a capacitor (A1B1) of capacitance C1 = 2 microfarad is charged under a voltage U1 = 20 V and then isolated.
a) Calculate, the voltage U A1B1 between its plates, its charge Q1 and its energy W1.
b) the plates A1 and B1 are connected to those A2 and B2 , of an uncharged capacitor of capacitance C2 = 4C1. At electric equilibrium, the respective charges of the capacitors become Q'1 and Q'2.
- establish a relation between Q1, Q'1, and Q'2
- compare the final energy of the system to the initial energy of the capacitor (A1B1). Deduce.
a) To calculate the voltage U A1B1 between the plates of the capacitor, we use the formula:
U = Q/C
where U is the voltage, Q is the charge, and C is the capacitance.
Given that the capacitance C1 = 2 microfarad and the voltage U1 = 20 V, we can substitute these values into the formula:
U A1B1 = Q1/C1 = 20 V
To calculate the charge Q1, we rearrange the formula as:
Q1 = U1 * C1 = 20 V * 2 microfarad = 40 microcoulombs.
To calculate the energy W1 stored in the capacitor, we use the formula:
W = (1/2) * C * U^2
Substituting the values:
W1 = (1/2) * C1 * U1^2 = (1/2) * 2 microfarad * (20 V)^2 = 400 microjoules.
Therefore, the voltage U A1B1 between the plates is 20 V, the charge Q1 is 40 microcoulombs, and the energy W1 stored in the capacitor is 400 microjoules.
b) When the plates A1 and B1 are connected to those A2 and B2 of an uncharged capacitor of capacitance C2 = 4C1, the two capacitors are effectively connected in parallel. At electric equilibrium, the charges on the capacitors equalize, so we can equate the charges:
Q'1 = Q'2
Now, let's relate the charges Q1, Q'1, and Q'2:
The charge Q1 on the initial capacitor A1B1 is the same as the charge Q'1 on the first capacitor in the new configuration.
Since the second capacitor has a capacitance C2 = 4C1, its charge Q'2 is related to the charge Q'1 as:
Q'2 = 4Q'1
So, the relation between Q1, Q'1, and Q'2 is:
Q1 = Q'1
Q'2 = 4Q'1
For the final energy of the system, we calculate the energy using the formula mentioned before:
W' = (1/2) * (C1 + C2) * U^2
Substituting the values:
W' = (1/2) * (2C1 + 4C1) * U1^2 = 14W1
The initial energy of the capacitor A1B1 is 400 microjoules, and the final energy of the system is 14 times the initial energy. Hence, the final energy is 14 times greater than the initial energy.
Overall, the relation between the charges Q1, Q'1, and Q'2 is that Q1 equals Q'1, and Q'2 is 4 times Q'1. Additionally, the final energy of the system is 14 times greater than the initial energy of the capacitor A1B1.