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

A solenoid with cross-sectional area \(A\) and length \(\ell\) is filled with a material having a relative permeability \(\mu_r = 2\). Determine the magnetic energy stored within the solenoid, expressed in terms of the magnetic field \(B\).

Concept Used:

The energy density of a magnetic field in a medium is defined as:

\[ u = \frac{B^2}{2\mu}. \]

When the medium has a relative permeability \(\mu_r\), the permeability \(\mu\) is given by:

\[ \mu = \mu_r \mu_0. \]

The total magnetic energy \(U\) stored in a solenoid of volume \(V = A\ell\) is calculated as:

\[ U = u \times V = \frac{B^2}{2\mu} \times A\ell = \frac{B^2 A \ell}{2\mu_r \mu_0}. \]

Step-by-Step Solution:

Step 1: Substitute the given value of \(\mu_r = 2\):

\[ U = \frac{B^2 A \ell}{2(2)\mu_0} = \frac{B^2 A \ell}{4\mu_0}. \]

Step 2: Assuming the question requires energy per unit length or a general dependency, and given that \(\ell\) is a constant parameter, the energy per unit length can be expressed as:

\[ \frac{U}{\ell} = \frac{B^2 A}{4\mu_0}. \]

Final Computation & Result

The magnetic energy stored in the solenoid is:

\[ \boxed{\frac{B^2 A}{4\mu_0}}. \]

Correct Option: \( \dfrac{B^2 A}{4\mu_0} \)