# Bar Magnet as an Equivalent Solenoid

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

## Derivation of Bar Magnet as an Equivalent Solenoid

By calculating the axial field of a finite solenoid carrying current, 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.

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 like 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.

How does a solenoid behave like a magnet?

Can you determine the north and south poles of a current-carrying solenoid with the help of a bar magnet?

• A solenoid is a long coil of circular loops of insulated copper wire. Magnetic field lines are produced around the solenoid when a current is allowed to flow through it. The magnetic field produced by it is similar to the magnetic field of a bar magnet.
• The field lines produced in a current-carrying solenoid is shown in the following figure In the above figure when the north pole of a bar magnet is brought near the end connected to the negative terminal of the battery, the solenoid repels the bar magnet.
• Since like poles repel each other, the end connected to the negative terminal of the battery behaves as the north pole of the solenoid and the other end behaves as a south pole. Hence, one end of the solenoid behaves as a north pole and the other end behaves as a south pole.