Capacitors and Dielectrics — Assertion and Reason Type

Assertion (A): Electric potential is scalar and electric intensity is vector quantity.

Reason (R): The value of electric potential and electric intensity at the middle point of the line joining an electron and a proton is zero.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Electric potential is defined as work done per unit charge and has magnitude only, so it is a scalar. Electric intensity is force per unit charge and has both magnitude and direction, so it is a vector.

Reason (R) is incorrect: At the middle point of the line joining an electron and a proton, the potential is the algebraic sum of the two potentials,

$\text V = \dfrac{1}{4\pi\epsilon_0}\sum\dfrac{\text q}{\text r} \\[1em] \text V = 0 \\[1em] \text{since} \quad \sum \text q = -\text q + \text q \\[1em] \sum \text q = 0$

The electric intensity, however, is a vector sum. A unit positive charge placed at the middle point is repelled by the proton and attracted by the electron with equal forces acting in the same direction, so the resultant field is not zero.

Therefore, assertion is true but reason is false.

Assertion (A): Positive charge always moves from a higher potential point to a lower potential point.

Reason (R): Electric potential is a vector quantity.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Electric lines of force originate from a positive charge and terminate on a negative charge, that is, they run from a higher potential towards a lower potential. Hence a positive charge left free in an electric field moves from a point of higher potential to a point of lower potential.

Reason (R) is incorrect: Electric potential is a scalar quantity, since it is defined as the work done per unit charge and possesses magnitude only.

Therefore, assertion is true but reason is false.

Assertion (A): The surface of a conductor is an equipotential surface.

Reason (R): Conductor allows the flow of charge.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: A surface on which the potential has the same value at every point is called an equipotential surface, and the surface of a conductor is such a surface.

Reason (R) is also correct and explains the assertion: A conductor allows charge to flow freely through it. If two points on a conductor were at different potentials, charge would flow from the point of higher potential to that of lower potential until the two potentials became equal. Hence the whole surface settles down at one and the same potential.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): Conductors having equal positive charge and volume, must also have same potential.

Reason (R): Potential depends only on charge and volume of conductor.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: The potential of a charged conductor depends on its shape as well, and not merely on the charge and the volume. Two conductors of equal charge and equal volume but of different shapes may be at different potentials.

Reason (R) is also incorrect: For the same reason, the potential does not depend only on the charge and the volume of the conductor. For a spherical conductor, for instance,

$\text V = \dfrac{\text Q}{4\pi\epsilon_0 \text R}$

so it is the linear dimension R, fixed by the shape, that enters the expression.

Therefore, both assertion and reason are false.

Assertion (A): Electric potential of the earth is zero.

Reason (R): No electric field exists on earth surface.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: The earth is a very good conductor of an extremely large size, so its capacitance is enormous. Giving or taking away a small charge therefore produces practically no change in its potential, and the potential of the earth is taken as zero. The potentials of all other bodies are measured with reference to it.

Reason (R) is incorrect: An electric field does exist at the surface of the earth; in fact the atmosphere possesses a significant electric field. The zero of potential is a matter of choosing a reference level, not of the absence of a field.

Therefore, assertion is true but reason is false.

Assertion (A): No work is done in taking a positive charge from one point to other inside a positively charged metallic sphere while outside the sphere work is done in taking the charge toward the sphere.

Reason (R): Inside the sphere electric potential is same at each potential, but outside it is different for different points.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Inside a charged metallic sphere every point is at the same potential, so in moving a charge from one point to another inside it,

$\text W = \text q\thinspace\Delta \text V \\[1em] \text W = 0$

Outside the sphere the potential varies from point to point, so a potential gradient exists and W ≠ 0.

Reason (R) is also correct and explains the assertion: It states precisely this property — the potential is the same at each point inside the sphere but different for different points outside it — which is the reason why the work is zero in the one case and non-zero in the other.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): On going away from a point charge or a small electric dipole, electric potential decreases at the same rate in both the cases.

