Combining Electric and Magnetic Forces

Claimed by Alana Kaplan (Fall, 2015); Claimed to Edit by Alexis Jacob (Spring, 2016) (Added middle and difficult examples, figures, and minor format changes)

Though the pattern in which electric and magnetic forces interact with particles is observably different, their effects can be quantitatively be compared. The principle of adding the two functions of force as a net force is one that now serves as a fundamental principle of electromagnetics. It serves as a building block for many important Laws such as Hall Effect, Motional Emf, Inductance and Magnetic Torque.

An easy way to conceptualize the net force principle, Lorentz Force, is when a charged particle is moving through a space with present electric and magnetic forces; if the forces are not equal but opposite, the particles trajectory will change.

The Main Idea

A Mathematical Model

Electric Forces

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Figure 1. An electric force acts in a pattern parallel to the electric field, pointing radially inward or outward of a particle. The direction depends on the signs of the interacting charged particles.

• A particle being acted upon by an electric force will move in a straight line, in the path, or negative path depending on charge, of the the electric field line (See Figure 1) .

• Electric fields point in a direction radially outward/ inward of a charged particle. There are four possible scenarios for the interaction of 2 charged particles:

1) A (-) charged Particle(1) is acting on a (-) charged particle(2)

• Particle(2) feels force pointing radially outward from Particle(1)

2) A (+) charged Particle(1) is acting on a (-) charged particle(2)

• Particle(2) feels force pointing radially inward toward Particle(1)

3) A (-) charged Particle(1) is acting on a (+) charged particle(2)

• Particle(2) feels force pointing radially inward toward Particle(1)

4) A (+) charged Particle(1) is acting on a (+) charged particle(2)

• Particle(2) feels force pointing radially outward from Particle(1)

The electric force formula is as follows:

Where 1 is the electric field from the source and 2 is the charge of the particle feeling the source

• Note that electric forces can perform work

Magnetic Forces

Figure 2. Magnetic Fields follow a helical pattern

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Figure 3. Magnetic Force Right Hand Rule

• The magnetic force on a charged particle is orthogonal to the magnetic field.

• The particle must be moving with some velocity for a magnetic force to be present.

• Particles move perpendicular to the magnetic field lines in a helical manner (See Figure 2)

• To find the magnetic force, you can use the Right Hand Rule as follows (See Figure 3):

1) Thumb in direction of the velocity

2)Fingers in the direction of the magnetic field

3) Your palm will face in the direction of the Magnetic Force

The magnetic force on an object is:

Note that if the velocity and magnetic field are parallel the magnetic force is zero; this is due to AxB = |A||B|sin(θ). If the angle between the two vectors is 0 or 180 degrees (parallel), then sin(θ) = 0. Therefore, AxB = 0.

Electric and Magnetic Forces Combined

Figure 4. The electric field, magnetic field, and velocity vector are all perpendicular to each other

The net force acting on a particle passing through a magnetic and electric field is:

This net force calculation is known as "Lorentz Force"

When the net force is equal to zero, the velocity stays constant. The net force is equal when:

As seen in Figure 4 , when the net forces acting on a particle are balanced the electric field, magnetic field, and velocity vector are all perpendicular to each other. The electric and magnetic forces are equal but opposite. When forces are not balanced the trajectory of the the particle will change.

The Lorentz Force calculation is now a fundamental principle of electromagnetism.

Examples

Simple

Q: A proton is moving with velocity 7e8 in the +x direction. The trajectory of the proton is constant. There is an electric field in the area of 3.6e7 in the +y direction. Calculate the direction and magnitude of the magnetic field acting on the particle?

A: |qE| = |qvB|

E = vB

B = E/v = 3.6e7/7e8

B = 0.051 T

The magnetic field is in the +z direction.

Middling

Q: At a particular instant, a proton is moving with velocity <0,5e5,0> m/s and an electron is moving with velocity <-4.2e2,0,0> m/s. The electron is located 1.4e-3 m below the proton (in the -y direction).

Determine the net force on the electron due to the proton.

A: Fnet = F(e) + F(B)

@ the electron's location, B = 0 because the velocity of the proton is parallel to r-hat.

Fnet = F(e) + 0 = q*E

E = (1/4πεo)*(q/r^2)*rhat

where rhat = <0,-1,0> and r = 1.4e-3 m

E = <0,-7.35e-4,0> N/C

Fnet = -e*E = <0,1.18e-22,0> N

Difficult

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Figure 5. Middle example diagram.

Q: A copper bar of length L and zero resistance slides at a constant velocity, v. There is a uniform magnetic field, B, directed into the page. A voltmeter is connected across a resistor, R, and reads ΔV. See Figure 5.

Determine the direction of the magnetic force on the diagram and the current through the resistor. Your answer should be in terms of the given variables.

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Figure 6. Middle example solution.

A: Direction of the Magnetic Force- Use right hand rule for v cross B. See Figure 6.

Current through the resistor-

ΔV round trip = 0

ΔV round trip = +motional emf - ΔV resistor = 0

motional emf = IR

I = (motional emf/R)

According to Lorentz force, |F(e)| = |F(B)|

q*E = q*v*B

E = v*B

|ΔV| = E*ΔL = vBΔL

I = (vBΔL)/R

Connectedness

The Lorentz Force principle has been a component in many modern day inventions and critical building block for many physics principles. With known forces, we can predict the very important figure, the velocity and trajectory of a moving particle.

Applications

Velocity Selector

Figure 7. Illustration of a Velocity Selector

The Velocity Selector is a device used to filter particles based on their velocity. A Velocity Selector uses controlled, perpendicular, electric and magnetic fields to filter certain charged particles (See Figure 7 ). Particles with the correct speed will be unaffected while other particles will be deflected. This technique is used in technologies such as electron microscopes and spectrometers.

Electric Motor

An electric motor is a device that uses the Lorentz force to convert electric energy into mechanical energy. Using the magnetic torque principle, electric energy is created by using the magnetic field of a magnet. The torque laws are based off the principles of the net electric and magnetic forces.

Other related topics

Here are other principles that use the net force of magnetic and electric forces as a building block:

Hall Effect

Lorentz Force

Motional Emf

Electric Motors

Motional Emf using Faraday's Law

Inductance

See also

Hall Effect

Lorentz Force

Motional Emf

Electric Motors

Motional Emf using Faraday's Law

Inductance

Further reading

Books, Articles or other print media on this topic

| MIT Physics notes on Lorentz force

External links

Great youtube videos on Lorentz Force Law: |Lorentz Force Law Video 1 | Lorentz Force Law Video 2

References

Boundless. “Electric vs. Magnetic Forces.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 05 Dec. 2015 from https://www.boundless.com/physics/textbooks/boundless-physics-textbook/magnetism-21/motion-of-a-charged-particle-in-a-magnetic-field-158/electric-vs-magnetic-forces-554-11176/

Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Vol. 2. Hoboken, NJ: Wiley, 2015. 812-814. Print.

All images found on google image search: https://en.wikipedia.org/wiki/Magnetic_field https://en.wikipedia.org/wiki/Wien_filter http://aplusphysics.com/wordpress/regents/em/electric-field/

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