91影视

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

Write the formula ofthe energy stored in the toroidal coil.

$\mathbf{W}=\frac{\mathbf{1}}{\mathbf{2}{渭}_{\mathbf{0}}}{鈭�}_{\mathbf{allspace}}{\mathbf{B}}^{\mathbf{2}}\mathbf{d}蟿$ 鈥︹�� (1)

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

Determine theenergy stored in the toroidal coil.

Substitute $\frac{{渭}_{0}nI}{2蟺s}$ for B and localid="1658742313069" $hrd蠒ds$ for $d蟿$ into equation (1).

Here, r = s

Then

$$\begin{gathered} W=\frac{1}{2{渭}_0}{鈭�}_{allspace}{\left(\frac{{渭}_{0}nI}{2蟺s}\right)}^{2}hrd蠒ds \\ =\frac{1}{2{渭}_0}\left(\frac{{渭}_0^2{n}^2{I}^2}{4蟺}鈭�\frac{1}{s^2}hrd蠒ds\right) \\ =\frac{1}{2{渭}_0}\left(\frac{{渭}_0^2{n}^2{I}^2}{4蟺}h\left(2蟺\right)\underset{a}{\overset{b}{鈭�}}\frac{ds}{s}\right) \\ =\frac{{渭}_0{n}^2{I}^2}{4蟺}h\ln \left(\frac{b}{a}\right) \end{gathered}$$

From reference equation as 7.27.

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

But $W=\frac{{渭}_0{n}^2{I}^2}{4蟺}h\ln \left(\frac{b}{a}\right)$ 鈥︹�� (2)

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

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

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