# Force between magnetic poles formula derivation class 12

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In this article, we will derive an expression for the magnetic force between two magnetic poles, so let’s get started…

## What is magnetic poles?

Magnetic poles are the region of apparently concentrated magnetic strength inside the magnet, where the magnetic attraction is maximum. The magnetic poles didn’t lie at the end of geometrical length but somewhat inside the magnet. [latexpage]

## What is pole strength?

Pole strength is the strength of the magnetic poles that attract magnetic material towards itself. It is a scaler quantity.

## Magnetic pole strength formula

Formula of pole strength is given as- $$P=\frac{W}{I}$$

where,

• P is the pole strength
• W is the work done while moving the magnet around the wire
• I is the electric current in the wire

## Force between two magnetic poles formula derivation

The force of repulsion or attraction F between two magnetic poles of strength $q_{m_1}$ and $q_{m_2}$ separated by a distance r is directly proportional to the product of pole strength and inversely proportional to the square of the distance between their centers i.e

$$F\propto \frac{ q_{m_1} q_{m_2} }{r^2}\;\text{or}\; F=K \frac{q_{m_1} q_{m_2}}{r^2}$$

Where K is the proportionality constant called magnetic force constant which depends on the nature of the medium as well as the system of units chosen. In SI units, $$K=\frac{\mu_0}{4\pi}=10^{-7}{N A^{-2}}$$

Where $\mu_0$ is the absolute permeability of the free space (air/vacuum). Combining all the components of the force, we get-

$$\boxed{F=\frac{\mu_0}{4\pi}\frac{q_{m_1} q_{m_2}}{r^2}}$$ This expression is called the coulomb’s law of magnetic force.

### SI unit of magnetic pole strength derivation

From the expression of coulomb’s law of magnetic force, the SI unit of magnetic pole strength is given as:

$$F=\frac{\mu_0}{4\pi}\frac{q_{m_1} q_{m_2}}{r^2}$$

In the form of SI units this expression is expressed as:

$$N=\frac{N}{A^{2}}\cdot\frac{{q_{m}}^2}{m^2}$$ $${q_m}^2=A^2 m^2$$ $$q_m =Am$$

So ampere-metre (Am) is the SI unit of pole strength.

## Definition of unit pole strength

If $q_{m_1}= q_{m_2}= 1\text{unit}$; $r=1m$, then $$F=\frac{m_0}{4\pi}=10^{-7}N$$

Hence, a unit magnetic pole strength can be defined as that the pole which when placed in the vacuum at the distance of one meter from an identical pole then, repels each other at the force of $10^{-7}N$

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