Example 22.1 from College Physics, 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

Calculating Magnetic Force: Earth’s Magnetic Field on a Charged Glass Rod

With the exception of compasses, you seldom see or personally experience forces due to the Earth’s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth’s magnetic field, if you throw it with a horizontal velocity of 10 m/s due west in a place where the Earth’s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in Figure 22.18.)

The effects of the Earth’s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.

Figure 22.18 A positively charged object moving due west in a region where the Earth’s magnetic field is due north experiences a force that is straight down as shown. A negative charge moving in the same direction would feel a force straight up.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation $F = qvB sin θ$ to find the force.
Solution
The magnetic force is

$F = qvB sin θ .$
22.4

We see that $sin θ = 1$ , since the angle between the velocity and the direction of the field is $90º$ . Entering the other given quantities yields
$\begin{array}{l} F & = & (20 \times 10^{–9} C) (10 m/s) (5 \times 10^{–5} T) \\ = & 1 \times 10^{–11} (C \cdot m/s) (\frac{N}{C \cdot m/s}) = 1 \times 10^{–11} N. \end{array}$
22.5
Discussion
This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth’s field varies with location and is given to only one digit.) The Earth’s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in Force on a Moving Charge in a Magnetic Field: Examples and Applications.

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