Lecture 14Lecture 14MagnetismMagnetism

MagnetsMagnets ... two poles: N and S ... two poles: N and S

Like poles repelLike poles repel

Unlike poles attractUnlike poles attract

Magnetic Field lines: Magnetic Field lines:

(defined in same way as electric field lines, direction and density)(defined in same way as electric field lines, direction and density)

NS

Broken Permanent MagnetBroken Permanent Magnet

If we break a permanent magnet in half, we do not get a If we break a permanent magnet in half, we do not get a separate north pole and south pole.separate north pole and south pole.When we break a bar magnet in half, we always get two new When we break a bar magnet in half, we always get two new magnets, each with its own north and south pole.magnets, each with its own north and south pole.

Source of Magnetic FieldsSource of Magnetic Fields

What is the source of magnetic fields?What is the source of magnetic fields?

Answer: electric charge in motionAnswer: electric charge in motion

e.g., current in wire surrounding cylinder (solenoid) produces very similar e.g., current in wire surrounding cylinder (solenoid) produces very similar

field to that of magnets.field to that of magnets. Therefore, understanding source of field generated by bar magnet lies in Therefore, understanding source of field generated by bar magnet lies in

understanding currents at atomic level within bulk matter.understanding currents at atomic level within bulk matter.

Orbits of electrons about nuclei

Intrinsic “spin” of electrons (more important effect)

Magnetic MaterialsMagnetic Materials

• Materials can be classified by how they respond to an applied magnetic field, Bapp.

• Paramagnetic (aluminum, tungsten, oxygen,…)

• Atomic magnetic dipoles (~atomic bar magnets) tend to line up with the field, increasing it. But thermal motion randomizes their directions, so only a small effect persists: Bind ~ Bapp •10-5

• Diamagnetic (gold, copper, water,…)

• The applied field induces an opposing field; again, this is usually very weak; Bind ~ -Bapp •10-5 [Exception: Superconductors exhibit perfect diamagnetism they exclude all magnetic fields]

• Ferromagnetic (iron, cobalt, nickel,…)

• Somewhat like paramagnetic, the dipoles prefer to line up with the applied field. But there is a complicated collective effect due to strong interactions between neighboring dipoles they tend to all line up the same way.

• Very strong enhancement. Bind ~ Bapp •10+5

Magnetic Field DirectionMagnetic Field DirectionA vector quantity: magnitude and direction…A vector quantity: magnitude and direction…The letter The letter BB is used to represent magnetic fields. is used to represent magnetic fields.

Magnetic Field of the EarthMagnetic Field of the Earth

A small magnetic bar should be said to have north and A small magnetic bar should be said to have north and south south seekingseeking poles. The north of the bar points towards poles. The north of the bar points towards the North of the Earth.the North of the Earth.

The geographic north corresponds to a south The geographic north corresponds to a south magnetic pole and the geographic south magnetic pole and the geographic south corresponds to a magnetic northcorresponds to a magnetic north..

The configuration of the Earth magnetic resemble that of The configuration of the Earth magnetic resemble that of a (big) magnetic bar one would put in its center.a (big) magnetic bar one would put in its center.

Magnetic Field of the EarthMagnetic Field of the Earth

Magnetic Fields in analogy with Electric FieldsMagnetic Fields in analogy with Electric Fields

Electric FieldElectric Field:: DistributionDistribution of charge creates an electric field of charge creates an electric field EE in in

the surrounding space.the surrounding space. Field exerts a force Field exerts a force FF=q =q EE on a charge on a charge qq

Magnetic FieldMagnetic Field:: MovingMoving charge or current creates a magnetic field charge or current creates a magnetic field

BB in the surrounding space.in the surrounding space. Field exerts a force Field exerts a force FF on a charge moving on a charge moving qq

The magnetic forceThe magnetic force

Observations show that the force is proportional toObservations show that the force is proportional to The fieldThe field The chargeThe charge The velocity of the particleThe velocity of the particle The sine of the angle between the field and the direction of the The sine of the angle between the field and the direction of the

particle’s motion.particle’s motion.

Strength and direction of the Strength and direction of the Magnetic ForceMagnetic Force on a charge on a charge in motionin motion

+q

F

B

v

[Q] The figure shows a charged particle with velocity v travels through a uniform magnetic field B. What is the direction of the magnetic force F on the particle?

Magnetic Field MagnitudeMagnetic Field Magnitude

[Q] An electron (q = -1.6x10-19 C) is moving at 3 x 105 m/s in the positive x direction. A magnetic field of 0.8T is in the positive z direction. The magnetic force on the electron is:

Magnetic Field UnitsMagnetic Field Units

[F] = Newton[F] = Newton[v] = m/s[v] = m/s[q] = C[q] = C[B] = tesla (T).[B] = tesla (T).

Also called weber (Wb) per square meter.Also called weber (Wb) per square meter. 1 T = 1 Wb/m1 T = 1 Wb/m22.. 1 T = 1 N s m1 T = 1 N s m-1-1 C C-1-1.. 1 T = 1 N A1 T = 1 N A-1-1 m m-1-1..

CGS unit is the Gauss (G)CGS unit is the Gauss (G) 1 T = 101 T = 1044 G. G.

