Right-Hand Rule 1
The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force on a charge moving at a speed in a magnetic field of strength is given by
5.1
where is the angle between the directions of and This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI units, we solve for
5.2
Because is unitless, the tesla is
5.3
Note that C/s = A.
Another smaller unit, called the gauss (G), where is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. Earth’s magnetic field on its surface is only about or 0.5 G.
The direction of the magnetic force is perpendicular to the plane formed by and as determined by the right-hand rule 1 (or RHR-1), which is illustrated in Figure 5.9. RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of the fingers in the direction of and a perpendicular to the palm points in the direction of One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.
Making Connections: Charges and Magnets
There is no magnetic force on static charges. However, there is a magnetic force on moving charges. When charges are stationary, their electric fields do not affect magnets. But, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic fields emerges—each affects the other.
Example 5.1 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 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 Earth’s magnetic field, if you throw it with a horizontal velocity of 10 m/s due west in a place where Earth’s field is due north parallel to the ground. (The direction of the force is determined with right-hand rule 1 (RHR-1) as shown in Figure 5.10.)
Strategy
We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation to find the force.
Solution
The magnetic force is
5.4
We see that since the angle between the velocity and the direction of the field is Entering the other given quantities yields
5.5
Discussion
This force is completely negligible on any macroscopic object, consistent with experience. It is calculated to only one digit, because Earth’s field varies with location and is given to only one digit. 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.