# Magnetic Force: Definition, Equation & Units (w/ Examples)

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One surprising discovery in early physics was that electricity and magnetism are two sides of the same phenomenon: electromagnetism. In fact, magnetic fields are generated by moving electric charges or changes in the electric field. As such, magnetic forces act, not just on anything magnetized, but also on moving charges.

## Definition of Magnetic Force

The magnetic force is the force on an object due to interactions with a magnetic field.

The SI unit for magnetic force is the newton (N) and the SI unit for magnetic field is the tesla (T).

Anyone who has held two permanent magnets near each other has noticed the presence of a magnetic force. If two magnetic south poles or two magnetic north poles are brought near each other, the magnetic force is repulsive and the magnets will push against each other in opposite directions. If opposite poles are brought near, it is attractive.

But the fundamental origin of the magnetic field is moving charge. On a microscopic level, this is happening due to movements of electrons in the atoms of magnetized materials. We can understand the origins of magnetic forces more explicitly, then, by understanding how a magnetic field affects a moving charge.

## Magnetic Force Equation

The Lorentz force law relates magnetic field to the force felt by a moving charge or current. This law can be expressed as a vector cross product:

\bold F=q\bold v \times\bold B

for a charge q moving with velocity v in magnetic field B. The magnitude of the result simplifies to F = qvBsin(θ) where θ is the angle between v and B. (So the force is maximum when v and B are perpendicular, and 0 when they are parallel.)

This can also be written as:

for electric current I in a wire of length L in field B.

This is because:

\bold IL=\frac{q}{\Delta t}L = q\frac{L}{\Delta t} = q\bold v

#### Tips

• If an electric field is also present, this force law includes the term qE to include the electric force as well, where E is the electric field.

The direction of the Lorentz force is determined by the right-hand rule. If you point the index finger of your right hand in the direction a positive charge is moving, and your middle finger in the direction of the magnetic field, your thumb gives the direction of the force. (For a negative charge, the direction flips.)

## Examples

Example 1: A positively charged alpha particle traveling to the right enters a uniform 0.083 T magnetic field with its magnetic field lines pointing out of the screen. As a result, it moves in a circle. What is the radius and direction of its circular path if the velocity of the particle is 2 × 105 m/s? (The mass of an alpha particle is 6.64424 × 10-27 kg, and it contains two positively charged protons.)

As the particle enters the field, using the right-hand rule we can determine that it will initially experience a downward force. As it changes direction in the field, the magnetic force ends up pointing towards the center of a circular orbit. So its motion will be clockwise.

For objects undergoing circular motion at constant speed, the net force is given by Fnet = mv2/r. Setting this equal to the magnetic force, we can then solve for r:

\frac{mv^2}{r}=qvB\implies r = \frac{mv}{qB}=\frac{(6.64424\times10^{-27})(2\times 10^5)}{(2\times 1.602\times 10^{-19})(0.083)}=0.05\text{ m}

Example 2: Determine the force per unit length on two parallel straight wires a distance r apart carrying current I.

Since the field and the current are at right angles, the force on the current carrying wire is F = ILB, so the force per unit length will be F/L = IB.

The field due to a wire is given by:

B=\frac{\mu_0I}{2\pi r}

So the force per unit length felt by one wire due to the other is:

\frac{F}{L}=IB=\frac{\mu_0I^2}{2\pi r}

Note that if the direction of the currents is the same, the right-hand rule shows us that this will be an attractive force. If the currents are anti-aligned, it will be repulsive.
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