A square parallel plate capacitor in a circuit with a 2.0 V battery is replaced with a new one. If the sides of the new plates are 3 × as large as those of the old, and the distance between the plates is 1/4 of the distance between the old plates, then the maximum potential energy that the new capacitor can store in that circuit is _____ the amount of energy that the old capacitor could store in that circuit.
To determine the maximum potential energy that the new capacitor can store in the circuit, we need to compare it to the energy stored in the old capacitor. The potential energy stored in a capacitor is given by the equation:
U = (1/2) * C * V^2
where U is the potential energy, C is the capacitance, and V is the voltage.
Let's denote the characteristics of the old capacitor as follows:
- Side length of the old plates: S
- Distance between the old plates: D
Now, we can calculate the maximum potential energy stored in the old capacitor.
1. Find the capacitance of the old capacitor:
The capacitance of a parallel plate capacitor is given by the formula:
C = ε₀ * (A / D)
where ε₀ is the permittivity of free space, A is the area of the plates, and D is the distance between the plates.
For the old capacitor:
- Side length of the old plates: S
- Area of the old plates: A = S^2
- Distance between the old plates: D
Therefore, the capacitance of the old capacitor is:
C_old = ε₀ * (S^2 / D) ...(1)
2. Calculate the maximum potential energy stored in the old capacitor:
Given that the voltage of the circuit is 2.0 V, we can substitute the value of C_old and V into the formula:
U_old = (1/2) * C_old * V^2 ...(2)
Now, let's determine the characteristics of the new capacitor:
- Sides of the new plates: 3 times as large as the old plates, so the new side length is 3S.
- Distance between the plates: 1/4 of the distance between the old plates, so the new distance is (1/4)D.
3. Find the capacitance of the new capacitor:
Using the same formula as before, with the new values of S and D:
C_new = ε₀ * [(3S)^2 / (1/4)D]
= ε₀ * (9S^2 / (1/4)D)
= 36 * ε₀ * (S^2 / D) ...(3)
4. Calculate the maximum potential energy stored in the new capacitor:
Substitute the values of C_new and V into the formula:
U_new = (1/2) * C_new * V^2 ...(4)
Now, we can compare the maximum potential energy stored in the new capacitor to the old capacitor.
To determine the relationship, let's compare Equations (2) and (4):
U_new = (1/2) * (36 * ε₀ * (S^2 / D)) * V^2
= 18 * ε₀ * (S^2 / D) * V^2
Comparing this equation to Equation (2), we can see that the maximum potential energy stored in the new capacitor is 18 times that of the old capacitor.
Therefore, the maximum potential energy that the new capacitor can store in the circuit is 18 times the amount of energy that the old capacitor could store in that circuit.