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Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

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 $F F size 12{F} {}$ on a charge $q q size 12{q} {}$ moving at a speed $v v size 12{v} {}$ in a magnetic field of strength $B B size 12{B} {}$ is given by

F = qvB sin θ, F = qvB, size 12{F= ital "qvB""sin"θ} {}

22.1

where $θ θ size 12{θ} {}$ is the angle between the directions of $v v$ and $B . B . size 12{B} {}$ This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength $B B size 12{B} {}$ —in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength $B B size 12{B} {}$ 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 $F = qvB sin θ F = qvB size 12{F= ital "qvB""sin"θ} {}$ for $B B size 12{B} {}$ .

B = \frac{F}{qv sin θ} B = \frac{F}{qv} size 12{B= {  {F}  over  { ital "qv""sin"θ} } } {}

22.2

Because $sin θ sin size 12{θ} {}$ is unitless, the tesla is

1 T = \frac{1 N}{C \cdot m/s} = \frac{1 N}{A \cdot m} 1 T = \frac{1 N}{C \cdot m/s} = \frac{1 N}{A \cdot m} size 12{"1 T"= {  {"1 N"}  over  {C cdot "m/s"} } = {  {1" N"}  over  {A cdot m} } } {}

22.3

(note that C/s = A).

Another smaller unit, called the gauss (G), where $1 G = 10^{- 4} T 1 G = 10^{- 4} size 12{1`G="10" rSup { size 8{ - 4} } `T} {}$ , 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 $5 \times 10^{- 5} T 5 \times 10^{- 5} size 12{5 times "10" rSup { size 8{ - 5} } `T} {}$ , or 0.5 G.

The direction of the magnetic force $F F size 12{F} {}$ is perpendicular to the plane formed by $v v size 12{v} {}$ and $B B$ , as determined by the right hand rule 1 (or RHR-1), which is illustrated in Figure 22.17. 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 $v v$ , the fingers in the direction of $B B$ , and a perpendicular to the palm points in the direction of $F F$ . 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.

The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.

Figure 22.17 Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by

v v

and

B B size 12{B} {}

and follows right hand rule–1 (RHR-1) as shown. The magnitude of the force is proportional to

q q size 12{q} {}

v v size 12{v} {}

B B size 12{B} {}

, and the sine of the angle between

v v size 12{v} {}

and

B B size 12{B} {}

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 22.1

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 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.

Strategy

We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation $F = qvB sin θ F = qvB size 12{F= ital "qvB""sin"θ} {}$ to find the force.

Solution

The magnetic force is

F = qvB sin θ . F = qvB . size 12{F= ital "qvb""sin"θ} {}

22.4

We see that $sin θ = 1 sin size 12{"sin"θ=1} {}$ , since the angle between the velocity and the direction of the field is $90º 90º size 12{"90" rSup { size 8{ circ } } } {}$ . 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} \begin{array}{l} F & = & (20 \times 10^{–9}) (10 m/s) (5 \times 10^{–5}) \\ = 1 \times 10^{–11} \end{array} alignl { stack {
 size 12{F= left ("20" times "10" rSup { size 8{ - 9 } } `C right ) left ("10"`"m/s" right ) left (5 times "10" rSup { size 8{ - 5} } `T right )}  {} #
"    "=1 times "10" rSup { size 8{ - "11"} } ` left (C cdot "m/s" right ) left ( {  {N}  over  {C cdot "m/s"} }  right )=1 times "10" rSup { size 8{ - "11"} } `N "."  {}
} } {}

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.