Question:Calculate the energy stored in the toroidal coil of Ex. 7.11, by applying Eq. 7.35. Use the answer to check Eq. 7.28.

Consider the magnetic field inside the toroid is $B=\frac{{\mu }_{0}nI}{2\pi s}$.

Write the formula ofthe energy stored in the toroidal coil.

$\mathbf{W}=\frac{\mathbf{1}}{\mathbf{2}{\mu }_{\mathbf{0}}}{\int }_{\mathbf{allspace}}{\mathbf{B}}^{\mathbf{2}}\mathbf{d}\tau$ …… (1)

Here, ${\mu }_{\mathbf{0}}$ is permeability and $\mathbf{B}$ is magnetic field inside the toroid

Determine theenergy stored in the toroidal coil.

Substitute $\frac{{\mu }_{0}nI}{2\pi s}$ for B and localid="1658742313069" $hrd\varphi ds$ for $d\tau$ into equation (1).

Here, r = s

Then

$$\begin{gathered} W=\frac{1}{2{\mu }_0}{\int }_{allspace}{\left(\frac{{\mu }_{0}nI}{2\pi s}\right)}^{2}hrd\varphi ds \\ =\frac{1}{2{\mu }_0}\left(\frac{{\mu }_0^2{n}^2{I}^2}{4\pi }\int \frac{1}{s^2}hrd\varphi ds\right) \\ =\frac{1}{2{\mu }_0}\left(\frac{{\mu }_0^2{n}^2{I}^2}{4\pi }h\left(2\pi \right)\underset{a}{\overset{b}{\int }}\frac{ds}{s}\right) \\ =\frac{{\mu }_0{n}^2{I}^2}{4\pi }h\ln \left(\frac{b}{a}\right) \end{gathered}$$

From reference equation as 7.27.

$L=\frac{{\mu }_0{n}^{2}h}{2\pi }\ln \left(\frac{b}{a}\right)$

But $W=\frac{{\mu }_0{n}^2{I}^2}{4\pi }h\ln \left(\frac{b}{a}\right)$ …… (2)

Substitute $\frac{{\mu }_0{n}^{2}h}{2\pi }\ln \left(\frac{b}{a}\right)$ for L into above equation (2).

$W=\frac{1}{2}L{I}^2$

Therefore, thevalue ofthe energy stored in the toroidal coil is $W=\frac{1}{2}L{I}^2$.