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The Lorentz Force Law

Instructor
Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Expert Contributor
Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Electric and magnetic fields are both capable of creating forces on charged particles. In this lesson you'll learn how these two forces are related through the Lorentz force law.

Electromagnetic Force

Our universe is filled to the brim with forces. These forces can be responsible for mundane things like you pushing the keys on your keyboard. They can also accomplish extraordinary feats such as keeping our planet spinning around the sun, or allowing atoms to exist by keeping the atomic nuclei from flying apart.

No matter how mundane or extraordinary the force, they all fall into four different categories. We call these the four fundamental forces of nature, and they are the gravitational force, electromagnetic force, strong force, and weak force. Let's look a bit at the electromagnetic force.

When you read the word electromagnetic, in your mind you probably break it down into two parts; electric and magnetic. The electromagnetic force governs all electric and magnetic forces, but how are electric and magnetic forces related to each other? It turns out that we can couple these two forces together through something called the Lorentz force law.

Electric Force

Before we get into the Lorentz force law, let's look at electric and magnetic forces separately. We'll start with electric force. Imagine you have a particle with some charge (q). For example, it could be a negatively charged electron or a positively charged proton.

Now, let's look at what happens when we put two charged particles next to each other. Any object with charge creates an electric field (E). Also, any charged particle put in an electric field feels a force created by this field. We call this an electric force (Fe).

Electric Field Lines from an Electron
electric field lines

You've probably already heard that like charges repel and opposite charges attract. These objects are creating forces on each other through their electric fields that are either pulling them together or pushing them apart.

When looking at the electric field acting on a single charged particle we find that the force it creates on that particle is equal to the charge of the particle multiplied by the electric field.


electric force


What happens when this force is put on our charged particle? One thing that can occur is that it can start moving. It turns out something really interesting happens when charged particles are on the move.

Magnetic Force

When you have a bunch of charged particles moving in one direction it creates an electrical current. Electric currents are everywhere in modern day life. For example, every electric appliance you plug into your wall has a current of charged electrons moving through it.

When you have two electric currents, such as the ones found in electric wires, next to each other they impart forces on each other. If the currents flow in the same direction the forces attract the two currents, but if they flow in opposite directions they repel each other.

Magnetic Fields Created by Current
magnetic field lines

What's happening is that the moving charged particles are creating a magnetic field (B). This magnetic field is not only being created by the moving charged particles, but also only moving charged particles feel the force it creates. Any stationary charged particle wouldn't be affected by this force at all.

Since a charged particle needs to be moving for it to feel a magnetic force, it makes sense that this force should be tied to that particle's velocity (v). When looking at the formula for magnetic force (Fb) we can see that this is indeed the case.


magnetic force


The magnetic force equals the charge of the particle times the cross product of the velocity and the magnetic field. It is important to note that the 'x' in the equation is very specifically a cross product sign between two vectors, v and B, and not the standard multiplication sign you see when multiplying two scalars.

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Additional Activities

The Lorentz Force Law

Electricity and magnetism are tightly intertwined, so to speak. Moving electric charges generate magnetic fields and changing magnetic fields induce electric current in conductors. Electric charges moving in magnetic fields experience a force, but only if the moving charge is not parallel or antiparallel to the magnetic field. The equation for the force a moving electric current experiences when moving in a magnetic field is F = qvBsinθ where:

  • F is the force the charge experiences
  • q is the magnitude of the charge
  • v is the velocity of the charge
  • θ is the angle between the velocity of the charge vector and the magnetic field vector

The right-hand rule is a very useful way to determine the direction of the force on a moving electric charge, the magnetic field orientation, and the velocity vector of the moving electric charge. This visual device depends on whether the charge is positive or negative.

For positive charges:

  • Right-hand fingers point in the direction of the magnetic field lines (from N to S), and thumb is extended.
  • Bend right-hand fingers perpendicular to the magnetic field lines
  • Thumb automatically points in the direction of the force

For negative charges:

Do the same as for positive charges, except the force is in the opposite direction of the way your thumb points.

Let's do some practice with this equation.

Practice Problems

  1. An electron is moving antiparallel to the magnetic field lines. Explain using the Lorentz force equation and right hand rule why this charge doesn't experience a force.
  2. What angle θ would any moving charge experience a maximum force?
  3. An electron moving in a magnetic field experiences a force of 5 x 10-20 N while it is moving at 5 x 107 m/s. The magnetic field strength is 1 x 10-8 T. At what angle is the charge moving relative to the magnetic field lines?

Answers

  1. The sin of 0 is 0. The right hand rule can't be done because the thumb is always is perpendicular to the fingers in the right hand rule.
  2. 90° because the sin of 90° is 1.
  3. F = qvBsinθ ..... 5 x 10-20 = (1.6 x 10-19)(5 x 107)(1 x 10-8)(sin θ) .... sinθ = 0.625 .... θ ≈ 39°

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