Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.
Electricity and magnetism are closely related. Electric charges generate electric fields, and moving charges generate magnetic fields. In this lesson, we will learn how to calculate the strength of a magnetic field and use the Lorentz force law, which deals with the two forces on a charge moving through a magnetic field.
Things That Go Great Together
Pen and paper, peanut butter and jelly, lock and key. These are classic pairs. One classic pair of physics-related phenomenon is electricity and magnetism. With the first set of examples I gave, one can exist without the other, and they definitely do not generate each other, but moving electric charges generate magnetic fields, and changing magnetic fields generate electric fields.
Generating a Magnetic Field
The magnetic field generated by a moving electric charge can be calculated using Equation 1, shown here:
μo is a the permeability of free space equal to 4π x 10-7 Tesla-meter-per-amp (Tm/A)
q is the magnitude of the charge in coulombs (C)
v is velocity in meters-per-second (m/s)
r is the distance from the charge in meters (m)
θ is the angle between the direction of the charge's velocity and the distance to the point in question
This equation involves the cross-product, which is a way to multiply vectors that results in a vector that is perpendicular to the velocity of the moving charge and distance from it.
The magnitude of the magnetic field is given by Equation 2, shown here:
Since the magnetic field is a vector, there must be a direction attached to its magnitude. We use the right-hand rule to determine the direction of the magnetic field at the point in question.
The right pointer finger points in the direction of the charged particle's velocity.
The right middle finger points in towards point where magnetic field is to be calculated.
The thumb points in the direction of the magnetic field at the point in question.
A dot and an X represent direction.
A representation of the magnetic field generated from a moving positive charge is given in Diagram 1.
If the charge is negative, the magnetic field points in the opposite direction.
Calculate the magnetic field at point P created by an electron moving at 2 x 104 m/s.
It is always a good idea to draw a diagram of the scenario.
We are now ready to use Equation 2, which was given earlier, to solve the problem:
Using the right-hand rule, we point the finger to the right, the middle finger up, which automatically makes the right thumb point out of the screen. This means the magnetic field at point P points out of the screen.
Two Forces Acting at Once
We've learned from previous lessons that charges exert forces on other charges via the electric fields they generate. A magnetic field also exerts a force on a charge moving through it as long as the charge is not moving parallel to the magnetic field. When these two forces are added together we get the Lorentz force law.
The electric force is given by Equation 3.
F is force in newtons (N)
q is the electric charge in coulombs (C)
E is the electric field in newtons-per-coulomb (N/C)
Magnetic Force on an Electric Charge
The only time an electric charge will not feel a force due to a magnetic field is if it is moving directly parallel to the magnetic field or it isn't moving in the field.
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From this information, we can tell that to calculate the magnetic force on a charged particle will involve the value of the charged particle, its velocity, and the sine function because it is a minimum at 0o, and a maximum at 90o. The force on a moving charge due to a magnetic field is given in Equation 4.
F and q are the same as before
v is the velocity of the electric charge in meters-per-second (m/s)
B is the magnitude of the magnetic field in tesla (T)
Equation 4 is the cross-product of the product of the charge's magnitude and its velocity and the magnitude of the magnetic field. We can rewrite this equation to give us the magnitude of the force as shown in Equation 5.
Since force is a vector, we have to indicate the direction the force acts. Since it involves the cross product, we must use the right-hand rule to determine the force's direction. It is slightly different when determining a force due to a magnetic field.
The Lorentz force law in all of its mathematical glory is given in Equation 6:
A 3.4 x 10-4 T magnetic field points at 000o. A 5.8 x 104 N/C electric field is perpendicular to the magnetic field pointing into the screen. What is the force on a proton moving at 4.4 x 104 m/s in the magnetic field at a 90o above the direction of the field?
Solution: Drawing a sketch of the scenario gives us a visual representation of the problem.
Plugging the given information into the Lorentz equation, Equation 6, we get:
The electric field is forcing the proton into the screen. To get the direction of the magnetic force, we use the right-hand rule. Point your right finger up, the middle finger points to the right, and the thumb points into the screen. This means the magnetic force on the proton is acting into the screen. Into the screen is represented in our equation by k hat.
Moving electric charges generate magnetic fields. The magnitude of the field is given by this equation
and the right-hand rule helps to determine the direction of the field at any point in relation to the moving charge.
The Lorentz force law is the sum of the electric field force on a charge and the magnetic field force on the charge if it is moving at any orientation to a magnetic field not parallel to the field.
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