91Ó°ÊÓ

The rate of change of flux is given by:

$$\begin{gathered} \frac{d\mathrm{Ï•}B}{dt}=A\frac{dB}{dt} \\ =\mathrm{Ï€}{r}_1^2\frac{dB}{dt}\frac{d\mathrm{Ï•}B}{dt} \\ =\mathrm{Ï€}{r}_1^2\frac{dB}{dt} \end{gathered}$$

Now, according to the faradays law we get:

$$\begin{gathered} \mathrm{Î\mu }=∫\stackrel{→}{E}â‹\dots \stackrel{¯}{dl} \\ =−\frac{d\mathrm{Ï•}B}{dt} \end{gathered}$$

Now, the magnetic flux is:$$\begin{gathered} \mathrm{Ï•}=AB \\ ={\mathrm{Ï€°ù}}_1^2\mathrm{B} \end{gathered}$$

Now, integrating LHS gives:

$\begin{array}{r}\stackrel{→}{E}â‹\dots \stackrel{→}{dl}\\ =E\left(2\mathrm{Ï€}{r}_1\right)\end{array}$

Therefore:

$E2\mathrm{Ï€}{r}_1=\frac{d\left(B\mathrm{Ï€}{r}_1^2\right)}{dt}=\mathrm{Ï€}{r}_1^2\frac{dB}{dt}$

Or

$E=\frac{r_1}{2}\frac{dB}{dt}$

The direction of the induced electric field is clockwise.

Since all flux is within r>R then the magnetic field within r_2>R is:

$\mathrm{Ï•}=AB=\mathrm{Ï€}{R}^{2}B$

Integrating LHS we get:

$$\begin{gathered} âˆ\circledR \stackrel{→}{E}â‹\dots \left(\stackrel{→}{dl}\right)=E\left(2\mathrm{Ï€}{r}_2\right) \\ ,E\left(2\mathrm{Ï€}{r}_2\right)=\mathrm{Ï€}{R}^2\frac{dB}{dt} \end{gathered}$$

So

$E=\frac{R^2}{2r_2}\frac{dB}{dt}$

To draw the graph of the magnitude of the induced electric field as a function as a function of the distance r from the axis $r=0tor=2r$. Is shown below:

To calculate the induced emf in the circular turn radius$r=R/2$. We can use the equation used above with$r1→R/2$

$$\begin{gathered} \mathrm{Î\mu }=\mathrm{Ï€}(R/2{)}^2\frac{dB}{dt} \\ =\frac{\mathrm{Ï€}{R}^2}{4}\frac{dB}{dt} \\ \mathrm{Î\mu }=\frac{\mathrm{Ï€}{R}^2}{4}\frac{dB}{dt} \end{gathered}$$

Now, the magnitude of the induced emf in a circular turn of radius$r=R.$ we use:

$\mathrm{Î\mu }=\frac{\mathrm{Ï€}{R}^2}{4}\frac{dB}{dt}$

Finally, we need to calculate the magnitude of the induced emf in a circular turn of radius$r=2R$. All the flux is within$r<R$, therefore:

$$\begin{gathered} \mathrm{Î\mu }=\frac{d\mathrm{Ï•}B}{dt} \\ =\mathrm{Ï€}{R}^2\frac{dB}{dt} \end{gathered}$$

Therefore, the induced emf is$\mathrm{Ï€}{R}^2\frac{dB}{dt}$.