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# Magnetic forces on a current carrying conductor

## Magnetic force on a current-carrying conductor

A conductor carrying a current inside a magnetic field experiences a force perpendicular to the current.

The electric current in a conductor is made up of many moving electrons. Each electron generates a magnetic field, which interacts with the external field. This interaction means that each electron experiences a force due to the magnetic field.

The combination of the forces on all the electrons inside the conductor gives the overall force on the conductor.

The direction of the force on the conductor is given by Fleming's left hand rule.

The wire swings to the right because of the magnetic force on it. The direction can be determined from Fleming's left hand rule.

## Calculating the force on a current-carrying conductor

The force on a current carrying conductor placed in a magnetic field (e.g. wires, electric motors) is given by: $$F=BI\ell\sin\theta$$ $F$ is the force on the conductor, $B$ is the magnetic flux density, $I$ is the current in the conductor, $\ell$ is the length of the conductor and $\theta$ is the angle between the current and the magnetic field vectors.

This formula can be obtained by taking the sum of forces on all the charges travelling (as an electric current) through the conductor.

This is done by applying the relations $I=Q/t$, $Q=Nq$ and $vt=\ell$ to the formula $F=Bqv\sin\theta$.

The direction of the force is given by Fleming's Left Hand Rule. This force is always perpendicular to the current and the magnetic field.

Force on a current carrying wire

## Forces on two current-carrying conductors

Two current-carrying wires placed next to each other exert a force on each other.

This force is due to the interaction of the individual magnetic fields generated by the electrons flowing in each wire.

If the current in each wire flows in the same direction, they attract each other.

If the currents in each wire are travelling in opposite directions, they repel each other.

Two parallel wires carrying current in the same direction will attract each other.

## Calculating the forces between two wires

When two parallel, current-carrying wires are placed in close proximity, they exert forces on each other due to the interactions of their individual magnetic fields.

If the current in each wire flows in the same direction, they attract each other. If the currents in each wire are travelling in opposite directions, they repel each other.

The ratio of the magnitude of the force to the length of the wires (which is assumed to be equal) is given by: $$\frac{F}{\ell}=\frac{\mu_{0}I_{1}I_{2}}{2\pi d}$$ $F$ is the force between the wires, $\ell$ is the length of the wires, $I_{1}$ and $I_{2}$ are the currents flowing in each wire and $d$ is the distance between the wires (not the sum of the lengths of the wire).

The term $\mu_{0}=1.26\times 10^{-6}\text{ m kg s}^{-2}\text{ A}^{-2}$ is a universal constant called the permeability of free space and is used in many electromagnetic calculations.

Permeability is the capacity of a medium to magnetise in the presence of a magnetic field. For the case of $\mu_{0}$, the medium in question is "free space" (i.e. a vacuum).