Question

If the magnetic field inside a solenoid is \(B\), then the magnetic energy stored in it per unit volume is (\(c\) - speed of light in vacuum and \(\epsilon_0\) is permittivity of free space)

• A. \(\epsilon_0 c^2 B^2\)
• B. \(\frac{\epsilon_0 c^2 B^2}{2}\)
• C. \(2 \epsilon_0 c^2 B^2\)
• D. \(\frac{\epsilon_0 c^2 B^2}{4}\)

Hint:

Magnetic energy density can be expressed using permittivity and speed of light as \( \frac{\epsilon_0 c^2 B^2}{2} \).

Answer 1

The Correct Option is B

Step 1: Use the energy density formula
Energy density (energy per unit volume) in a magnetic field is given by: \[ u = \frac{B^2}{2 \mu_0}, \] where \(\mu_0\) is permeability of free space.
Step 2: Express \(\mu_0\) in terms of \(\epsilon_0\) and \(c\)
\[ \mu_0 = \frac{1}{\epsilon_0 c^2} \] Step 3: Substitute and simplify
\[ u = \frac{B^2}{2} \times \epsilon_0 c^2 = \frac{\epsilon_0 c^2 B^2}{2} \] Step 4: Conclusion
Energy stored per unit volume is \(\frac{\epsilon_0 c^2 B^2}{2}\).