Magnetic Force on Moving Charges

A charge has electric field around it. When the charge movies it also has magnetic field. The magnetic field exerts force on other moving charges. A stationary charge does not have magnetic field but a moving charge has both electric and magnetic fields. The magnetic field only exerts force on other moving charges not on stationary charges.

Since moving charge is a current, the electric current has a magnetic field and it exerts force on other currents. The expression of magnetic force is based on the experimental evidence, that is the equation for the magnetic force we are about to determine is completely experimental not theoretical.

The experiments with a moving charge $q$ in a magnetic field reveal proofs similar to those of electric force. The magnetic force is directly proportional to the moving charge $q$. It means if you double the charge, the magnetic force doubles. The magnetic force is directly proportional to the velocity $\vec v$ of the charge, and it is directly proportional to the magnetic field $\vec B$. If $\theta $ is the angle between $\vec v$ and $\vec B$, the magnetic force is also directly proportional to $\sin \theta$.

Figure 2 Magnetic force on a moving charge.

The direct proportionality to $\sin \theta$ means that the magnetic force is directly proportional to the component of $\vec v$ or $\vec B$ perpendicular to $\vec B$ or $\vec v$ respectively. In words the magnetic force is proportional to the component of velocity perpendicular the magnetic field or the component of magnetic field perpendicular to the velocity if the velocity vector makes an angle with the field. You'll see how we arrange the definition of magnetic force as a cross product so its direction is given by the right hand rule.

The observations that are different from similar experiments involved to determine electric force are the magnetic force is proportional to the velocity of the charge and the magnetic force is proportional to $\sin \theta$. Note that the direction of magnetic force is perpendicular to the plane containing velocity vector and magnetic field.

Figure 2 Right hand rule of vector cross product.

The direction of magnetic force is determined by the right hand rule of vector cross product. To determine the direction, imagine $\vec v$ is moving into $\vec B$, and curl the fingers of your right hand in that direction and the thumb then points to the direction of magnetic force for a positive charge. For negative charge, the direction is opposite to the direction the thumb points. The magnitude of magnetic force $\vec F$ based on the experimental observations is

\[F = |q|vB\sin \theta \tag{1}\label{1}\]

The value $B\sin \theta$ is the component of magnetic field perpendicular the velocity vector. You can also rearrange the above equation as $|q|Bv\sin \theta$ and the quantity $v\sin \theta$ is the component of $\vec v$ perpendicular to $\vec B$. In vector form we can represent the above equation as the cross product of two vectors (if you are not familiar with the cross product of vectors you may need to review article on cross product first).

\[\vec F = q \vec v \times \vec B \tag{2} \label{2}\]

The direction of $\vec F$ as already noted is perpendicular to the plane containing $\vec v$ and $\vec B$ also given by the right hand rule (curl the fingers in the sense of $\vec v$ moving into $\vec B$). Note that the cross product is not commutative. It is because of the direction of the vector (result of the cross product).

From Equation \eqref{1}, $B = F/qv\sin \theta$, so the SI unit of magnetic field is $\text{N} \cdot s/\text{m} \cdot \text{C}$. You know the SI unit of electric current is Ampere (A) and one Ampere is one coulomb per second, that is $1\text{A}= 1 \text{C/s}$, so the SI unit of magnetic field is $\text{N/A}\cdot \text{m}$. This unit is called Tesla, that is $1\text{ T} = 1\text{N}/\text{A}\cdot\text{m}$.

The magnetic field can be measured by determining force on known charges moving in known speed and then using equation Equation \eqref{2} to solve for the magnetic filed. For example a magnetic field is applied along with a cathode ray tube which deflects the charges under the action of magnetic force. The direction of deflection of electron beam also provides the sense of direction of magnetic force.

If a particle of charge $q$ moves in space in the presence of both electric and magnetic fields, the total force on the moving charge is the sum of both forces due to electric and magnetic fields, that is

\[\vec F = q\vec E + q\vec v \times \vec B \]

This total force is called Lorentz force and this relationship for this total force is called Lorentz force law.

Electromagnetism
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