Question

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.

Short answer

Answer

The value of the energy stored in the toroidal coil is $W=\frac{1}{2}L{I}^2$.

Step-by-step solution

Write the given data from the question

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

Determine the formula of the energy stored in the toroidal coil

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

(a) Determine the value of the energy stored in the toroidal coil

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