Question

A solenoid having area $ A $ and length $ \ell $ is filled with a material having relative permeability 2. The magnetic energy stored in the solenoid is:

• A. \( \frac{B^2A}{\mu_0} \)
• B. \( \frac{B^2A}{2\mu_0} \)
• C. \( \frac{B^2A}{\mu_0} \)
• D. \( \frac{B^2A}{4\mu_0} \)

Hint:

For problems involving energy stored in a magnetic field, use the formula for the energy density in the magnetic field \( U/V = \frac{B^2}{2 \mu_0} \), and apply the volume of the solenoid to find the total energy.

Answer 1

The Correct Option is D

We are given the energy stored in a solenoid \( U \), and the relation for magnetic energy density \( U/V \) is: \[ U/V = \frac{B^2}{2\mu_0} \] This implies that: \[ U = \frac{B^2}{2\mu_0} \times V \] Where \( V = A \ell \), the volume of the solenoid. Substituting: \[ U = \frac{B^2}{2\mu_0} \times A \ell \] Thus, the magnetic energy stored in the solenoid is: \[ U = \frac{B^2 A \ell}{4 \mu_0} \] Hence, the correct answer is: \[ \frac{B^2 A \ell}{4 \mu_0} \]