Question

Two identical capacitors of capacitance \( C \) are connected in series. If the space between the plates of one of the capacitors is filled with a medium of dielectric constant \( k \), what is the effective capacitance?

Hint:

When capacitors are connected in series, the reciprocal of the effective capacitance is the sum of the reciprocals of the individual capacitances.

Answer 1

Step 1: Capacitance of a capacitor with dielectric.
The capacitance of a capacitor filled with a dielectric of dielectric constant \( k \) is given by: \[ C' = kC \] where \( C \) is the capacitance of the capacitor without the dielectric, and \( C' \) is the capacitance with the dielectric.
Step 2: Equivalent capacitance in series.
When two capacitors \( C_1 \) and \( C_2 \) are connected in series, the effective capacitance \( C_{\text{eff}} \) is given by the formula: \[ \frac{1}{C_{\text{eff}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Let the capacitances of the two capacitors be \( C \) and \( kC \) (since one is filled with the dielectric).
Step 3: Calculate the effective capacitance.
The effective capacitance is: \[ \frac{1}{C_{\text{eff}}} = \frac{1}{C} + \frac{1}{kC} \] Now, simplify the expression: \[ \frac{1}{C_{\text{eff}}} = \frac{k + 1}{kC} \] Thus, the effective capacitance is: \[ C_{\text{eff}} = \frac{kC}{k + 1} \]