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 Volts. (C is the capacitance and U is the stored energy.) Select true or false for each statement.

1) With the capacitor connected to the battery, decreasing d decreases Q.
2) After being disconnected from the battery, inserting a dielectric with kappa > 1 will increase C.
3) After being disconnected from the battery, inserting a dielectric with kappa > 1 will increase U.
4) With the capacitor connected to the battery, decreasing d increases C.
5) After being disconnected from the battery, increasing d decreases V.
6) With the capacitor connected to the battery, decreasing d increases U.

1) False

2) True
3) False
4) True
5) False
6) True

1) False. The charge Q on a capacitor is determined by the voltage V and the capacitance C, according to the formula Q = C * V. The plate separation d does not directly affect the charge on the capacitor when connected to a battery.

2) True. When a dielectric material is inserted between the plates of a capacitor, it increases the capacitance C. The dielectric constant (kappa) is a measure of how much a material can increase the capacitance, with values greater than 1 indicating an increase in capacitance.

3) False. The stored energy U in a capacitor is given by the formula U = (1/2) * C * V^2. Inserting a dielectric with a higher dielectric constant (kappa > 1) does not directly affect the stored energy, as it only increases the capacitance and not the voltage.

4) False. The capacitance C of a parallel plate capacitor is determined by the plate area A and the plate separation d, according to the formula C = ε0 * (A / d), where ε0 is the vacuum permittivity. Decreasing the plate separation d would decrease the capacitance C, not increase it.

5) True. The voltage V on a capacitor is inversely proportional to the plate separation d. Increasing the plate separation d will decrease the voltage V, assuming the charge remains constant.

6) False. The stored energy U in a capacitor is determined by the capacitance C and the voltage V, according to the formula U = (1/2) * C * V^2. Decreasing the plate separation d does not directly affect the stored energy; it only affects the capacitance and voltage.

To determine whether each statement is true or false, let's analyze each statement one by one.

1) With the capacitor connected to the battery, decreasing d decreases Q.

False: The charge on a capacitor is directly proportional to the voltage across it and the capacitance. The equation that relates charge (Q), capacitance (C), and voltage (V) is Q = C * V. Since the battery provides a constant voltage, decreasing the plate separation (d) does not affect the charge on the capacitor.

2) After being disconnected from the battery, inserting a dielectric with kappa > 1 will increase C.

True: Capacitance (C) is directly proportional to the dielectric constant (kappa) when a dielectric material is inserted between the plates of a capacitor. By increasing the dielectric constant (kappa > 1), the capacitance of the capacitor will also increase.

3) After being disconnected from the battery, inserting a dielectric with kappa > 1 will increase U.

True: The stored energy (U) in a capacitor is given by the equation U = (1/2) * C * V^2. Since the capacitance (C) increases when a dielectric with kappa > 1 is inserted, the stored energy will also increase.

4) With the capacitor connected to the battery, decreasing d increases C.

False: Capacitance (C) is a constant property of a capacitor that depends on factors such as the area and separation of the plates, and the properties of the medium between the plates. Changing the plate separation (d) while the capacitor is connected to a battery does not affect the capacitance.

5) After being disconnected from the battery, increasing d decreases V.

True: When a capacitor is disconnected from the battery, the charge on the plates remains constant. Voltage (V) is directly proportional to the charge (Q) and inversely proportional to the capacitance (C), so increasing the plate separation (d) reduces the capacitance and therefore reduces the voltage.

6) With the capacitor connected to the battery, decreasing d increases U.

False: The stored energy (U) in a capacitor is given by the equation U = (1/2) * C * V^2. Since the voltage (V) is provided by the battery and does not change with the plate separation (d), decreasing d does not impact the stored energy (U).

Summary:
1) False
2) True
3) True
4) False
5) True
6) False