Question

If 27 identical charged conducting spheres, each of capacitance \( 10 \mu F \), combine to form a big sphere, then the capacitance of the big sphere is:

• A. \( 30 \mu F \)
• B. \( 270 \mu F \)
• C. \( 90 \mu F \)
• D. \( 10 \mu F \)

Hint:

For capacitance of merged spheres, use: \[ C_{\text{new}} = C_{\text{old}} \times N^{1/3} \] where \( N \) is the number of identical spheres.

Answer 1

The Correct Option is C

Step 1: Understanding Capacitance Scaling The capacitance \( C \) of a sphere is given by: \[ C = 4\pi \epsilon_0 R \] When multiple spheres combine to form a single larger sphere, the radius of the larger sphere scales as: \[ R_{\text{new}} = R_{\text{old}} \times N^{1/3} \] where \( N = 27 \) (number of spheres). Step 2: Scaling Capacitance Since capacitance is proportional to radius: \[ C_{\text{new}} = C_{\text{old}} \times N^{1/3} \] \[ = 10 \times 27^{1/3} \] \[ = 10 \times 3 = 30 \mu F \] Thus, the correct answer is: \[ 90 \mu F \]