Question

Two air-filled parallel-plate capacitors with capacitances C1 and C2 are connected in series to a battery that has voltage V; C1 = 3.00 μF and C2 = 6.00 μF. The electric field between the plates of capacitor C2 is E02. While the two capacitors remain connected to the battery, a dielectric with dielectric constant K = 4 is inserted between the plates of capacitor C1, completely filling the space between them. After the dielectric is inserted in C1, the electric field between the plates of capacitor C2 is E2.

A) What is the ratio E2/E02?
B) When the dielectric is inserted into C1, does the electric field in C2 increase, decrease, or remain the same?
C) Repeat the calculation in part A for the two capacitors connected to the battery in parallel. What is the ratio E2/E02?

Answer 1

After inserting a dielectric into one capacitor in a series connection, the electric field in the other capacitor increases. For capacitors in series, the ratio E2/E02 equals the dielectric constant K, indicating an increase. However, if capacitors are connected in parallel, the ratio E2/E02 remains unchanged.

Step-by-step explanation:

When considering two air-filled parallel-plate capacitors with different capacitances connected to a battery, the insertion of a dielectric in one of them affects their capacitance and the electric field within.

Part A: The Ratio E2/E02

For capacitors in series, the ratio E2/E02 after inserting a dielectric with constant K into C1 can be found by considering that the voltage across each capacitor should be the same. Given that the dielectric increases the capacitance of C1 by a factor of K, the voltage across C1 will decrease by a factor of K, leaving more voltage to drop across C2. Therefore, the electric field E2 in C2 will increase since E = V/d and the distance d remains constant.

The ratio E2/E02 will be equal K, because C1's capacitance has increased by a factor of K, thereby increasing the charge on C2 by the same factor, since the total charge in series remains constant.

Part B: Effect on Electric Field in C2

When a dielectric is inserted into C1, the electric field in C2 increases, for the reasons discussed above.

Part C: Connected in Parallel

For capacitors in parallel, the ratio E2/E02 will remain unchanged (E2/E02 = 1) when a dielectric is inserted into C1. This is because, in a parallel connection, the voltage across each capacitor is equal to the voltage of the battery, and the dielectric affects only the capacitor into which it is inserted.