1. Magnetism

2. Objectives:
Understand the meaning of magnetic field and how to find its magnitude and direction in simple situations involving straight line conductors (𝐵= 𝜇 0 2𝜋 𝐼 𝑟 ) and solenoids (𝐵= 𝜇 0 𝑁𝐼 𝐿 ), using the appropriate right hand rule.
Find the force on moving charges (𝐹=𝑞𝑣𝐵 sin 𝜃 ) and currents (F=B𝐼L sin θ ) in magnetic fields and appreciate the definition of the ampere as a fundamental SI unit, using the right hand rule for forces where appropriate.

3. Magnetic Field Created by a charge
Electric Fields
Magnetic Fields
Created by a charge
Any charge that moves into this field will experience an electric force.
This concept can be extended to magnetism.
Both magnets and electric currents create magnetic fields around themselves.
When another magnet or electric current (or moving charge in general) enters this field, it experiences a magnetic force.
The magnetic field (like the electric field) is a vector quantity (it has magnitude and direction).
The magnetic field is represented by the symbol B, and has units of teslas (T).
The Tesla is a big unit (the Earth’s magnetic field is about 10-4 T on the surface).

4. Magnetic Field Direction
The magnetic field direction is determined by the effect it has on a compass needle (i.e. a small bar magnet). The magnetic needle aligns itself in the direction of the field. Magnetic field lines exit the north pole and enter the south pole.

5. Sketch the magnetic field lines
N S
N S
N S

6. Magnetic force on a current
If a current is placed in a region of magnetic field, it will experience a magnetic force.
The magnitude of this force is determined by the equation F=kB𝐼L sin 𝜃
k is a constant of proportionality that is often made to be one by saying that when the force on 1m of wire carrying a 1A current equals 1 N, then the magnetic field is defined to be 1 tesla (T).
So, for all intents and purposes, the formula can be simplified to F=B𝐼L, provided that the magnetic field is perpendicular to the current and B is in teslas, I is in amps, and L is in meters.

7. #1 Right hand rule for magnetic force on a current:
With your thumb, index finger, and middle finger at right angles, place your thumb in the direction of the current, your index finger in the direction of the field and your middle finger points in the direction of the force.

8. Practice RHR #1 B I What direction is the force?
What direction is the magnetic field?
What direction is the current? B I

9. Magnetic force on a moving charge
Current experiences a magnetic force when placed in a magnetic field, but current is just moving charges so a moving charge will also experience a magnetic force.
Consider a positive charge q that moves to the right with speed v. In time ∆t the charge moves a distance L so, 𝐿= 𝑣×∆𝑡. The current created by this charge is 𝐼= 𝑞 ∆𝑡 , so the force on this current is F=B𝐼𝐿 sin 𝜃 = 𝐵 𝑞 ∆𝑡 𝑣∆𝑡 sin 𝜃=𝑞𝑣𝐵 sin 𝜃 , where θ is the angle between the direction of the velocity and magnetic field.

10. #2 Right hand rule for the direction of the magnetic force on a moving charge:
With your thumb, index finger, and middle finger at right angles, place your thumb in the direction of the charge’s velocity, your index finger in the direction of the field and your middle finger points in the direction of the force.

11. Motion of charges in a magnetic field
The fact that the magnetic force on a moving charge is always perpendicular (normal) to the velocity, indicates that the charge must travel in a circular motion (circle or helix).
Remember circular motion and centripetal force?

12. Motion of charges in a magnetic field
Therefore, 𝑞𝑣𝐵=𝑚 𝑣 2 𝑅 ∴ 𝑅= 𝑚𝑣 𝑞𝐵 .
Very massive or fast charges will move on large circles. Large charges and large magnetic fields will move in small circles.
The time to make one full revolution in a magnetic field is found from 𝑇= 2𝜋𝑅 𝑣 = 2𝜋 𝑣 𝑚𝑣 𝑞𝐵 = 2𝜋𝑚 𝑞𝐵 . It is important to note that the time required to make a revolution is independent of speed.
This is an important result in experimental particle physics and forms the basis for and accelerator called the cyclotron.

