Two capacitors are identical, except that one is empty and the other is filled with a dielectric (k = 3.6). The empty capacitor is connected to a 15 -V battery. What must be the potential difference across the plates of the capacitor filled with a dielectric so that it stores the same amount of electrical energy as the empty capacitor?

Question

Two capacitors are identical, except that one is empty and the other is filled with a dielectric (k = 3.6). The empty capacitor is connected to a 15 -V battery. What must be the potential difference across the plates of the capacitor filled with a dielectric so that it stores the same amount of electrical energy as the empty capacitor?

Answer 1

C₂=kC₁

Energy = CV²/2
C₁V₁²/2=C₂V₂²/2
V₂=V₁/√(k)=15/√(3.6) = 7.9 V

Answer 2

To find the potential difference across the plates of the capacitor filled with a dielectric, we need to understand the relationship between capacitance and energy stored in a capacitor.

The formula for the energy stored in a capacitor is given by:

E = 1/2 * C * V^2

Where:
E is the energy stored in the capacitor
C is the capacitance of the capacitor
V is the potential difference across the plates of the capacitor

Since both capacitors are identical in all aspects except for the dielectric, their capacitances will be equal when the dielectric is inserted.

Let's denote the capacitance of each capacitor as C. Since the empty capacitor is connected to a 15-V battery, the potential difference across its plates is 15 V.

For the capacitor filled with a dielectric to store the same amount of electrical energy, we need to solve for the potential difference (V').

Using the formula for energy stored in a capacitor, we can write the following equation for the empty capacitor:

E_empty = 1/2 * C * (15 V)^2

Since the capacitors are identical, the energy stored in the empty capacitor should be equal to the energy stored in the capacitor filled with a dielectric:

E_empty = E_dielectric

Substituting the formula for energy into the equation, we have:

1/2 * C * (15 V)^2 = 1/2 * C * (V')^2

Simplifying the equation:

(15 V)^2 = (V')^2

225 V^2 = (V')^2

Taking the square root of both sides, we find:

V' = ±15 V

Since we know that the potential difference can't be negative, the potential difference across the plates of the capacitor filled with a dielectric must be 15 V, the same as the empty capacitor.

Therefore, the potential difference across the plates of the capacitor filled with a dielectric must be 15 V to store the same amount of electrical energy as the empty capacitor.