Lecture 23Magnetic field and current.

dl

B field by a current

Strategy a) consider small length of wire dl b) calculate B-field from this small segment (dB)c) integrate over full wire, B dB

02

ˆ

4v r

dB dqr

Charge that goes through it in time dt :This charge moves with velocity:

Consider small length of wire dl

dq I dt

dqdl dl dlv I

dt dq dt dq

02

ˆ 4

dl rdB I

r

dqv I dl

Biot-Savart law

02

ˆ 4

dl rI

r

Handy trick for direction

• RH thumb in direction of I• Fingers at P, curl with fingers• Tips of fingers give direction of B

02

ˆ 4

dl rdB I

r

DEMO: Magnetic

field around a wire

Magnetic field by an infinite straight wire

02

ˆ 4

dl rdB I

r

I

P

• 2 2r x y

y

x z

dl

r̂

dB

ˆ ˆˆwhere siny

dl r dx k dxkr

0

3/ 22 24z

ydB I dx

x y

03/ 2

2 24z

dxB I y

x y

2

2

y

0

2

I

y

0 distance f rom wire2

IB R

R

In-class example: Perpendicular wires

Two infinitely long wires form the x and y axis of a 2D coordinate system. They each carry current as shown. What is the magnetic field at a point P (2.0 m, 2.0 m)?

A. 710−7 T into the page

B. 710−7 T out of the page

C. 110−7 T into the page

D. 110−7 T out of the page

E. 0

I1 = 3.0 A

I2 = 4.0 A

P

0

1

3 A into the page

2 2 mB

0 7

total

1 A10 T out of the page

2 2 mB

0

2

4 A out of the page

2 2 mB

Superposition works just like for electric fields:

total 1 2B B B

ACT: Force between two parallel wires

Two very long straight, parallel wires have currents I running through them as shown in the figure. The force on wire 2 due to wire 1 is:

A. Down

B. Up

C. Into the page

D. Out of the page

I1

I2

d

Magnetic field by 1 at location of wire 2 is out of the page.B1

Force on I2 by B1 is up.

F

0 11 out of the page

2

IB

dComplete calculation:

2 1

0 1 2

Force f or section of lenght :

or f orce per unit length: 2

L F I LB

I IFL d

Parallel currents: attraction

Antiparallel currents: repulsion

DEMO: Parallel wires

Magnetic field along the axis of a circular loop

dB

r̂

02

ˆ 4

dl rdB I

r

ˆwith dl r dlx

z

A circular loop of radius R carries a current I . What is B on the z axis?

dl

dl

2 2r R z

03/ 2

2 2 4

RdlI

R z

2 2

sinR Rr R z

20

3/ 22 22z

I RB

R z

20

3/ 2 02 2

4

R

z z

RB dB I dl

R z

0

2 2 sin4z

dldB I

R z

Because of the symmetry, only dBz matters:

3

1 f or B z R

zNormal dipole behavior!

Magnetic field by a dipole

B field lines for a current loop

Somehow, it must be the same thing…

B field lines for a magnet

Interaction between two magnetic dipoles

=

=Parallel currents: attraction

N

S

N

S

N-S: attraction

=Antiparallel currents: repulsion

N

S

S

N

S-S (or N-N): repulsion

N

S=

DEMO: Magnets

Magnetism in materials

Orbits of electrons about nuclei

Intrinsic “spin” of electrons (more important effect)

Currents at atomic level within bulk matter:

In general, these atomic magnetic dipoles point in random direction, so material does not produce a net magnetic field.

Magnetic materials (at a glance)

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 ~ Bapp10−5

• Diamagnetic (gold, copper, water,…)• The applied field induces an opposing field; again, this is usually

very weak; Bind ~ −Bapp10−5 [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 105

Ferromagnets, continued:

• Even in the absence of an applied B, the dipoles tend to strongly align over small patches – “domains”. Applying an external field, the domains align to produce a large net magnetization.

• “Soft” ferromagnets• The domains re-randomize when the field is removed

• “Hard” ferromagnets• The domains persist even when the field is removed• “Permanent” magnets

• Domains may be aligned in a different direction by applying a new field

• Domains may be re-randomized by sudden physical shock

• If the temperature is raised above the “Curie point” (770°C for iron), the domains will also randomize paramagnet

MagneticDomains

Fridge magnets

How does a magnet attract screws, paper clips, refrigerators, etc?

The materials are all “soft” ferromagnets. The external field temporarily aligns the domains so there is a net dipole, which is then attracted to the bar magnet.

- The effect vanishes with no applied B field- It does not matter which pole is used.

End of paper clipS N

Compare to: Electric polarization of materials, Phys 221 lecture 30, balloon that sticks to the wall