13. Motion of charges in a magnetic field
Positives – clockwise Negatives Counter-clockwise
Angle between v and B (θ) in degrees
Path
0° or 180°
Straight line
90°
Circle
Anything else
Helix

14. Work done & Magnetic forces
The magnetic force is always perpendicular to the direction of motion, therefore, it cannot do any work.
The big magnets in particle accelerators are used only to deflect particles, not to increase their kinetic energy (this job is done by electric fields).
𝑊=𝐹×∆𝑥×𝑐𝑜𝑠𝜃
MAGNETIC FIELDS DO NOT DO WORK ON MOVING CHARGES IN MAGNETIC FIELDS

15. Reminder
Permission slips for Tucson Trip to Titan Missile Silo & Flandrau Science Center (and maybe some Physics labs on the UA campus) are due TOMORROW
10 spots left!

16. Ørsted’s discovery
The current in a straight, long wire produces a magnetic field around it.
The magnitude of the magnetic field B created by the current in a wire varies linearly with the current in the wire and inversely with the perpendicular distance from the wire; 𝐵= 𝜇 0 2𝜋 𝐼 𝑟 .

17. 𝐵= 𝜇 0 2𝜋 𝐼 𝑟
The constant of proportionality involves the new physical constant µ0, which is called the permeability of vacuum (or sometimes ‘free space’).
If the wire is surrounded by something other than a vacuum, you must use the permeability of that medium (water, air, etc.).
The value of µ0 is 4π x 10-7 N/A2. It is the magnetic analog to the electric permittivity, 𝜀.

18. #3 Right hand rule for the direction of the magnetic field due to a current-carrying wire:
Place your thumb in the direction of the current and the direction that your fingers curl is the direction of the magnetic field.

19. Practice RHR #3
Draw the magnetic field lines for the following current- carrying wires. X

20. The single current loop
The magnetic field of a single current loop is shown here. The magnetic field strength, B at the center of a particular loop of radius R carrying current I is 𝐵= 𝜇 0 𝐼 2𝑅 .

21. The solenoid
Solenoids (wire wound tightly many times around an axis) can create a uniform magnetic field (meaning it has the same magnitude and direction in a region of space).
In the interior of a solenoid, the magnetic field is uniform in magnitude and direction and is given by 𝐵= 𝜇 0 𝑁𝐼 𝐿 ; where N is the number of turns of wire, L is the length of the solenoid, and I is the current through it.
The magnetic field is stronger if the solenoid has an iron core. Outside of the solenoid, the magnetic field resembles a bar magnet.

22. #4 Right hand rule for the direction of the magnetic field in a solenoid:
Wrap your fingers in the direction of the current and your thumb points in the direction of the magnetic field.

23. Force between 2 current-carrying wires
Consider now two long, straight, parallel wires each carrying current (I1 and I2).
The first wire creates a magnetic field in space, and in particular at the position of the second wire. Thus, wire 2 will experience a magnetic force. Similarly, wire 2 creates a magnetic field at the position of wire 1, so that wire 1 will also experience a force from wire 2.
These forces must be equal and opposite due to Newton’s 3rd law. If the currents are parallel, the forces are attractive. If the currents are antiparallel (running opposite directions), the forces are repulsive.

24. Practice
Draw the forces for the anti-parallel (left) & parallel (right).

25. Re-definition of the Ampere
The fact that two parallel current carrying wires exert equal and opposite magnetic forces on each other has been used to redefine the ampere.
It is still equal to a coulomb per second, but it is now defined through the magnetic force between two parallel wires.
Definition: If the force on a 1m length of two wires that are 1 m apart and carrying equal currents is 2 x 10-7 N, then the current passing through each wire is defined to be 1 ampere.
The coulomb is defined in terms of the ampere as the amount of charge that flows past a certain point in a wire when a current of 1A flows for 1 second.

26. Finds the direction of the magnetic [force/field] on/for [this] object
Recap
Right hand rule
Finds the direction of the magnetic [force/field] on/for [this] object
What do you do? 1 2 3 4