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

The magnetic energy stored in any region is obtained from the energy density formula for a magnetic field inside a linear medium.

Energy density in a magnetic material is given by
$$u = \frac{1}{2} B H$$

For a linear medium, $$B = \mu H$$, where $$\mu = \mu_0 \mu_r$$. Substituting $$H = \frac{B}{\mu}$$ in the energy density expression:

$$u = \frac{1}{2} B \left(\frac{B}{\mu}\right) = \frac{B^2}{2\mu}$$ $$-(1)$$

The solenoid is completely filled with a material of relative permeability $$\mu_r = 2$$, so
$$\mu = \mu_0 \mu_r = 2\mu_0$$.

Substituting this value of $$\mu$$ in equation $$(1)$$:

$$u = \frac{B^2}{2(2\mu_0)} = \frac{B^2}{4\mu_0}$$ $$-(2)$$

The total magnetic energy $$U$$ stored in the solenoid equals the energy density multiplied by the volume of the field region.
Volume of the solenoid, $$V = A l$$.

Therefore,
$$U = u \, V = \frac{B^2}{4\mu_0} \, (A l) = \frac{B^2 A l}{4\mu_0}$$.

Thus, the magnetic energy stored in the solenoid is $$\frac{B^2 A l}{4\mu_0}$$.

Correct option: Option D