Question

Obtain the expression of the magnetic energy stored in a solenoid in terms of magnetic field B, area A and length l of the solenoid.

Answer 1

Hint: The expression of the magnetic energy stored in a solenoid in terms of magnetic field B, area A and length l of the solenoid can be given using the following formulas which are as follows:
$${\epsilon }_{\circ }={\displaystyle \frac{1}{c^2{\mu }_{\circ }}}$$
$$E=Bc$$
$${\displaystyle \frac{U}{Al}}={\displaystyle \frac{1}{2}}\times {\epsilon }_{\circ }\times E^2$$

Complete step-by-step answer:
The permittivity of a particular medium is given by,
$${\epsilon }_{\circ }={\displaystyle \frac{1}{c^2{\mu }_{\circ }}}$$ …(1)
Now, the electric field generated by a magnetic field of strength B is given by,
$$E=Bc$$ ….(2)
The value of the energy stored in a capacitor for per volume is given as follows,
$${\displaystyle \frac{U}{Al}}={\displaystyle \frac{1}{2}}\times {\epsilon }_{\circ }\times E^2$$
Substituting values from equation (1) and equation (2) we get,
$$U={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{c^2{\mu }_{\circ }}}\times {\left(Bc\right)}^2\times Al$$
$$U={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{{\mu }_{\circ }}}\times B^2\times Al$$
Therefore, the magnetic energy stored in a solenoid in terms of magnetic field B, area A and length l of the solenoid is $${\displaystyle \frac{1}{2}}{\displaystyle \frac{B^2}{{\mu }_{\circ }}}\times Al$$ which is the correct answer.

Note: A solenoid is a type of electromagnet which has a purpose of generating a controlled magnetic field through a coil wound into a tightly packed helix. The coil can be arranged in a given manner to produce a uniform magnetic field in a volume of space when an electric current is passed through it. The solenoid works on the principle of electromagnetism. When the current flow through the coil magnetic field is generated in it and then if you place a metal core inside the coil the magnetic lines of flux are concentrated on the core which increases the induction of the coil as compared to the air core.