Obtain and expression for magnetic energy …
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Sia ? 4 years, 5 months ago
W = {tex}\int d W=\int \mathfrak{E} I d t=\int L \frac{d I}{d t} I d t=\int_{0}^{I} L I d I=\frac{1}{2} L I^{2}{/tex}
The magnetic energy stored in the solenoid is
UB = W = {tex}\frac{1}{2} L I^{2}=\frac{1}{2} L\left(\frac{B}{\mu_{0} n}\right)^{2}{/tex} {tex}\left[\because B=\mu_{0} n I\right]{/tex}
= {tex}\frac{1}{2}\left(\mu_{0} n^{2} A l\right)\left(\frac{B}{\mu_{0} n}\right)^{2}{/tex} {tex}\left[\because L=\mu_{0} n^{2} A l\right]{/tex}
{tex}U_{B}=\frac{1}{2 \mu_{0}} B^{2} A l{/tex}
{tex}u_{B}=\frac{U_{B}}{V}=\frac{U_{B}}{A l}{/tex} [Here V is the volume that contains flux]
[Here V is volume that contains flux]
or {tex} u_{B}=\frac{B^{2}}{2 \mu_{0}}{/tex} ....(i)
We know that the electrostatic energy stored per unit volume in a parallel plate capacitor is
{tex}u_{E}=\frac{1}{2} \varepsilon_{0} E^{2}{/tex} .....(ii)
In both the cases energy is proportional to the square of the field strengths. Eqs. (i) and (ii) are general and valid for any region of space in which a magnetic field or/and an electric field exist.
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