# PHYS 2212 Module 7.2

## Magnetic Fields and Lines

7.2 Magnetic Fields and Lines

### Learning Objectives

By the end of this section, you will be able to:

• Define the magnetic field based on a moving charge experiencing a force
• Apply the right-hand rule to determine the direction of a magnetic force based on the motion of a charge in a magnetic field
• Sketch magnetic field lines to understand which way the magnetic field points and how strong it is in a region of space

### Magnetic Field Lines

First, let’s define something. You know how we call electric fields ? Well, magnetic fields get their own symbol too, and it’s . I know, it’s not M, for magnetic? It’s not. This is due to historical reasons that I won’t go too deep into here. But when James Clerk Maxwell was developing his theory of electromagnetism, he was naming the vector fields and he did so alphabetically. There’s an field, a field, and so on. It just so happens that the electric field was conveniently named the field and the magnetic field got stuck with the name: the field.

We know that permanent magnets exist in nature, and we also now know from Oersted’s experiments that an electric current is a source of magnetic field too. In this image you can see the magnetic field lines (blue lines with arrows labeled ) for different magnets.

One thing to note about magnetic field lines is that they are always closed loops. This is very different from electric field lines (E-fields start at positive charge and end at negative charge) because B-field lines don’t start or end, they are continuous. Look at the bar magnet (a) in the figure above. Outside the magnet, the B-field lines go from N to S and inside the magnet the B-field lines go from S to N. You can see a similar pattern in the other B-fields too.

### Magnetic Force (Lorentz Force)

There are three experimental observations about the magnetic force, which are essentially rules about the force:

1. An electric charge (q) can experience a magnetic force only if it is moving in a magnetic field.
2. If the electric charge is moving parallel to the B-field, the charge will not experience a magnetic force.
3. The magnetic force on an electric charge moving through a B-field is always perpendicular to both the velocity of the charge and the B-field vector.

Rule 3 requires we use something called the Right Hand Rule (RHR). We use the RHR to determine the direction of the magnetic force on a charge if we know the directions of the velocity and the B-field. I’ll use the textbook’s version of the RHR (there are lots of different ones, but they all work generally the same). By the way, when you learned about the RHR in physics 1, most likely when you learned about torque, this is the same RHR. This RHR works for any cross product.

### Vectors that Point Into the Page or Out of the Page

With the magnetic force, we are dealing with vectors in three dimensions, which can get a little confusing and also difficult to draw on a two-dimensional page. So we have symbols we use to denote a vector that points into the page (or into the screen as you read this on a computer) and out of the page.

Imagine we use a dart to represent a vector, where the pointy end is the tip and the flight (the other end) is the tail.

If you rotated the dart so that the pointy end was coming right at you, this would be like a vector pointing out of the page. This would look like:

where the dot is the tip of the vector. This is the symbol we use to denote a vector field that points out of the page, towards you.

If you rotated the dart so that the pointy end was pointing away from you, like you’re about to throw the dart, you would look at the flight and it would look something like:

where the X is the flight of the dart. This symbol denotes a vector field that points into the page, away from you.

In this example,

(1) use your right hand, (2) holding your right hand flat, orient your right hand so that your fingers point upward along the direction of and rotate your hand so your fingers curl and point to the right, in the direction of , the magnetic field. (3) Then your thumb indicates the direction of the force. What direction is your thumb pointing?

Your thumb points into the page, so the magnetic force on this charge would be .

What if that charge was negative instead of positive? Do you think that would make a difference? Sure it would. If we replaced the positive charge with a negative charge, but kept everything else the same, then the force would reverse direction. So the magnetic force on a negative charge would be out of the page: .

Try a few of these on your own. What is the direction of the magnetic force on the charge? Or is the force zero?

Practice!

Example: The electron gun in a very old TV (called a CRT display) accelerates the electrons through about 25,000 Volts. Estimate the acceleration of such an electron due to the magnetic field of the earth. Hint: In Atlanta, the Earth’s magnetic field has a magnitude of about 5.2 × 10-5 T.

You should try working through this problem on your own, then watch my video solution.