## SAT Physics Subject Test

## Chapter 11 Magnetic Forces and Fields

In __Chapter 8__, we learned that electric charges are the source of electric fields and that other charges experience an electric force in those fields. The charges generating the field were assumed to be at rest. Electric charges *that move* are the source of **magnetic fields**, and other charges that move can experience a magnetic force in these fields. As you may recall, magnetic forces and fields make up a significant portion of the SAT Physics Subject Test; they are the focus of this chapter.

### THE MAGNETIC FORCE ON A MOVING CHARGE

If a particle with charge *q* moves with velocity **v** through a magnetic field **B**, it will experience a magnetic force, **F**_{B}, with magnitude

F_{B} = |*q*| vBsin *θ*

where *θ* is the angle between **v** and **B**. From this equation, we can see that if the charge is at rest, then *v* = 0 immediately gives us *F*_{B} = 0—magnetic forces only act on moving charges. Also, if **v** is parallel (or antiparallel) to **B**, then **F**_{B} = 0 since, in either of these cases, sin *θ* = 0. So only charges that cut across the magnetic field lines will experience a magnetic force. Furthermore, the magnetic force is maximized when **v** is perpendicular to **B**, since if *θ* = 90°, then sin *θ* is equal to 1, its maximum value.

The direction of **F**_{B} (given by the right-hand rule) is always perpendicular to both **v** and **B** and depends on the sign of the charge *q*.

**Right-Hand Rule:**

With your right hand (palm up), point your thumb in the direction of **V** and your fingers in the direction of **B**. If *q* is positive, **F**_{B} points out of the palm. If *q* is negative, **F**_{B} points into the palm.

Notice that there are fundamental differences between the electric force and magnetic force on a charge. First, a magnetic force acts on a charge only if the charge is moving; the electric force acts on a charge whether it moves or not. Second, the direction of the magnetic force is always perpendicular to the magnetic field, while the electric force is always parallel (or antiparallel) to the electric field.

The SI unit for the magnetic field is the **tesla** (abbreviated **T**), which is one newton per ampere-meter. Another common unit for magnetic field strength is the **gauss** (abbreviated **G**); 1 G = 10^{–4} T.

1. A charge +*q* = +6 × 10^{–6} C moves with speed **v** = 4 × 10^{5} m/s through a magnetic field of strength **B** = 0.5, T, as shown in the figure below. What is the magnetic force experienced by *q* ?

Here”s How to Crack It

The magnitude of F_{B} is

F_{B} = *q***v**B sin *θ* = (6 × 10^{–6} C)(4 × 10^{5} m/s)(0.5 T) sin 30° = 0.6 N.

The right-hand rule on the previous page was designed for situations in which the charged particle moves perpendicular to **B**. We can use it for our example if we point the thumb in the direction of the component of **v** that is perpendicular to **B** (up). **B** points to the right and therefore **F**_{B} is into the page.

2. A particle of mass *m* and charge +*q* is projected with velocity **v** (in the plane of the page) into a uniform magnetic field **B** that points into the page. How will the particle move?

Here”s How to Crack It

Since **v** is perpendicular to **B**, the particle will feel a magnetic force of strength *qvB*, which will be directed perpendicular to **v** (and to **B**) as shown.

Since **F**_{B} is always perpendicular to **v**, the particle will undergo uniform circular motion; **F**_{B} will provide the centripetal force. Notice that, because **F**_{B} is always perpendicular to **v**, the magnitude of **v** will not change, just its direction.

**The Skinny on Magnetic Fields**

**F**_{B} is always perpendicular to both **v** and **B**.

Magnetic forces cannot change the speed of an object, only its direction.

The magnetic field does no work on any charge.

The radius of the particle”s circular path is found from the equation *F*_{B} *= F*_{C}.

3. A particle of charge –*q* is shot into a region that contains an electric field, **E**, crossed with a perpendicular magnetic field, **B**. If *E* = 2 × 10^{4} N/C and *B* = 0.5 T, what must be the speed of the particle if it is to cross this region without being deflected?

Here”s How to Crack It

If the particle is to pass through undeflected, the electric force it feels has to be canceled by the magnetic force. In the diagram above, the electric force on the particle is directed upward and the magnetic force is directed downward. So **F**_{E} and **F**_{B} point in opposite directions, and for their magnitudes to balance, *q*E must equal *q***v**B, so **v** must equal E/B, which in this case gives

4. The figure below shows a uniform magnetic field, **B**, whose field lines point up in the plane of the page, and three particles, all with the same positive charge, *q*, and all moving with the same speed, **v**. Which particle—X, Y, or Z—will experience the greatest magnetic force?

Here”s How to Crack It

Since the velocity of particle Y is parallel to the direction of **B**, particle Y feels no magnetic force (the angle *θ* between **v**_{Y} and **B** is 0, and sin 0 = 0). Since the velocity of particle Z is antiparallel to the direction of **B**, particle Z also feels no magnetic force (the angle *θ* between **v**_{Z} and **B** is 180°, and sin 180° is also 0). However, the velocity of particle X is perpendicular to the direction of **B** (that is, *θ* = 90°), so the magnetic force on particle X is *qv*B sin *θ* = *qv*B sin 90° = *qv*B. Therefore, particle X feels the greatest magnetic force.