A parallel plate capacitor with plate separation d is connected to a battery. The capacitor is fully charged to Q Coulombs and a voltage of V. (C is the capacitance and U is the stored energy.) Answer the following questions regarding the capacitor charged by a battery. For each statement below, select True or False.
After being disconnected from the battery, decreasing d decreases C.
After being disconnected from the battery, increasing d increases V.
With the capacitor connected to the battery, inserting a dielectric with κ will decrease C.
After being disconnected from the battery, inserting a dielectric with κ will decrease V.
After being disconnected from the battery, inserting a dielectric with κ will increase U.
With the capacitor connected to the battery, decreasing d increases U.
To answer each statement, let's break down the scenario and analyze it step by step:
1. After being disconnected from the battery, decreasing d decreases C.
- True. The capacitance of a parallel plate capacitor is given by the formula C = ε₀A/d, where A is the area of the plates and d is the separation between them. Decreasing the separation distance (d) will result in an increase in capacitance, not a decrease. Therefore, this statement is False.
2. After being disconnected from the battery, increasing d increases V.
- False. The voltage (V) across a capacitor is determined by the potential difference between the plates, which is in turn determined by the charge (Q) on the plates and the capacitance (C), i.e., V = Q/C. Increasing the separation distance (d) without changing the charge or the capacitance will not affect the voltage. Therefore, this statement is False.
3. With the capacitor connected to the battery, inserting a dielectric with κ will decrease C.
- False. When a dielectric material with a relative permittivity (κ) is inserted between the plates of a capacitor, the capacitance increases. The formula for the new capacitance is C' = κC, where C' is the new capacitance and C is the initial capacitance. Therefore, inserting a dielectric with κ will increase C, not decrease it. Therefore, this statement is False.
4. After being disconnected from the battery, inserting a dielectric with κ will decrease V.
- True. When a dielectric material is inserted after disconnecting the capacitor from the battery, the charge on the plates remains constant. The voltage (V) across the capacitor is given by V = Q/C, where Q is the charge and C is the capacitance. Since the capacitance (C) increases when a dielectric with κ is inserted, the voltage will decrease for a fixed charge (Q). Therefore, this statement is True.
5. After being disconnected from the battery, inserting a dielectric with κ will increase U.
- True. The stored energy (U) in a capacitor is given by U = (1/2)QV, where Q is the charge and V is the voltage. When a dielectric with κ is inserted after disconnecting the capacitor from the battery, the charge (Q) remains constant, and the voltage decreases. As a result, the stored energy (U) increases because U ∝ QV. Therefore, this statement is True.
6. With the capacitor connected to the battery, decreasing d increases U.
- False. The stored energy (U) in a capacitor is given by U = (1/2)QV, where Q is the charge and V is the voltage. Decreasing the separation distance (d) between the plates does not change the charge (Q) or voltage (V) if the capacitor remains connected to the battery. Hence, the stored energy (U) will not increase. Therefore, this statement is False.