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# Lorentz Force

## Charged particles in motion create magnetic fields.

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Practice Lorentz Force
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Force on an Electric Current in a Magnetic Field

[Figure1]

Powerful electromagnets are commonly used for industrial lifting.  In this image, the magnet is lifting scrap iron and loading it into a flat car for transporting to a scrap iron recovery plant.  Other uses for lifting magnets include moving cars in a junk yard, lifting huge rolls of steel sheeting, and lifting large steel parts for various machines. Electromagnets are usually used for these jobs because they are magnets only when the electric current is on.  The magnet will hold the iron object when the current is on and release it when the current is off.

### Force on an Electric Current in a Magnetic Field

A moving charged particle creates a magnetic field around it. When a moving charged particle moves through another magnetic field, the magnetic field of the particle and the other magnetic field will interact. The result is a force exerted on the moving charged particle. When a charged particle moves through a magnetic field at right angles to the field, the field exerts a force on the charged particle and changes its direction.

In the case sketched below, an electron is moving downward through a magnetic field. The motion of the electron is perpendicular to the magnetic field. The force exerted on the electron can be calculated by the equation

Credit: CK-12 Foundation - Richard Parsons

$F=Bqv$

where  $B$ is the strength of the field,  $q$ is the charge on the particle and  $v$ is the velocity of the particle.

$B \ \text{is expressed in} \ \frac{Newtons}{Amp \cdot meter}$  while  $q$ is expressed in coulombs and  $v$ is expressed in m/s.

You can see that the product of these three units is Newtons, the appropriate unit for force.

$F=Bqv=\frac{Newtons}{Amp \cdot meter} \cdot coulomb \cdot \frac{meter}{s}$ and since amperes are $\frac{coulomb}{second}$ , amps cancels coulombs/second and meters cancels meters. The only unit remaining is Newtons.

#### Hand Rules

The direction of the force is determined by “hand rules.” When the charge on the moving particle is negative, the left hand rule is used. The fingers of the left hand are pointed in the direction of the magnetic field and the thumb points in the direction of the electron movement. Then the palm of the left hand indicates the direction of the force on the electron. The direction of the magnetic field, the direction of the moving charge, and the direction of the force on the particle are all perpendicular to each other.

Credit: Hand: Image copyright violetkaipa, 2013; modified by CK-12 Foundation - Samantha Bacic
Source: http://www.shutterstock.com

If the charged particle moving through the magnetic field is positively charged, then the right hand rule is used.

Credit: Hand: Image copyright kriangkrai wangjai, 2013; modified by CK-12 Foundation - Samantha Bacic
Source: http://www.shutterstock.com

Example Problem:  An electron traveling at  $3.0 \times 10^6 \ m/s$ passes through a  $0.0400 \ N/amp \cdot m$ uniform magnetic field. The electron is moving at right angles to the magnetic field. What force acts on the electron?

Solution: $F=Bqv=(0.0400 \ N/amp \cdot m)(1.6 \times 10^{-19} \ C)(3.0 \times 10^6 \ m/s)=1.9 \times 10^{-14} \ N$

When the current is traveling through a magnetic field while inside a wire, the magnetic force is still exerted but now it is calculated as the force on the wire rather than on the individual charges in the current.

The equation for the force on the wire is given as $F = BIL$ , where  $B$ is the strength of the magnetic field,  $I$ is the current in amps and  $L$ is the length of the wire in the field and perpendicular to the field.

Example Problem: A wire 0.10 m long carries a current of 5.0 A. The wire is at right angles to a uniform magnetic field. The force the field exerts on the wire is 0.20 N. What is the magnitude of the magnetic field?

Solution: $B=\frac{F}{IL}=\frac{0.20 \ N}{\left(5.0 \ A \right) \left(0.10 \ m\right)}=0.40 \ \frac{N}{A \cdot m}$ ( $N/A \cdot m$ are also known as Tesla )

### Summary

• A moving charged particle creates a magnetic field around it.

• When a moving charged particle moves through another magnetic field, the magnetic field of the particle and the other magnetic field will interact and exert a force exerted on the moving charged particle.

• The force exerted on the electron can be calculated by the equation $F = Bqv$ .

• The direction of the force is determined by “hand rules.”

• When the current is traveling through a magnetic field while inside a wire, the magnetic force is still exerted but now it is calculated as the force on the wire rather than on the individual charges in the current.

• The equation for the force on the wire is given as $F = BIL$ .

### Review Questions

1. Find the force on a 115 m long wire at right angles to a  $5.0 \times 10^{-5} \ N/A \cdot m$ magnetic field if the current through the wire is 400. A.

2. Find the force on an electron passing through a 0.50 T magnetic field if the velocity of the electron is $4.0 \times 10^6 \ m/s$ .

3. A stream of doubly ionized particles  $(\text{charge}= 2+)$ moves at a velocity of  $3.0 \times 10^4 \ m/s$ perpendicularly to a magnetic field of 0.0900 T. What is the magnitude of the force on the particles?

4. A wire 0.50 m long carrying a current of 8.0 A is at right angles to a 1.0 T magnetic field. What force acts on the wire?

5. Suppose a magnetic field exists with the north pole at the top of the computer monitor and the south pole at the bottom of the monitor screen. If a positively charged particle entered the field moving from your face to the other side of the monitor screen, which way would it bend?

1. left
2. right
3. up
4. down
5. none of these
6. Suppose the surface of your dining room table is a magnetic field with the north pole at the north edge and the south pole at the south edge. If an electron passes through this field from ceiling to floor, which way will the path of the electron bend?

1. west
2. east
3. north
4. south
5. toward the ceiling

After practicing the questions above, here is a quiz on the subject matter.

Here is a Real World Application of how the futuristic weapon called the "Rail Gun" works.

### Practice

In this video, a wire is attached to a battery so that a current flows through the wire. The wire and battery combination is placed in a magnetic field. Use this resource to answer the questions that follow:

1. What happens to the wire when the current begins to flow?
2. What difference would it make if the magnetic field were stronger?
3. What difference would it make if the battery were 3.0 V instead of 1.5 V?