91Ó°ÊÓ

The given data is listed as follows,

• The radial distance of induced magnetic field from the central axis of a circular parallel plate capacitor is,$r=6.0mm×\frac{1m}{1000mm}=6.0×{10}^{-3}m$
• The induced magnetic field is,$B=2.0×{10}^{-7}T$
• The radius of the plate is,$R=3.0mm×\frac{1m}{1000mm}=3.0×{10}^{-3}m$

The expression for Maxwell’s law of induction is as follows,

$\mathbf{∫}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}\mathbf{·}\mathit{d}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{A}}\mathbf{=}{\mathit{μ}}_{\mathbf{0}}{\mathit{Î\mu }}_{\mathbf{0}}\mathit{A}\frac{\mathbf{d}\mathbf{E}}{\mathbf{d}\mathbf{t}}$

Here, B is the induced magnetic field, A is the area enclosed which is given by the following expression $A=\mathrm{Ï€}{R}^2$(Here, R is the radius of the plate),${\mathrm{μ}}_0$is the magnetic permeability of the free space with the value of $\left(4\mathrm{Ï€}×{10}^{-7}\mathrm{N}/{\mathrm{A}}^2\right),{\mathrm{Î\mu }}_0$, is the electric permeability of the free space with the value of $\left(8.85×{10}^{-12}{C}^2/N·m^2\right)$, and$\frac{dE}{dt}$is then the rate of change of electric field.

Substitute the$2\mathrm{Ï€°ù}$for dA, and${\mathrm{Ï€¸é}}^2$ for A in the Maxwell’s law of induction and rearrange the expression.

$\begin{array}{rcl}B\left(2\mathrm{Ï€°ù}\right)& =& {\mathrm{μ}}_0{E}_0\mathrm{Ï€}{R}^2\frac{dE}{dt}\\ B& =& \frac{{\mathrm{μ}}_0{E}_0\mathrm{Ï€}{R}^2}{2r}\frac{dE}{dt}\\ \frac{dE}{dt}& =& \frac{2Br}{{\mathrm{μ}}_0{E}_0{R}^2}\end{array}$

Substitute all the values in the above expression.

$\begin{array}{rcl}\frac{dE}{dt}& =& \frac{2×2.0×{10}^{-7}T×6.0×{10}^{-3}m}{\left(4\mathrm{π}×{10}^{-7}\mathrm{N}/{\mathrm{A}}^2\right)\left(8.85×{10}^{-12}{C}^2/N·m^2\right){\left(3.0×{10}^{-3}m\right)}^2}\\ & =& \frac{24.0×{10}^{-10}}{1001×{10}^{-25}}T·m·A^2/C^2×\frac{1N/A·m}{1T}\\ & =& 2.3975×{10}^{13}N·A/C^2×\frac{1C/s·m}{1A}\\ & =& 2.4×{10}^{13}N/C·s×\frac{1V/m}{1N/C}\\ & =& 2.4×{10}^{13}V/m·s\end{array}$

Thus, the value of$\frac{d\stackrel{â‡\P Ä}{E}}{dt}$between the charging plates is $2.4×{10}^{13}V/m·s$.