# Bar Magnet As An Equivalent Solenoid

In order to understand how a bar magnet is equivalent to a solenoid, it becomes important to understand what is a bar magnet and what is a solenoid and what are their similarities and differences.

## What is a Bar Magnet?

A bar magnet is a rectangular or a square piece of an object made from iron or steel having permanent magnetic properties with two poles: north and south.

## What is a Solenoid?

A solenoid is a coil with a length greater than its diameter and is a type of electromagnet to produce controlled magnetic fields by passing an electric current through it.

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## Derivation of Bar Magnet as an Equivalent Solenoid

By calculating the axial field of a finite solenoid carrying current, a bar magnet can be demonstrated as a solenoid. Consider a solenoid of radius a and length 2l with n number of turns per unit length that has current I passing through the solenoid. Considering a small element of thickness dx of the solenoid at distance x from O such that OP = r.

Bar magnet as an Equivalent Solenoid

Magnetic field due to n turns at axis of solenoid

$dB=\frac{\mu _{0}ndxIa^{2}}{2\left [ (r-x)^{2}+a^{2} \right ]^{\frac{3}{2}}}$

Integrating x from -I to +I to get the magnitude of the total field

$B=\frac{\mu _{0}nIa^{2}}{2}\int_{-I}^{I}\frac{dx}{\left [ (r-x)^{2}+a^{2} \right ]^{\frac{3}{2}}}$ $\left [ (r-x)^{2} \right ]^{\frac{3}{2}}=r^{3}$

And $B=\frac{\mu _{0}nIa^{2}}{2r^{3}}\int_{-I}^{I}dx$ $=\frac{\mu _{0}nI}{2}\frac{2Ia^{2}}{r^{3}}$

Therefore, $B=\frac{\mu _{0}}{4\pi }\frac{2m}{r^{3}}$

From the above expression, it is understood that the magnetic moment of a bar magnet is equal to the magnetic moment of a solenoid.

### Difference between a Bar Magnet and a Solenoid

• The bar magnet is a permanent magnet whereas a solenoid is an electromagnet ie, it acts as a magnet only when an electric current is passed through.
• When a bar magnet is cut into two halves, both the pieces act as a magnet with the same magnetic properties whereas when a solenoid is cut into two halves, they will have weaker fields.
• The poles of the bar magnet are fixed whereas for a solenoid the poles can be altered.
• The strength of the magnetic field of a bar magnet is fixed ie, unaltered whereas the strength of the magnetic field of a solenoid depends on the electric current that is passed through it.

### Similarities between a Bar Magnet and a Solenoid

• Bar magnet and solenoid both have attractive and directive properties ie, to align itself along the external magnetic field.
• The magnetic field at the axial point is the same for both.
• The magnetic moment is the same for both.