Lorentz Force Law

Lorentz Force Law Definition

The Lorentz force can be defined as the following.

In particular, the Lorentz force considers the action of both electric and magnetic fields on charged particles.

Lorentz Force Law Equation

The Lorentz force takes the mathematical equation

\[ \vec{F} = q\vec{E} + (q \vec{v} \times \vec{B} ),\]

where \(\vec{F}\) is the force vector exerted on the charged particles measured in Newtons \(\mathrm{N}\), \(q\) is the charge of the particles measured in coulombs \(\mathrm{C}\), \(\vec{v}\) is the velocity vector of the charged particle measured in \(\mathrm{\frac{m}{s}}\), \(\vec{B}\) is the magnetic field vector measured in teslas \(\mathrm{T}\), and \(\vec{E}\) is the electric field vector measured in \(\mathrm{\frac{V}{m}}\).

We can see that this equation is made up of two components; the first term on the right-hand side is the electric force, whereas the second term is the magnetic force.

In order to find the magnitude of the Lorentz force, we take the magnitude of the vector quantities appearing in the equation above. For the electric force term, this is relatively simple as we need only take the magnitude of the electric field \(|\vec{E}|\) multiplied by the magnitude of the charge \(q\).

On the other hand, the magnitude of the cross-product is slightly trickier. Recall that when taking the magnitude of a cross-product, we must multiply the magnitudes of the two vectors by the sine of the angle between the vectors. This ensures that we are taking the perpendicular components of both vectors. One arrives at the equation

\[ | \vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin(\theta),\]

where \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors \(\vec{a}\) and \(\vec{b}\) respectively, and \(\theta\) is the angle between the two vectors.

We can now apply this to our equation for the Lorentz force to find that the magnitude of the Lorentz force is given by

\[ |\vec{F}| = q |\vec{E}|+ q|\vec{v}||\vec{B}|\sin(\theta),\]

where \(\theta\) is the angle between the magnetic field and the velocity of the charged particle, measured in radians \(\mathrm{rad}\). Since \(q\) is a scalar quantity, we don't have to do anything to it.

Lorentz Force Law Derivation

One derivation that can be done from the definition of the Lorentz force, is the velocity of a charged particle when moving in a magnetic field. If we assume that there is no electric field present, only a magnetic field, we can see from the cross-product that the resultant Lorentz force on the charged particle is always perpendicular to the particle's direction of motion. The consequence of this is that the trajectory of the charged particle attains curvature. What force have we come across before that also acts in the perpendicular direction to the motion of the object?

Yes! It is indeed the centripetal force. A charged particle traveling in a magnetic field will undergo circular motion. Recall that our equation for the centripetal force is

\[ F_{\mathrm{cent}} = \frac{mv^2}{r},\]

where \(F_{\mathrm{cent}}\) is the centripetal force measured in newtons \(\mathrm{N}\), \(m\) is the mass of the object measured in \(\mathrm{kg}\), \(v\) is the velocity of object measured in \(\mathrm{\frac{m}{s}}\), and \(r\) is the radius of the rotation measured in \(\mathrm{m}\).

Now that we know that the charged particle is in rotational motion, we can equate the magnitude of the Lorentz force and the centripetal force to find the magnitude of the resultant velocity due to the charged particle's interaction with the magnetic field. Equating and rearranging we find

\[ \begin{align} Bqv\sin(\theta) &= \frac{mv^2}{r} \\ \frac{Bqr \bcancel{v} \sin(\theta)}{m} & = v^{\bcancel{2}} \\v &= \frac{Bqr\sin(\theta)}{m}. \end{align}\]

When solving problems involving any of these quantities, we can rearrange this equation to isolate the quantity for which we're interested in solving. Just what that quantity is, will vary from problem to problem.

Fig. 2 - An electron traveling in a magnetic field will experience a Lorentz force that causes circular motion.

Applications of Lorentz Force Law

During physics experiments in school, we often come across a piece of equipment called a cathode ray tube, or an electron gun. These devices allow us to see the path of an electron beam being deflected due to an external electric field being applied. A metal filament is heated up on one end such that the electrons in the metal gain enough kinetic energy to break free. Since the electrons are negatively charged, they are attracted to the positively charged anode at the other end of the vacuum tube. Furthermore, the vacuum chamber in which the electrons pass through is lined with a fluorescent material such that when the electrons collide with the walls, they appear as light which can be seen by the human eye. Finally, the curvature of the electron beam is due to the Lorentz force interaction between the charged electrons and the surrounding electric field.

Fig. 3 - An electron gun where the green light represents the path of the electrons.

Lorentz Force Law Example

Finally, let's consider an example where we use the Lorentz force equation.