Reason (R): Electric potential is inversely proportional to square of distance from the charge in all cases.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: The potential falls off at different rates in the two cases. For a point charge

$\text V \propto \dfrac{1}{\text r}$

whereas for a small electric dipole

$\text V \propto \dfrac{1}{\text r^2}$

so the potential of a dipole decreases much more rapidly.

Reason (R) is also incorrect: The inverse square relation does not hold in all cases; it applies to a small dipole, while for a point charge the potential is inversely proportional to the first power of the distance.

Therefore, both assertion and reason are false.

Assertion (A): At a point in space, the electric field points towards north. In the region, surrounding this point the rate of change of potential will be zero along the east and west.

Reason (R): Electric field due to a point charge in the space surrounds the charge.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: The electric field is directed towards the north, so the potential changes only along the north–south direction. Along the east and west directions there is no component of the field, so

$\dfrac{\text{dV}}{\text{dr}} = 0$

that is, the rate of change of potential is zero along east and west.

Reason (R) is also correct: The electric field due to a point charge does surround the charge in the space around it.

However, this general statement about the field of a point charge does not account for the vanishing of the potential gradient along the east–west direction, which follows from the direction of the field at that point.

Therefore, both are true but the reason is not the correct explanation of the assertion.

Assertion (A): For practical purposes, the earth is used as a reference at zero potential in electrical circuits.

Reason (R): The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by $\dfrac{\text Q}{4\pi\epsilon_0 \text R}$ .

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: The earth is used as the reference at zero potential in electrical circuits, since its capacitance

$\text C = 4\pi\epsilon_0 \text R$

is very large on account of its very large radius. Hence V = q/C is extremely small and may be treated as zero.

Reason (R) is also correct: The potential of a sphere of radius R carrying a charge Q uniformly distributed on its surface is indeed Q/4πε₀R.

However, the reason as stated merely gives the expression for the potential of a charged sphere; it does not by itself explain why the earth is chosen as the zero of potential, which follows from its very large capacitance.

Therefore, both are true but the reason is not the correct explanation of the assertion.

Assertion (A): For a charged particle moving from point P to point Q, the net work done by an electrostatic field on the particle is independent of the path connecting point P to point Q.

Reason (R): The net work done by a conservative force on an object moving along a closed loop is zero.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: The electrostatic force is a conservative force, so the work done by it in taking a charge from P to Q depends only on the initial and the final positions and not on the path followed.

Reason (R) is also correct: The work done by a conservative force on an object moving along a closed loop is zero, since the initial and the final positions coincide.

However, the statement about a closed loop is a separate property of conservative forces and is not itself the explanation of the path independence between two distinct points P and Q.

Therefore, both are true but the reason is not the correct explanation of the assertion.

Assertion (A): Work done by electric field on moving a positive charge on an equipotential surface is always zero.

Reason (R): Electric lines of forces are always perpendicular to equipotential surfaces.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The work done in moving a charge is

$\text W = \text{Fd}\cos\theta$

and on an equipotential surface this work is always zero.

Reason (R) is also correct and explains the assertion: Electric lines of force are always perpendicular to an equipotential surface, so the displacement along the surface makes an angle θ = 90° with the force. Hence

$\text W = \text{Fd}\cos 90^\circ \\[1em] \text W = 0$

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): An electric field is preferred in comparison to magnetic field for deflecting the electron beam in a television picture tube.

Reason (R): Electric field requires low voltage.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: A magnetic field, and not an electric field, is preferred for deflecting the electron beam in a television picture tube.

Reason (R) is also incorrect: If an electric field were used for the deflection, a very high voltage would have to be applied and a very long tube would have to be taken. Hence it is not true that the electric field requires a low voltage.

Therefore, both assertion and reason are false.

Assertion (A): Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point.

Reason (R): Electric potential is constant within and at the surface of each conductor.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The electric field just outside a charged conductor is perpendicular to its surface at every point. If it had a component along the surface, the free charges would move along the surface, which is not possible in the static condition.

