# Learning Objectives

### Learning Objectives

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

• 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

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

5.1 $F=qvBsinθ,F=qvBsinθ, size 12{F= ital "qvB""sin"θ} {}$

where $θθ size 12{θ} {}$ is the angle between the directions of $vv$ 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 $BB size 12{B} {}$ 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 $F=qvBsinθF=qvBsinθ size 12{F= ital "qvB""sin"θ} {}$ for $B.B.size 12{B} {}$

5.2 $B=FqvsinθB=Fqvsinθ size 12{B= { {F} over { ital "qv""sin"θ} } } {}$

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

5.3 $1 T = 1 N C ⋅ m/s = 1 N A ⋅ m . 1 T = 1 N C ⋅ m/s = 1 N A ⋅ m . size 12{"1 T"= { {"1 N"} over {C cdot "m/s"} } = { {1" N"} over {A cdot m} } } {}$

Note that C/s = A.

Another smaller unit, called the gauss (G), where $1 G=10−4T,1 G=10−4T,size 12{1G="10" rSup { size 8{ - 4} } T} {}$ is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. Earth’s magnetic field on its surface is only about $5×10−5T,5×10−5T,size 12{5 times "10" rSup { size 8{ - 5} } T} {}$ or 0.5 G.

The direction of the magnetic force $FF size 12{F} {}$ is perpendicular to the plane formed by $vv size 12{v} {}$ and $B,B,$ 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 $v,v,$ 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.

Figure 5.9 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 $vv$ and $BB size 12{B} {}$ 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 $vv size 12{v} {}$ 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 5.1Calculating 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.)

Figure 5.10 A positively charged object moving due west in a region where 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 experience 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 $F=qvBsinθF=qvBsinθ size 12{F= ital "qvB""sin"θ} {}$ to find the force.

Solution

The magnetic force is

5.4 $F=qvbsinθ.F=qvbsinθ. size 12{F= ital "qvb""sin"θ} {}$

We see that $sinθ=1,sinθ=1,size 12{"sin"θ=1} {}$ 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

5.5 F = 20 × 10 –9 C 10 m/s 5 × 10 –5 T = 1 × 10 –11 C ⋅ m/s N C ⋅ m/s = 1 × 10 –11 N. F = 20 × 10 –9 C 10 m/s 5 × 10 –5 T = 1 × 10 –11 C ⋅ m/s N C ⋅ m/s = 1 × 10 –11 N. 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 "." {} } } {}

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.