Right Hand RuleRight Hand Rule

Example:Example: Proton traveling in Earth’s magnetic field.Proton traveling in Earth’s magnetic field.

A proton moves with a speed of 1.0 x 10A proton moves with a speed of 1.0 x 1055 m/s through the Earth’s m/s through the Earth’s magnetic field which has a value of 55 magnetic field which has a value of 55 T a particular location. When T a particular location. When the proton moves eastward, the magnetic force acting on it is a the proton moves eastward, the magnetic force acting on it is a maximum, and when it moves northward, no magnetic force acts on it. maximum, and when it moves northward, no magnetic force acts on it. What is the strength of the magnetic force? And what is the direction of What is the strength of the magnetic force? And what is the direction of the magnetic field?the magnetic field?

Northward or southward.

Magnetic Force on Current- carrying conductor.Magnetic Force on Current- carrying conductor.

A magnetic force is exerted on a single charge in motion A magnetic force is exerted on a single charge in motion through a magnetic field.through a magnetic field.

That implies a force should also be exerted on a That implies a force should also be exerted on a collection of charges in motion through a conductor I.e. a collection of charges in motion through a conductor I.e. a current.current.

The force on a current is the sum of all elementary The force on a current is the sum of all elementary forces exerted on all charge carriers in motion.forces exerted on all charge carriers in motion.

Magnetic Force on CurrentMagnetic Force on Current

If B is directed into the page we If B is directed into the page we use blue crosses representing the use blue crosses representing the tail of arrows indicating the tail of arrows indicating the direction of the field,direction of the field,

If B is directed out of the page, we If B is directed out of the page, we use dots.use dots.

If B is in the page, we use lines If B is in the page, we use lines with arrow heads. with arrow heads.

x x x xx x x x x

x x x x x xx x x x xx x x x

. . . .. . . . .

. . . . . .. . . . .. . . .

Force on a wire carrying current in a magnetic field.Force on a wire carrying current in a magnetic field.

x x x xx x x x x

x x x x x xx x x x xx x x x

x x x xx x x x x

x x x x x xx x x x xx x x x

I = 0 I

Bin Binx x x x

x x x x x x x x x x x

x x x x xx x x x

I

Bin

x x x x x x x x x

x x x x x x x x x x x x x xx x x x x x xx x x x x x x

x x x x x

Force on a wire carrying current in a magnetic field.

Avdq

ivd

F

L

. . .

. . .B (Out of the paper)

. . .

Force on a wire carrying current in a magnetic field.Force on a wire carrying current in a magnetic field.

General Case: field at angle General Case: field at angle relative to current. relative to current.

I

B

B sin

ExampleExample: : Wire in Earth’s B FieldWire in Earth’s B Field

A A wire carries a current of 22 A from east to west. Assume that at this wire carries a current of 22 A from east to west. Assume that at this location the magnetic field of the earth is horizontal and directed from location the magnetic field of the earth is horizontal and directed from south to north, and has a magnitude of 0.50 x 10south to north, and has a magnitude of 0.50 x 10-4-4 T. Find the magnetic T. Find the magnetic force on a 36-m length of wire. What happens if the direction of the force on a 36-m length of wire. What happens if the direction of the current is reversed?current is reversed?

Motion of Charged Particle in magnetic fieldMotion of Charged Particle in magnetic field

Consider positively charge Consider positively charge particle moving in a uniform particle moving in a uniform magnetic field.magnetic field.

Suppose the initial velocity of Suppose the initial velocity of the particle is perpendicular to the particle is perpendicular to the direction of the field.the direction of the field.

Then a magnetic force will be Then a magnetic force will be exerted on the particle and exerted on the particle and make follow a circular path.make follow a circular path.

Fv

q

r

Bin

The magnetic force produces a centripetal acceleration.

The particle travels on a circular trajectory with a radius:

What is the period of revolution of the motion?What is the period of revolution of the motion?

The frequency

ExampleExample : Proton moving in uniform magnetic field : Proton moving in uniform magnetic field

A proton is moving in a circular orbit of radius 14 cm in a uniform A proton is moving in a circular orbit of radius 14 cm in a uniform magnetic field of magnitude 0.35 T, directed perpendicular to the magnetic field of magnitude 0.35 T, directed perpendicular to the velocity of the proton. Find the orbital speed of the proton.velocity of the proton. Find the orbital speed of the proton.

Use

ExampleExample: If a proton moves in a circle of radius 21 cm perpendicular to : If a proton moves in a circle of radius 21 cm perpendicular to a B field of 0.4 T, what is the speed of the proton and the frequency of a B field of 0.4 T, what is the speed of the proton and the frequency of motion?motion?

v

r

x x

x xm

qBf

2

kg

TCf

27

19

1067.12

4.0106.1

HzHzf 68 101.61067.128.6

4.06.1

Hzf 6101.6

m

qBrv

kg

mTCv

27

19

1067.1

21.04.0106.1

s

m

s

mv 68 101.81067.1

21.04.06.1

s

mv 6101.8

[Q]: [Q]:

Consider the mass spectrometer. The electric field between the plates of the velocity selector is 950 V/m, and the magnetic fields in both the velocity selector and the deflection chamber have magnitudes of 0.930 T. Calculate the radius of the path in the system for a singly charged ion with mass m=2.18×10-

26 kg.