Question

C_o is the capacitance of a parallel plate capacitor with air as a medium between the plates (as shown in Fig. 1). If half space between the plates is filled with a dielectric of relative permittivity \( \varepsilon \)_r (as shown in Fig. 2), the new capacitance of the capacitor will be :

• A. \( {{{C_o}} \over 2}(1 + {\varepsilon _r}) \)
• B. \( {C_o} + {\varepsilon _r} \)
• C. \( {{{C_o}{\varepsilon _r}} \over 2} \)
• D. \( {C_o}(1 + {\varepsilon _r}) \)

Hint:

In Fig. 2 the dielectric fills half the plate area side by side, so the capacitor is two capacitors (air and dielectric parts) in parallel.

Answer 1

The Correct Option is A

Step 1: The dielectric fills half of the plate area (the figure shows it occupying the right half, over the full plate separation), so the two halves are in parallel.
Step 2: Air part: area A/2, \( C_1=\frac{\varepsilon_0(A/2)}{d}=\frac{C_o}{2} \).
Step 3: Dielectric part: \( C_2=\frac{\varepsilon_r\varepsilon_0(A/2)}{d}=\frac{\varepsilon_rC_o}{2} \).
Step 4: \( C=C_1+C_2=\frac{C_o}{2}(1+\varepsilon_r) \).
Answer: (A) \( \frac{C_o}{2}(1+\varepsilon_r) \)