# 22.4 Magnetic field strength: force on a moving charge in a magnetic

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## Learning objectives

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

• Describe the effects of magnetic fields on moving charges.
• Use the right-hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of magnetic force on a moving charge.
• Calculate the magnetic force on a moving charge.

The information presented in this section supports the following AP® learning objectives and science practices:

• 2.D.1.1 The student is able to apply mathematical routines to express the force exerted on a moving charged object by a magnetic field. (S.P. 2.2)
• 3.C.3.1 The student is able to use right-hand rules to analyze a situation involving a current-carrying conductor and a moving electrically charged object to determine the direction of the magnetic force exerted on the charged object due to the magnetic field created by the current-carrying conductor. (S.P. 1.4)

What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. Magnetic fields exert forces on moving charges , and so they exert forces on other magnets, all of which have moving charges.

## 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     $F$ on a charge $q$ moving at a speed $v$ in a magnetic field of strength $B$ is given by

$F=\text{qvB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \text{,}$

where $\theta$ is the angle between the directions of $\mathbf{\text{v}}$ and $\mathbf{\text{B}}.$ This force is often called the Lorentz force    . In fact, this is how we define the magnetic field strength $B$ —in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength $B$ 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 $F=\text{qvB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ for $B$ .

$B=\frac{F}{\text{qv}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta }$

Because $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ is unitless, the tesla is

$\text{1 T}=\frac{\text{1 N}}{C\cdot \text{m/s}}=\frac{\text{1 N}}{A\cdot m}$

(note that C/s = A).

Another smaller unit, called the gauss    (G), where $1 G={\text{10}}^{-4}\phantom{\rule{0.25em}{0ex}}T$ , is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about $5×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}T$ , or 0.5 G.

The direction of the magnetic force $\mathbf{\text{F}}$ is perpendicular to the plane formed by $\mathbf{\text{v}}$ and $\mathbf{\text{B}}$ , as determined by the right hand rule 1 (or RHR-1), which is illustrated in [link] . 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 $\mathbf{\text{v}}$ , the fingers in the direction of $\mathbf{\text{B}}$ , and a perpendicular to the palm points in the direction of $\mathbf{\text{F}}$ . 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.

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