# Solved Problems-5

Problems-5

Find the field inside a solenoid of length

*L*having*N*turns uniformly wound round a cylinder of radius*a*and carrying a current*I*.or,

Derive a general expression for the field at any point along the axis of a solenoid (uniform cylindrical coil wound on a nonmagnetic frame). Sketch the variation of from point to point along the axis.

or,

A solenoid of length

*L*consists of*N*turns of wire carrying a current*I*. Show that the field at any point along the axis,where

*Î¸*_{1}and*Î¸*_{2}are the angles subtended at the point by the end turns. Also, show that if*L*is very large compared to the radius of the solenoid then at the centre of the solenoid, .Solution

We consider the cross section of the solenoid (Figure).

**Solenoid of finite length**

Since the solenoid consists of circular loops, the contribution to the magnetic field by an element of the solenoid of length

*dz*is,From the figure,
âˆ´

âˆ´
So, the field is given as,
âˆ´
We consider three cases:

The magnetic field is given as,

The magnetic field is given as,
Here,

The magnetic field is given as

âˆ´

**When***P*is at One End of the Solenoid:

The magnetic field is given as,

**Note:**If

*L*>>

*a*then

**When***P*is at the Centre of the Solenoid:

The magnetic field is given as,

**Note:**If

*L*>>

*a*then

**When***P*is at the Midway between One End and Centre of the Solenoid:

**Note:**If

*L*>>

*a*then and the magnetic field is given as,

The variation of the field from point to point is shown in Figure.

**Variation of magnetic field from point to point along the axis of a finite solenoid**