Question 359: Electromagnetic Induction

Question

A long solenoid of radius a and number of turns per unit length $$n$$ is enclosed by cylindrical shell of radius R, thickness $$d$$ $$(d < < R)$$ and length L. A variable current $$\mathrm{I}=\mathrm{I}_{0} \sin \omega t$$ flows through the coil. If the resistivity of the material of cylindrical shell is $$\mathrm{P}$$, find the induced current in the shell.

Accepted Answer

According to the question, radius of the solenoid is $$r=a$$,
Number of turns per unit length $$n$$,
Radius of the cylinder shell is $$\mathrm{R}$$,
Thickness of shell is $$d$$, length is L,
Resistivity of cylindrical shell is $$ho$$.
The expression of the current through solenoid is given as,
$$\mathrm{I}=\mathrm{I}_{0} \sin \omega t$$
In the given case the magnetic field outside the solenoid is zero. Therefore, magnetic field inside the solenoid is $$\mu_{0} n_{i}$$
Now, magnetic flux can be written as $$\phi=\mu_{0} n \mathrm{I} 	imes \pi a^{2}$$ Therefore, induced emf can be written as
$$\begin{aligned}
e & =-\frac{d \phi}{d t} \
e & =-\frac{d}{d t}\left(\mu_{0} n \mathrm{I} \pi a^{2}ight)
\end{aligned}$$
As current through the solenoid is given as
$$\begin{aligned}
& \mathrm{I}=\mathrm{I}_{0} \sin w t \
& \Rightarrow \quad e=-\frac{d}{d t}\left(\mu_{0} n \mathrm{I}_{0} \sin \omega t \pi a^{2}ight) \
& =-\left(\mu_{0} n \pi a^{2} \mathrm{I}_{0}ight) \frac{d}{d t} \sin \omega t \
& \Rightarrow \quad e=\left(\mu_{0} n \pi a^{2}ight)\left(\mathrm{I}_{0} l_{0} \omega \cos \omega tight)
\end{aligned}$$
Resistance of the cylindrical vessel,
$$\mathrm{R}_{c}=\frac{ho}{\mathrm{S}}=\frac{ho(2 \pi \mathrm{R})}{\mathrm{L} d}$$
$$	herefore$$ Induced current, $$\mathrm{I}=\frac{e}{\mathrm{R}_{c}}$$
$$=\frac{\left(\mu_{0} n \pi a^{2}ight)\left(\mathrm{I}_{0} \omega \cos \omega tight)}{\frac{ho(2 \pi \mathrm{R})}{\mathrm{L} d}}$$
$$\mathrm{I}=\frac{\mu_{0} \mathrm{~L} d n a^{2} i_{0} \omega \cos \omega t}{2 ho R}$$

Answer

$$\mathrm{I}=\frac{\left(\mu_{0} \mathrm{~L} d n a^{2} \mathrm{I}_{0} \omega \cos \omega t\right)}{\rho \mathrm{R}}$$

Answer

$$\mathrm{I}=\frac{\left(3\mu_{0} \mathrm{~L} d n a^{2} \mathrm{I}_{0} \omega \cos \omega t\right)}{2 \rho \mathrm{R}}$$

Answer

$$\mathrm{I}=\frac{\left(\mu_{0} \mathrm{~L} d n a^{2} \mathrm{I}_{0} \omega \cos \omega t\right)}{2 \rho \mathrm{R}}$$

Answer

$$\mathrm{I}=\frac{3\left(\mu_{0} \mathrm{~L} d n a^{2} \mathrm{I}_{0} \omega \cos \omega t\right)}{ \rho \mathrm{R}}$$

Electromagnetic Induction

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