Assertion (A): After charging a capacitor of capacitance (C) from a battery, it is connected to the same battery of potential difference (V) with reverse polarity. Loss of energy in this process is \(2CV^2\).
Reason (R): Work done by the battery is equal to loss of energy in the given case.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Initially, the capacitor stores energy \(U_i = \frac{1}{2}CV^2\) with charge \(Q_i = CV\). When connected with reverse polarity, the capacitor eventually charges to (-V\), and final stored energy is \(U_f = \frac{1}{2}C(-V)^2 = \frac{1}{2}CV^2\). The net change in stored energy is \(0\). The charge that flows from the battery is \(Q_f - Q_i = (-CV) - (CV) = -2CV\), meaning (2CV) charge flows. The work done by the battery is \(W_B = (2CV) \cdot V = 2CV^2\). Since \(W_B = \Delta U + Q_{\text{loss}}\), and ( \Delta U = 0\), the loss of energy is \(Q_{\text{loss}} = W_B = 2CV^2\). Both A and R are true and R explains A in this specific case.
Assertion (A): In a system of two concentric shell of inner radius \(a\) and outer radius \(b\). If outer is grounded and inner shell is given charge has less capacitance than inner has grounded and outer is given charge.
Reason (R): Electric field is zero outside outer shell when inner shell is grounded.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. For inner charged, outer grounded, \(C_1 = 4\pi\epsilon_0 \frac{ab}{b-a}\). For inner grounded, outer charged, \(C_2 = 4\pi\epsilon_0 \frac{b^2}{b-a}\). Since \(b>a\), \(C_1 < C_2\).\nR: False. If the outer shell is given charge, there will be a net charge creating an external electric field.\nTherefore, (A) is true and (R) is false.
Assertion (A): Two parallel plates having unequal charges have same capacitance as that of equal and opposite charges on same plates and same configuration.
Reason (R): Capacitance of system/ configuration is independent of charge on plates.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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A: True. The capacitance of a parallel plate capacitor, \(C = \frac{\epsilon_0 A}{d}\), is a geometric property and does not depend on the specific charge values on the plates, only their configuration.\nR: True. Capacitance is an intrinsic property dependent on geometry and dielectric, not on charge or potential.\n(R) correctly explains (A).
Assertion (A): When a dielectric slab is gradually inserted between the plates of an isolated parallel-plate capacitor, the energy of the system decreases.
Reason (R): The force between the plates decreases.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. For an isolated capacitor, charge \(Q\) is constant. Energy \(U = \frac{Q^2}{2C}\). Inserting a dielectric increases capacitance \(C\), so energy \(U\) decreases.\nR: False. The force between plates, \(F = \frac{Q^2}{2\epsilon_0 A}\), depends on \(Q\) and plate area \(A\), not on the dielectric constant when \(Q\) is constant.\nTherefore, (A) is true and (R) is false.
Assertion (A): A parallel plate capacitor is connected across battery through a key. A dielectric slab of dielectric constant \(K\) is introduced between the plates. The energy which is stored becomes \(K\) times.
Reason (R): The surface density of charge on the plate remains constant or unchanged.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. When connected to a battery, potential \(V\) is constant. Energy \(U = \frac{1}{2}CV^2\). As dielectric \(K\) is inserted, \(C\) becomes \(KC_0\), so \(U\) becomes \(KU_0\).\nR: False. Charge \(Q = CV\). Since \(C\) increases by \(K\) and \(V\) is constant, \(Q\) also increases by \(K\). Thus, surface charge density \(\sigma = Q/A\) also increases. Therefore, (A) is true and (R) is false.
Assertion (A): A dielectric slab is slightly inserted in charged parallel plate capacitor and then released slab will execute oscillation.
Reason (R): Electrostatic field is conservative field.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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A: True. For an isolated charged capacitor, inserting a dielectric reduces potential energy, creating an attractive force. With inertia, this can lead to oscillation.\nR: True. Electrostatic fields are conservative, meaning work is path-independent and potential energy can be defined. This is fundamental for oscillations derived from potential energy.
(R) is a fundamental basis explaining how (A) can occur.
Assertion (A): Capacitor reduces sparks in induction coil.
Reason (R): Capacitor provides alternative path to current when circuit is broken.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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A: True. In an induction coil, breaking the circuit induces a high back EMF, causing sparks across the switch. Capacitors are used to mitigate this.
R: True. A capacitor connected across the switch provides a path for the induced current, absorbing the inductive energy and preventing excessive voltage buildup that leads to sparks.\n(R) correctly explains how (A) works.
Assertion (A): If temperature is increased, the dielectric constant of a polar dielectric decreases whereas that of a non-polar dielectric does not change significantly.
Reason (R): The magnitude of dipole moment of individual polar molecule decreases significantly with increase in temperature.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. Increased temperature reduces the alignment of polar molecules, decreasing their dielectric constant. Non-polar dielectrics are less affected.
R: False. The magnitude of an *individual* dipole moment is largely temperature independent. It is the *average alignment* of these dipoles that decreases due to thermal agitation. Therefore, (A) is true and (R) is false.
Assertion (A): A capacitor of a certain capacity, whenever charged, will always store the same amount of charge.
Reason (R): A definite capacity implies always a same definite value of charge.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: False. The charge stored by a capacitor is \(Q = CV\). For a given capacitance \(C\), the charge \(Q\) depends on the applied voltage \(V\), which can vary.\nR: False. A definite capacity \(C\) does not imply a definite charge \(Q\), as \(Q\) is also proportional to the voltage \(V\) across the capacitor, which can be varied.\nTherefore, both (A) and (R) are false.
Assertion (A): Two protons placed at different distances, between the plates of a parallel plate capacitor experience the same force.
Reason (R): The electric field between the plates of parallel plate capacitor is constant.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true because the electric field (E) in a parallel plate capacitor is uniform. The force on a proton (q) is F = qE , which is constant. Reason (R) is true as the electric field between plates of an ideal parallel plate capacitor is constant. (R) correctly explains (A).