Reason (R) is also correct and explains the assertion: The electric field is zero inside a conductor, so from

$\text E = -\dfrac{\text{dV}}{\text{dr}}$

the potential is constant within the conductor and at its surface. The surface is therefore an equipotential surface, and the field just outside an equipotential surface must be normal to it.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): If a dielectric is placed in external field then field inside dielectric will be less than applied field.

Reason (R): Electric field will induce dipole moment opposite to field direction.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: When a dielectric is placed in an external field, it becomes polarised. The field due to the induced charges opposes the applied field, so the net field inside the dielectric is

$\text E = \dfrac{\text E_0}{\text K}$

which is less than the applied field E₀, since K > 1.

Reason (R) is incorrect: The induced dipole moment is set up in the same direction as the external field, not opposite to it. It is the field produced by the collective effect of these dipole moments that opposes the external field.

Therefore, assertion is true but reason is false.

Assertion (A): The capacity of a conductor, under given circumstances, remains constant irrespective of the charge present on it.

Reason (R): Capacity depends on size and shape of conductor and also on the surrounding medium.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The charge present on a conductor determines its potential, and the ratio Q/V remains constant. Hence the capacity of a conductor remains the same whatever the charge given to it.

Reason (R) is also correct and explains the assertion: The capacity is a function of the size and the shape of the conductor and of the surrounding medium. For a parallel-plate capacitor, for instance,

$\text C = \dfrac{\text K \epsilon_0 \text A}{\text d}$

where K is the dielectric constant of the medium, A the area of the plates and d the distance between them. As no charge term appears here, the capacity is independent of the charge.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): Capacity of a parallel plate capacitor increases when distance between the plates is decreased.

Reason (R): Capacitance of a capacitor is inversely proportional to distance between them.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: On decreasing the distance between the plates, the capacity of the parallel-plate capacitor to store charge increases.

Reason (R) is also correct and explains the assertion: The capacitance of a parallel-plate capacitor is

$\text C = \dfrac{\epsilon_0 \text A}{\text d}$

so that C is inversely proportional to d. Hence a decrease in d produces an increase in C.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): When air between the plates of a parallel plate condenser is replaced by an insulating medium of some dielectric constant, its capacity increases.

Reason (R): Electric field intensity reduces on inserting the dielectric in between the plates of the capacitor.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Replacing air by an insulating medium of dielectric constant K increases the capacity of the condenser to K times its original value.

Reason (R) is also correct and explains the assertion: The capacity of a parallel-plate condenser is

$\text C = \dfrac{\text q}{\text V} \qquad \dots(\text i)$

On inserting the dielectric, the electric field intensity becomes 1/K times its value, since

$\text K = \dfrac{\text E_0}{\text E}$

and therefore the potential difference V also becomes 1/K times. Hence from equation (i) the capacity becomes K times. Thus the electric field decreases and the capacitance increases when the condenser is filled with an insulating medium of some dielectric constant.

Therefore, both are true and the reason correctly explains the assertion.

Assertion (A): Charge on all the condensers connected in series is the same.

Reason (R): Capacitance of capacitor is directly proportional to charge on it.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Let two capacitors be connected in series. If a charge +q is given to the left plate of the first capacitor, a charge −q is induced on its right plate. This charge comes from the electrons drawn from the left plate of the second capacitor, so an equal charge +q appears on the left plate of the second capacitor and −q is induced on its right plate. Hence each capacitor carries the same charge q.

Reason (R) is incorrect: Capacitance is a quantity which depends on the construction of the capacitor, that is, on the area of the plates, their separation and the medium between them. It is independent of the charge on the capacitor.

Therefore, assertion is true but reason is false.

Assertion (A): A capacitor is connected to a battery. If we move its plate further apart, work will be done against the electrostatic attraction between the plates hence the energy of the capacitor gets decreased.

