How Do Magnets Behave?

• A Magnet has two poles: “North” and “South”. Example: The Earth.• For two magnets:

–Unlike poles attract; like poles repel.

• The North Pole of a magnet points North (towards the Arctic) by definition.•Where is Earth’s South Pole?

–In Canada.

•Where is Earth’s North Pole?–In Antarctica.

•Magnetism acts through a Magnetic Field

(a “ B-field ”) (S.I. Unit: “Tesla” (T))

How can we Visualize Magnetic Fields?

• Magnetic Field Lines– Lines start at North Pole and end at South Pole.– Lines must have a start and end; a North Pole cannot

exist without a South Pole.– Direction of a line at a point is in the direction of

the B-field.– Density of lines is proportional to the magnetic field

strength.– Patterns Similar to E-field lines can be found:

• Magnetic Dipole• Repulsive poles• Magnetic Quadrupole.

How is Magnetism caused?• Magnetism is

– caused by moving charges.– felt by moving charges.

• The charges must be moving to cause or be affected by magnetism.

• A stationary charge will cause and feel an E-field.

• A moving charge will cause and feel an E-field and a B-field.

• Electric Currents (moving charges)– Cause B-Fields (Hans Oersted, 1820)

– Feel B-Fields

Question

• But isn’t a charge’s motion relative to an observer?– The magnetic and electric fields observed depend

on your reference frame.

B-Fields caused by Currents• A current causes a B-field that is perpendicular to current

flow.• Point your right thumb in the direction of the current; your

fingers will curl in the direction of the B-field. (“Right Hand Rule #1”).

B-Fields Caused by Currents

How do Magnetic Materials cause B-fields?

• Electrons in atoms have “spin”; an intrinsic fixed quantity of angular momentum.– Spin can be “up” (+) or “down” (-)– Unpaired electrons in atom give the atom a net spin.

• A net spin– acts an effective current associated with the atom.– B-Field due to this current is in same direction as spin.

• If all atoms have their spins in same direction– An effective BOUND surface current exists and ….– the material is magnetic

Magnetic Materials (cont’d)

Nonmagnetic

Material(spins randomly

directed)

Permanent Magnet

(spins aligned)

Ferromagnetism• A ferromagnetic material

has magnetic domains, regions in which spins line up, but domains cancel on average.

• Applying an external B-field causes the domains to line up and the sample becomes a magnet.

(Example: electromagnet)

Magnetic Materials (cont’d)

• Permanent Magnet- Spins aligned on neighboring atoms (“ordered”), appears magnetic.

• Nonmagnetic – Spins randomly distributed; B-fields cancel out on average.

• Antiferromagnet – Spins on neighboring atoms aligned in opposite directions (“ordered”); B-fields cancel locally.

Forces between Current Carrying Wires.

Wires with currents in the same direction attract.

Wires with currents in opposite directions repel.

The opposite B-fields between the wires are as two unlike poles.

The B-fields between the wires are in the same direction and are as two like poles.

Magnitude and Direction of Force on a Current Carrying Wire due to a B-Field

• Magnitude: For a Wire having current I and length (l ) making an angle (θ) with the B-field: FB = I l B sin θ

• Direction: – Right Hand Rule #2– Thumb in direction of Current.– Fingers in direction of B-Field.– Palm points to direction of

Force.

NOTE: FB is PERPENDICULAR TO BOTH I and B.

Examples

1. Verify Right Hand Rule #2 For these wires.

2. Suppose the current in a wire is in the same direction as the B-Field. In what direction is the Force?

Answer: The Force is zero because sin θ is zero.

Magnitude and Direction of Force on a Charged Particle due to a B-Field

• Suppose we follow one charged (+q) particle in a current:– Particle has velocity (v) in the direction of the

current.– In time (Δt) the charge covers a distance: l= vΔt

• Thinking of one charged particle as a small current, the force isFB = IlB sinθ = (q/Δt) (vΔt) B sinθ

• Magnitude: FB = qvBsinθ

• Direction: Use RHR#2 with v instead of I

Examples

1. A wire of length l =.12 m and current I =30A makes an angle of 60o with a B-field having a magnitude of .90 T in the x-direction as shown. What is the Force on the wire?

θ

I

B

Examples (cont’d)2. A square wire loop has a mass m = 0.5kg and a

current I=10.0A. The loop hangs from a spring scale measuring in newtons. If the bottom of the loop is in a uniform B-field of 0.5T coming out of the page, what is the reading on the spring scale?

.

Examples (cont’d)

3. A positron (charge: +e) enters a region of uniform magnetic field of B = .10T directed into the page. The positron’s initial velocity is v =106m/s in the +x direction.

a. What path does the particle follow?

b. What is the rate at which work is done on the particle by the B-field?

c. How would the path change if the particle were an electron?

