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  • A uniform but time-varying magnetic field (Bt) exists in a cylindrical region of radius a and is directed into the plane of the paper as shown. The magnitude of the induced electric field at point

A uniform but time-varying magnetic field (Bt) exists in a cylindrical region of radius a and is directed into the plane of the paper as shown. The magnitude of the induced electric field at point P at a distance r from the centre of the circular region:

 

Option: 1

is zero


Option: 2

decreses as \frac{1}{r}


Option: 3

increses as r


Option: 4

decreases as \frac{1}{r^{3}}


Answers (1)

best_answer

Draw a concentric circle of radius r. The induced electric field € at any point on the circle is equal to that at P. For this circle, induced emf

\mathrm{\begin{aligned} & \mathrm{e}=\int \mathrm{E} \cdot \mathrm{dl}=\left|\frac{\mathrm{d} \phi}{\mathrm{dt}}\right|=\mathrm{S}\left|\frac{\mathrm{dB}}{\mathrm{dt}}\right| \\ & \therefore \mathrm{E}=\iint \mathrm{d} l=\pi \mathrm{a}^2\left|\frac{\mathrm{dB}}{\mathrm{dt}}\right| \quad \quad\left[\text { but } \int \mathrm{d} l=2 \pi \mathrm{r}\right] \\ & \therefore \mathrm{E} \times(2 \pi \mathrm{r})=\pi \mathrm{a}^2\left|\frac{\mathrm{dB}}{\mathrm{dt}}\right| \\ & \therefore \mathrm{E}=\frac{\mathrm{a}^2}{2 \mathrm{r}}\left|\frac{\mathrm{dB}}{\mathrm{dt}}\right| \\ & \Rightarrow \mathrm{E} \propto \frac{1}{\mathrm{r}} \end{aligned}}

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