Module 3 : MAGNETIC FIELD
Lecture 19 : Mutual Inductance
  Consider the magnetic field in the toroid at a distance $r$ from the axis. We have seen that the magnetic field $B$ is given by $\mu_0NI/2\pi r$. Thus the value of $B^2/2\mu_0$ at this distance is $\mu_0N^2I^2/8\pi^2 r^2$. Considering the toroid to consist of shells of surface area $2\pi rh$ and thickness $dr$, the volume of the shell is $2\pi rhdr$. The volume integral of $B^2/2\mu_0$ is therefore,
 
\begin{eqnarray*} \int \frac{B^2}{2\mu_0} d^3 r &=& \int_a^b \frac{\mu_0N^2I^2}{... ...\pi}\frac{1}{r}dr\\ &=& \frac{\mu_0N^2hI^2}{4\pi}\ln\frac{b}{a} \end{eqnarray*}
   
  which is exactly the expression for the stored energy derived earlier.
   
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