Magnetic Forces on a Current-Carrying Wire
When a wire carries a current in a magnetic field, there is a force on the wire that is equal to the sum of the magnetic forces on the charged particles whose motion produced the current.
L
The force on this wire in the presence of a magnetic field B can be written as:
F = ILB sin
Magnetic Field Lines
Just as the electric field, electric-field lines can represent E, the magnetic field can be represented by magnetic-field lines.
There are two important differences however:
1. The first has to do with the direction of the force the field exerts on the charge. The electric force on a positive charge is in the direction of the electric field, and thus, the electric field line. The magnetic force is perpendicular to a moving charge so the field lines are not in the direction of the moving charge. 2. The electric field lines began on positive charges and ended on negative charges. However, because there are no monopoles, there are no points where magnetic fields begin or end. Instead they form closed loops.
The motion of charged particles in a Magnetic Field
The motion of a charged particle in an electric field is much different than its motion in a magnetic field.
This has many implications and many applications to areas of medicine, astronomy and nuclear physics. To study the applications of particles in magnetic fields we will try to make an analogy to something you, hopefully, already know.
The Force of Gravity
If two massive particles are orbiting each other, such as the Earth orbiting the Sun,
In what direction is the force pointing? In what direction is the velocity?
F
V
In this scenario, the force and the velocity are perpendicular to each other so the Earth will move in a circle (orbit) around the Sun. Application to Magnetic Fields
In order to express things such as forces and velocities and magnetic fields going into and out of the paper, we will use the following conventions:
Dots represent a direction out of the page.
Crosses represent motion out of the paper.
Using this convention, please tell me the direction of the force if a particle is moving to the right of the page and the magnetic field is pointing into the page:
The force is directed to the “center of the page”!
The circular trajectory and the velocity selector
Recall the equation for the force of a charged particle moving through a magnetic field: F = qvB sin
However, as the particle enters the field, the magnetic force always remains perpendicular to the velocity and is directed toward the center of the circular path.
To find the radius of the path that the particle will trace out, recall from mechanics, the formula for the centripetal force:
F = mv2/r
Because the force and the velocity are always perpendicular, sin= 1 so we can equate the two equations to find the radius of the path:
r = mv/qB
Why is this important?
One of the most exciting, yippee, areas in physics today is the study of elementary particles, the building blocks of all types of matter.
Furthermore, it was found at laboratories such as Fermilab in Batavia Illinois, that the basic protons and neutrons are made up of even more fundamental units of matter called quarks. Even stranger still, Fermilab collides protons into antiprotons releasing a TREMENDOUS amount of energy. This energy is able to create strange massive particles such as muons, pions, etc…
One of the things that the physicists want to find out is the charge on these particles. So after they collide, they are sent into a bubble chamber where a magnetic field is set up. The direction the particle spirals tells the physicist the charge on that particle.
The Mass Spectrometer
After a physicist is able to determine the charge on a particle, the next thing he/she might try doing is to find the mass of the particle.
Physicists use a mass spectrometer to find the relative masses of particles and isotopes of elements.
The mass can be expressed in terms of r, B, and v by recalling the last equation. By setting values for v and B, we can measure r and thus find the mass of the particles. To measure the mass of isotopes of elements, the electrons are stripped off the atoms leaving only the positively charged nucleus. Atoms of higher mass will not follow the same path.
Sources of Magnetic Fields
We already know that the source of a magnetic field is moving charge or current. So then we can ask the questions: What type of magnetic fields do wires of different sizes and shapes produce? What will be the magnitude of the magnetic field?
To answer this question, we must use the other right hand rule: Point your thumb in the direction of the current, then as you curl your fingers, you are tracing out the direction of the magnetic field.
For example, what is the direction of the magnetic field for a straight wire:
The strength of the field is given as:
B = o I/2 r The constant is known as the permeability of free space and has a value of 4 x 10-7 T-m/A
Correction from last time, if two wires are carrying a current in opposite directions, they will repel one another and NOT attract each other.
If you have a loop of wire, what direction is the magnetic field pointing?
The strength of the field is given as:
B = N (o I/2 R)
Where N is the number of “turns” in the loop.
The Solenoid
A solenoid is a long coil of wire in the shape of a helix: The direction of the magnetic field is along the axis of the solenoid and nearly constant all the way through.
The strength of the field inside is given as:
B = o nI
Where n is the number of turns per unit length of the solenoid and I is the current.
Solenoids are also referred to as electromagnets, and they have several advantages over permanent magnets. For one thing, by changing the current in the wire, you can change the magnetic field.
Application: The MRI
MRI stands for: Magnetic Resonance Imaging. With this technique, detailed pictures of the internal structure of the body can be taken without placing the patient at risk by exposing them to X-rays.
A MRI machine is a huge solenoid and the patient is inserted into one end of the machine. Inside a huge magnetic field is produced that make the nuclei of certain atoms to act like tiny radio transmitters and emit radio waves. The hydrogen atom is made to behave this way.
With the hydrogens acting in this way, different parts of the body can be seen by adjusting the “dial” to the right “station”. With that, different types of tissue should look a certain way.
Abnormal tissue, or foreign intruders will look different than the normal healthy tissue and thus abnormalities can be detected.