Question

What is self inductance? Establish expression for self inductance of a long solenoid.

Answer 1

Self inductance: When the current is passed through a coil, the magnetic flux linked with the coil is directly proportional is the current.
If the magnetic flux is $\Phi$ for the current I, then
$\Phi \propto I$
or $\Phi =L.I$
Where L is constant, its value depends upon shape and size of the coil meidum and number of turns. It is called self inductance or coefficient of self induction of the coil.
Expression- Consider a solenoid whose length l is very large in comparison to its radius r. A uniform magnetic filed is induced within it at every point when a current I is passed through it. The intensity of this induced magnetic field.
$B=\frac{\mu BI}{l}$ ..........(1)
Here $\mu$ is the relative permeability of the material of core and N is the number of turns of the conducting wire on the solenoid. The magnetic flux linked with the solenoid is.
$\Phi =B\times$ total effective area of solenoid.
or $\Phi =B\times N\pi r^2$ ........(2)
If self inductance is L, then
$\Phi =LI$ ........(3)
Comparing eqns. (2) and (3),
$LI=B\times N\pi r^2$
or $L=\frac{B\times N\pi r^2}{I}$ ........(4)
Substituting B from eqn. (1) in eqn. (4),
$L=\frac{\mu NI}{l}\cdot \frac{N\pi r^2}{I}$
or $L=\frac{\mu \pi N^2{r}^2}{l}$
This is the required expression.
Factors affecting the self inductance of the solenoid-
(i) On N the number of turns in the solenoid: $L\propto N^2$ i.e., the self inductance increases on increasing the number of turns in the solenoid.
(ii) On the radius(r) of the solenoid: $L\propto r^2$ i.e., the self inductance increases on increasing the radius.
(iii) On the length (l) of the solenoid: $L\propto \frac{1}{l}$ i.e., self inductance decreases on increasing the length of solenoid.
(iv) On the relative permeability $\left(\mu \right)$ of the material placed inside the solenoid: $L\propto \mu$ i.e., if a soft iron rod is placed inside the solenoid its self inductance increases.