Q7.29P - Introduction to Electrodynamics

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.

Step-by-Step Solution

Answer

Answer

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

Step 1: Write the given data from the question

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

Step 2: Determine the formula of the energy stored in the toroidal coil

Write the formula of the 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

Step 3: (a) Determine the value of the energy stored in the toroidal coil

Determine the energy stored in the toroidal coil.

Substitute $\frac{{\mu }_{0}nI}{2\pi s}$  for B and  $hrd\varphi ds$   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, the value of the energy stored in the toroidal coil is  $W=\frac{1}{2}L{I}^2$.