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Physics · 2025 · 3 marks
CBSE 2025 · Region 1 · Set 1 · Q24
(a) Show that the energy required to build up the current I in a coil of inductance L is 1/2 L^2. (b) Considering the case of magnetic field produced…
(a)
Show that the energy required to build up the current I in a coil of inductance $\displaystyle L$ is $\displaystyle \frac{1}{2} L^{2}$.
(b)
Considering the case of magnetic field produced by air-filled current carrying solenoid, show that the magnetic energy density of a magnetic field $\displaystyle B$ is $\displaystyle \frac{B^{2}}{2 \mu_{0}}$.
Marking-scheme solution
(a)
Induced emf in an inductor
$$|\varepsilon| = L\frac{dI}{dt}$$
Rate of work done at any instant
$$\frac{dW}{dt} = |\varepsilon|\,I$$
Total Amount of work done in establishing current I
$$W = \int dW = \int_{0}^{I} L I\, dI$$
Energy required to build up current I is
$$W = \tfrac{1}{2} L I^{2}$$
(b)
The Magnetic Energy is
$$W = U_{B} = \tfrac{1}{2} L I^{2}$$
$$= \tfrac{1}{2} L \left(\frac{B}{n\mu_{0}}\right)^{2} \qquad \text{as } B = n\mu_{0} I$$
Using $\displaystyle L = \mu_{0} n^{2} A l$
$$U_{B} = \frac{1}{2}\left(\mu_{0} n^{2} A l\right)\left(\frac{B^{2}}{\mu_{0}^{2} n^{2}}\right)$$
$$\text{Energy density} = \frac{U_{B}}{\text{volume}}$$
$$\frac{U_{B}}{\text{volume}} = \frac{1}{2} \times \mu_{0} n^{2} A l \times \frac{B^{2}}{\mu_{0}^{2} n^{2}} \times \frac{1}{A l}$$
$$= \frac{1}{2}\frac{B^{2}}{\mu_{0}}$$
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