Northern Lights(Example 5)

Solenoid with Ferromagnetic Core• Suppose we have a solenoid with a current I

through the wire.

• What happens to the B-Field inside if we put a ferromagnetic core in it?

The original B-Field inside causes the Spins in the core to line up.The B-field in the core is reinforced to become MUCH greater.Application: Electromagnets.

Magnetic Flux

• Magnetic Field Strength (B): – S.I. Unit: Tesla (T)– Proportional to density of field lines

• Magnetic Flux (Φm)– Φm = (B┴ × A) = BAcos(θ)– Proportional to number of field lines

through an area.– S.I. Unit: Weber (Wb) (1 Wb = 1T×m2)

A B

Electromagnetic Induction• Faraday’s Law: A voltage (ε) (called “emf”

(“electromotive force”) is induced when a conductor or conducting loop experiences a change in Magnetic Flux.Vinduced = ε = - (Δ Φm )/Δt

Lenz’s Law: The current and emf induced by the changing magnetic flux is in a direction so as to oppose the change in flux.

Applications1. Motional emf : A conducting rod rides on

conducting rails with velocity (v) to the right as shown, in a uniform magnetic field as shown. Given B, L, and v, what is the emf generated?

Motional Emf Applet

Applications (cont’d)

2. Electric Generators: How power companies generate electricity for cities and homes.

• Relative motion of magnet and coil is what produces emf.

• Moving a magnet in and out of a coil produces an alternating current.

• Rotating a loop (or coil) between the poles of a magnet causes AC.

Result: Alternating Current (AC)

STOP HERE

• BEYOND HERE IS AP LEVEL

Examples (cont’d)

4. Mass Spectrometer:

Using Applied Electric and Magnetic Fields to find a particle’s mass (see pg. 642 of text):

Magnetic Field due to a Long Wire(WITHOUT PROOF)

Magnetic Permeability of Free Space:

Force Between Two Wires Carrying Current

d

L

Force on wire 1 by wire 2: F1 = I1 B2 (to the right)Force on wire 2 by wire 1: F2 = I2 B1 (to the left)

(So the wires attract)

Levitating Wire

• Two wires, each with weight per length of 1.00*10-4 N/m are parallel with one a height .10m above the other. The net Force on the upper wire is zero. The currents in the wires have the same magnitude and are opposite in direction. Find the currents.

Magnetic Fields of Loops, Coils, and Solenoids

(WITHOUT PROOF)

• For a wire loop of radius (r) carrying a current (I), the B-field at the center of the loop is:

B = (μ0 I)/2r• For a coil of N-loops of radius (r):

B = (μ0 N I)/2r (WHY?)• For a solenoid of N-loops and length (L):

B = (μ0 N I)/L

or: B = μ0 n I ; n = N / LSo, we can create a uniform B-Field with a solenoid.

Convenient Alternate RHR1 for Loops, Coils, and Solenoids

Curl your fingers in the direction of the current then thumb points in the direction of the B-field.

Electric Generators (cont’d)• Current is induced in lower and upper sides of the wire loop.•Upper side has v coming out of board, resulting in a current to the right.•Lower side has v going into the board, resulting in a current to the left.•Currents add to cause clockwise current.

•

Electric Generators (cont’d)• Angle of Area Vector and B-field fluctuates

between θ = 0° and 180 °

• Current and emf reverses direction with angular frequency (ω): θ = ωt

• ε = - (Δ Φm )/Δt = -BA Δ(cos (ωt))/ Δt = BA ω sin (ωt) (using calculus)

You are not responsible for this formula; just the concept of how AC is produced.

Power Generation

• Fuel (Nuclear, coal, water, wind) is used to create steam

• Steam drives a turbine.

• The turbine rotates a coil of wires between a the poles of a strong magnet.

• Alternating Current is produced

• Faster Rotation causes larger current.

• AC produced is carried to cities and homes.

The Big Picture of Electrical Power Creation

The Big Picture of Electrical Power Delivery

Transformers (Devices with Two Coils): Used to adjust Voltage/Current CharacteristicsPower is conserved: P = VHIGHILOW = VLOWIHIGH

Power is Transmitted more efficiently at: VHIGHILOW

E&M• A changing E-field (or voltage) causes a changing

B-field.– We saw this using an electromagnet.

• A changing B-field causes a changing E-field (or voltage)– We saw this by moving a magnet in a coil

• Maxwell’s Equations: – Four equations summarizing Classical Electrodynamics

• Electromagnetic Waves:– Transfer Electric and Magnetic energy (Radiation)– Have oscillating E-fields and B-fields.

Maxwell’s Equations:E&M Summarized