Name: Electric Field: Time Varying Magnetic Field
Uploaded: 2019-07-19
Duration: 8 min 32 s
Description: Did you know that the cause of the movement of charges in a stationary circular conducting coil placed in a time-varying magnetic field is due to an induced...

Question

Derive the expression for the self inductance of a section of a long solenoid and hence show that self inductance is proportional to the square of number of turns per unit length.

Accepted Answer

We are considering a solenoid with $N$ turns with length $l$ . The area of cross-section is $A$ . The solenoid carriers current $I$ and $B$ is the magnetic field inside the solenoid.

Magnetic field generated at inside the solenoid is, $B = \frac{μ_{\circ} NI}{l}$ ,

Where, $μ_{\circ}$ is the permeability of free space.

The magnetic flux is the product of the magnetic field and area of the cross-section. Here the magnetic flux per turn is given as,

$ϕ_{B} = N (\vec{B} . \vec{A})$ , ( $\vec{B}$ and $\vec{A}$ are along same direction).

$∴ ϕ_{B} = \frac{μ_{\circ} N^{2} IA}{l} . . . . . . (i)$

Now, flux $(ϕ_{B})$ is related to inductance $(L)$ as,

$ϕ_{B} = LI \Rightarrow L = \frac{ϕ_{B}}{I}$

From equation $(i)$ we get,

$\Rightarrow L = \frac{μ_{\circ} N^{2} A}{l}$

Hence, the expression for self inductance is given as, $\frac{μ_{\circ} N^{2} A}{l}$ .

Now, for second part of the question,

Let $N$ be a total number of turns in solenoid, $l$ length of solenoid, $R$ radius of cross-section of solenoid.

Magnetic field inside solenoid, $B = μ_{\circ} \frac{N}{l} i$ , where, $\frac{N}{l}$ is number of turns per unit length.

Now, e.m.f. induced, $E = NBA \Rightarrow E = N \times μ_{\circ} \frac{N}{l} i \times {πR}^{2}$

$\Rightarrow E = \frac{μ_{\circ} N^{2} {πR}^{2}}{l} i = Li$ , where, $L =$ Self inductance

Thus, $L = \frac{μ_{\circ} N^{2} {πR}^{2}}{l}$

Hence, $L \propto N^{2}$ .

Derive the expression for the self inductance of a section of a long solenoid and hence show that self inductance is proportional to the square of number of turns per unit length.

Important Questions on Electromagnetic Induction

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