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 these questions, let's go through each statement and explain why it is true or false.

1. After being disconnected from the battery, decreasing d decreases C.
True. The capacitance (C) of a parallel plate capacitor is given by the equation C = ε₀(A/d), where ε₀ is the vacuum permittivity, A is the area of the plates, and d is the plate separation. Since decreasing d in the equation results in a smaller denominator, the capacitance (C) will increase. Therefore, the statement is false, not true.

2. After being disconnected from the battery, increasing d increases V.
False. The voltage (V) across the capacitor is determined by the charge (Q) and capacitance (C) through the formula V = Q/C. When the capacitor is disconnected from the battery, the charge (Q) remains constant. As the plate separation (d) increases, the capacitance (C) decreases (as explained in the previous question). Since the numerator (Q) remains constant and the denominator (C) increases, the voltage (V) decreases. Therefore, the statement is false, not true.

3. With the capacitor connected to the battery, inserting a dielectric with κ will decrease C.
False. When a dielectric material is inserted between the plates of a parallel plate capacitor, it increases the capacitance (C) of the capacitor. The equation for the capacitance with a dielectric is C = κε₀(A/d), where κ is the dielectric constant of the material. As κ increases, C increases. Therefore, the statement is false, not true.

4. After being disconnected from the battery, inserting a dielectric with κ will decrease V.
False. When a dielectric material is inserted between the plates of a capacitor, the voltage (V) across the capacitor remains constant. The equation V = Q/C shows that since the charge (Q) and the capacitance (C) both increase (due to the presence of the dielectric), the voltage (V) remains the same. Therefore, the statement is false, not true.

5. After being disconnected from the battery, inserting a dielectric with κ will increase U.
False. The stored energy (U) in a capacitor is given by the equation U = (1/2)QV. When a dielectric material is inserted between the plates of a capacitor, the voltage (V) remains constant, and the charge (Q) increases (due to the presence of the dielectric). Thus, since neither V nor Q change, the energy (U) remains constant. Therefore, the statement is false, not true.

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)QV. When the plate separation (d) decreases, the capacitance (C) increases, which implies that the charge (Q) also increases (since C = Q/V). However, the voltage (V) across the capacitor remains constant when connected to the battery. Therefore, since neither Q nor V changes, the energy (U) remains constant. Therefore, the statement is false, not true.

To summarize:
1. False
2. False
3. False
4. False
5. False
6. False