Reason (R): The energy stored in capacitor is dissipated in the form of heat energy.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: When the plates are moved further apart, the capacitance decreases. The battery remains connected, so V is fixed and the charge q = CV on the plates decreases. Hence the energy

$\text U = \dfrac{1}{2}\text{CV}^2$

also decreases. Some charge flows back from the plates to the battery, that is, a part of the energy of the capacitor is transferred to the battery.

Reason (R) is also correct: The work done against the electrostatic attraction between the plates is used up in this transference of energy and is dissipated in the form of heat energy in the connecting wires.

However, the decrease in the energy of the capacitor follows from the decrease in its capacitance at constant voltage, and not from the dissipation of energy as heat.

Therefore, both are true but the reason is not the correct explanation of the assertion.

Assertion (A): The force between the plates of a parallel plate capacitor is proportional to single power of charge on it.

Reason (R): Force is equal to charge per unit area per ε₀ .

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: The potential energy stored between the plates of a capacitor, when the separation is x, is

$\text U = \dfrac{1}{2}\dfrac{\text Q^2}{\text C} \\[1em] \text U = \dfrac{1}{2}\dfrac{\text Q^2 \text x}{\epsilon_0 \text A} \qquad \left(\because\ \text C = \dfrac{\epsilon_0 \text A}{\text x}\right)$

The force between the plates is obtained by differentiating this with respect to x,

$\text F = -\dfrac{\partial \text U}{\partial \text x} \\[1em] = -\dfrac{\partial}{\partial \text x}\left(\dfrac{1}{2}\dfrac{\text Q^2 \text x}{\epsilon_0 \text A}\right) \\[1em] \text F = -\dfrac{1}{2}\dfrac{\text Q^2}{\epsilon_0 \text A}$

The negative sign shows that the force is attractive. The force is therefore proportional to the square of the charge, and not to its single power.

Reason (R) is also incorrect: Writing the same result in terms of the surface density of charge,

$\text F = -\dfrac{1}{2}\dfrac{\sigma^2}{\epsilon_0}\text A$

so the force is not simply the charge per unit area divided by ε₀; a factor of one-half and the square of σ are involved.

Therefore, both assertion and reason are false.

Assertion (A): Two condensers of same capacity are connected first in parallel and then in series. The ratio of resultant capacitance in the two cases will be 4 : 1.

Reason (R): In parallel capacitance increases and in series capacitance decreases.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: For two condensers of the same capacity C, the equivalent capacitance in series is

$\dfrac{1}{\text C_s} = \dfrac{1}{\text C} + \dfrac{1}{\text C} \\[1em] \dfrac{1}{\text C_s} = \dfrac{2}{\text C} \\[1em] \text C_s = \dfrac{\text C}{2}$

and in parallel it is

$\text C_p = \text C + \text C \\[1em] \text C_p = 2\text C$

Hence the required ratio is

$\dfrac{\text C_p}{\text C_s} = \dfrac{2\text C}{\text C/2} \\[1em] \dfrac{\text C_p}{\text C_s} = 4 : 1$

Reason (R) is also correct: In a parallel combination the capacitance increases and in a series combination it decreases.

However, this general statement only tells us which combination gives the larger capacitance; it does not by itself yield the numerical ratio 4 : 1, which follows from the actual calculation above.

Therefore, both are true but the reason is not the correct explanation of the assertion.

Assertion (A): Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have same charge as on hollow sphere.

Reason (R): Capacitance of metallic spheres depend on the radii of spheres.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true but reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Answer

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The charge given to a conductor resides entirely on its outer surface, so the potential of a conducting sphere, whether hollow or solid, is

$\text V = \dfrac{\text{KQ}}{\text R}$

If V₁ = V₂ and R₁ = R₂, then Q₁ = Q₂. Hence the solid sphere carries the same charge as the hollow one.

Reason (R) is also correct and explains the assertion: The capacitance of a sphere is

$\text C = 4\pi\epsilon_0 \text R$

which depends only on the radius. The two spheres have equal radii, hence equal capacitances, and being at the same potential they carry equal charges Q = CV.

Therefore, both are true and the reason correctly explains the assertion.