I am also having problems understanding how to complete this problem, any help would be greatly appreciated.
Two capacitors are identical, except that one is empty and the other is filled with a dielectric ( = 4.20). The empty capacitor is connected to a 18.0 V battery. What must be the potential difference across the plates of the capacitor filled with a dielectric such that it stores the same amount of electrical energy as the empty capacitor?
_________ V
The capacitor with the dielectric (#2) will have 4.2 times the C-value of the air-gap capacitor (#1). Let's call the capacitances C1 and C2 = 4.2 C1.
Since the energies (E) are equal,
E1 = (1/2) C1 V^2 = E2 = (1/2) C2 V2^2
C1 * 18^2 = (4.2 C1) * V2^2
V2^2 = (1/4.2) * 324
Solve for V2
To solve this problem, we need to use the formula for the energy stored in a capacitor, which is given as:
U = 1/2 * C * V^2
Where:
U 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
In this case, we have two capacitors of the same capacitance, but one is empty and the other is filled with a dielectric. Let's denote the empty capacitor as C1 and the dielectric-filled capacitor as C2.
Given that the potential difference across the plates of C1 is 18.0 V, we can calculate the energy stored in C1 using the formula above.
Now, we need to find the potential difference across the plates of C2 such that it stores the same amount of electrical energy as C1.
The capacitance of a capacitor filled with a dielectric is given by:
C2 = k * C1
Where:
C2 is the capacitance of the capacitor filled with a dielectric
k is the dielectric constant of the material filling the capacitor
C1 is the capacitance of the empty capacitor
In this problem, the dielectric constant is given as 4.20, so we can substitute this value into the equation to find the capacitance of C2.
Now, we can use the formula for energy stored in a capacitor to find the potential difference across the plates of C2. Since we want the energy stored in C2 to be the same as C1, we can set U1 = U2 and solve for V2.
1/2 * C1 * (V1)^2 = 1/2 * C2 * (V2)^2
Substituting the values and solving for V2 will give us the potential difference across the plates of C2.
I hope this explanation helps you understand how to solve this problem. Let me know if you have any further questions!