91Ó°ÊÓ

Radius of circular plate is $\mathrm{R}=16\mathrm{mm}$

Width is $\mathrm{d}=5.0\mathrm{mm}$

$\mathrm{V}=\left(100\mathrm{V}\right){\mathrm{e}}^{−\frac{\mathrm{t}}{\mathrm{τ}}}$

Radius is $\mathrm{r}=0.8\mathrm{R}$

Time constant, $\mathrm{Ï„}=12\mathrm{ms}$

We have to use the formula for magnetic field induced by electric field, and then we use the formula for electric field in terms of voltage and distance and substitute it in the equation of magnetic field.

Formula:

$\mathrm{B}=\frac{{\mathrm{μ}}_0{\mathrm{Î\mu }}_0\mathrm{r}}{2}×\frac{\mathrm{dE}}{\mathrm{dt}}$

$\mathrm{E}=\mathrm{V}/\mathrm{d}$

Magnetic field induced by changing electric field is given as

$\mathrm{B}=\frac{{\mathrm{μ}}_0{\mathrm{Î\mu }}_0\mathrm{r}}{2}×\frac{\mathrm{dE}}{\mathrm{dt}}$

We know $\mathrm{E}=\mathrm{V}/\mathrm{d}$.

Differentiating the above with respect to time:

$\begin{array}{c}\frac{\mathrm{dE}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{V}}{\mathrm{d}}\right)\\ =\frac{1}{\mathrm{d}}×\frac{\mathrm{d}}{\mathrm{dt}}\left(\left(100\right){\mathrm{e}}^{−\frac{\mathrm{t}}{\mathrm{Ï„}}}\right)\\ =\frac{−100×{\mathrm{e}}^{−\frac{\mathrm{t}}{\mathrm{Ï„}}}}{\mathrm{Ï„»å}}\end{array}$

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{μ}}_0{\mathrm{Î\mu }}_0\mathrm{r}}{2}×\frac{−100×{\mathrm{e}}^{−\frac{\mathrm{t}}{\mathrm{Ï„}}}}{\mathrm{Ï„»å}}\\ =−\frac{4\mathrm{Ï€}×{10}^{−7}×8.85×{10}^{−12}×16×{10}^{−3}×100×0.8×{\mathrm{e}}^{−\frac{\mathrm{t}}{12\text{ }\mathrm{ms}}}}{2×12×{10}^{−3}×5×{10}^{−3}}\\ =−(1.2×{10}^{−13}\mathrm{T}){\mathrm{e}}^{−\frac{−\mathrm{t}}{12\mathrm{ms}}}\end{array}$

The magnitude is$\mathrm{B}=(1.2×{10}^{−13}\mathrm{T}){\mathrm{e}}^{−\frac{−\mathrm{t}}{12\mathrm{ms}}}$

$\begin{array}{c}\mathrm{B}=(1.2×{10}^{−13}\mathrm{T}){\mathrm{e}}^{−\left(\frac{3\mathrm{τ}}{\mathrm{τ}}\right)}\\ =5.9×{10}^{-15}\mathrm{T}\end{array}$

Magnetic field as a function of time at $\mathrm{t}=3\mathrm{τ}$ is $5.9×{10}^{-15}\mathrm{T}$