for an electric dipole.
2. Two point charges q₁ and q₂ are located at r₁ and r₂ respectively in an external electric field E. Obtain an expression for the potential energy of the system.
3. The dipole moment of a molecule is 10^–30 Cm. It is placed in an electric field E of 10⁵ V/m such that its axis is along the electric field. The direction of E is suddenly changed by 60° at an instant. Find the change in the potential energy of the dipole, at that instant.

1. Consider two identical point charges located at points (0, 0) and (a, 0).
   1. Is there a point on the line joining them at which the electric field is zero ?
   2. Is there a point on the line joining them at which the electric potential is zero ?
   Justify your answers for each case.
2. State the significance of negative value of electrostatic potential energy of a system of charges.
   Three charges are placed at the corners of an equilateral triangle ABC of side 2.0 m as shown in figure. Calculate the electric potential energy of the system of three charges.

   

(i) Define electric potential at a point and write its SI unit.

(ii) Two capacitors are connected in series. Derive an expression of the equivalent capacitance of the combination.

(iii) Two point charges + q and – q are located at points (3a, 0) and (0, 4a) respectively in x-y plane. A third charge Q is kept at the origin. Find the value of Q, in terms of q and a, so that the electrostatic potential energy of the system is zero.