Lecture 20Magnetic fields, field lines,

moving charges.

Quick reminder: E field

Phys 221: Electric force between two charges 1 22

0

1 ˆ4

qqF r

r

Huge question: How is this force communicated over a distance?How does charge 2 know about charge 1?

Partial answer: Charge 1 sets up an electric field

I f a second charge is placed in electric fi eld

then it experiences a f orce

F qE

Remarkable conceptual leap…Electric field extends over ALL spaceIf charge 1 moves, field changes

Similarly a magnet sets up a magnetic fieldBIn a few weeks time we will show that it is the moving charges (electrons in atomic orbitals) that create the B-field

For now we will assume we can somehow create a B-field and look at the force it exerts on charged particles

B-field • extends to all points in space• at every point, there is a vector B that has a particular magnitude and direction• magnetic field lines: like E-field lines:

• B-field is tangent to them at all points• magnitude of B-field is indicated by the density of lines

Magnetic field B

Magnetic dipole field

B-field leaves N-pole and go to S-pole

B-field is continuous, no monopoles (“magnetic charge”) – There is nothing special about a “pole”: it’s just where the material

ends

Build from the basics

• For now, build from basics– Magnetic force experienced by a single moving charge in a

pre-existing B-field

• Next week– How to create a B-field from a moving charge

• Few weeks time– Interaction between two current loops

• Reason why iron is attracted to magnets

ACT: Direction of magnetic force

The B field is directed toward the board. What is the direction of the force when the electrons are just coming out of the emitter (gray part)?

A. Toward you

B. To the left

C. To the right

DEMO: Electron beam in

Helmholtz coils

Helmholtz coil

electron beam

Weird

The magnetic force on a particle is perpendicular to B-field

Magnetic force is also perpendicular to v

electricF qE

Very different to E-field, whereF is parallel to E

e-

BF

v

Also, magnetic force is observed to be• proportional to the charge • flip direction if charge of opposite sign is used

The motion is circular: F changed direction as v changed direction

e-

B

F

v

Magnetic force on a moving charge

F qv B

Direction: With your right hand1. place tails of v and B together2. point fingers in direction of v3. curl fingers towards B4. thumb gives vB

sin angle between F F qvB v, B

Units of B field: SI Tesla (T) 1 T = 1 N/(A·m)

Gauss (G) 1 G = 10-4 T

DEMO: Oscilloscope and

magnets

ACT: Magnetic force

What is the direction of the magnetic field if this is the force it produces on the moving negative charge?

A. +xB. −xC. Other

F

B

Noteworthy

• No force on a stationary charge

• No force on a charge that moves parallel to the B field

• This force is always perpendicular to v

magnetic field never changes the speed (magnitude) of a charge!

magnetic field never changes the kinetic energy of a particle

magnetic field does no work!

Hmm. Then what if I look at this from a different reference frame?

F qv B

Lorentz force

F q E v B

A charge feels both electric and magnetic forces.

Lorentz force

In-class example: Lorentz force

− − − − −

+ + + + +

E0 B0

y

x z

v

A positive charge q with velocity v as shown enters a region with electric and magnetic fields as in the figure (magnitudes are shown). What is the net force on the charge? (Ignore gravity)

A. 0

B. −vB i + qE0 j

C. −vB k + qE0 j

D. vB i + qE0 j

E. qE0 j

B and v are antiparallel: vxB = 0

Circular motion in a uniform B field

Back to the Helmholtz demo:• uniform B• v is initially perpendicular to B

Uniform circular motion

F qvB

centripetalF ma2v

mR

2vqvB m

R

mvR

qB

Helix

Now the initial velocity of the charges is NOT perfectly perpendicular to B

F

Magnetic flux

B da

In general (for any field and surface):

How many magnetic field lines go through a given surface?

eff ectiveBA

Line density is proportional to the magnitude of the magnetic field:

For uniform B and flat surfaces only

cosBA B A

2Units : Weber 1 Wb 1 Tm

Convention: For closed surfaces, vector A points out.

Gauss’s law for magnetic fields

B-field lines are always continuous: no beginning or end.

For any closed surface,

# of lines entering the surface = # of lines leaving the surface

closed surf ace 0B da

For E-fields: enclosed

closed surf ace0

q

Electric charge (“electric monopoles”)