Lecture 22Current loops.

Sources of magnetic field.

Force on a square loop of current

The square loop below has side length L and carries a current I. The magnetic field B points out of the screen and is uniform. What is the net force on the loop?

B

I

Magnitude of force is the same on all four sides: F = ILB

F

F

F

F

Net force is zero.

Current loop in a uniform B field

(B in any direction, not necessarily in the plane of the screen)

B

The net force on a closed loop of current in a uniform B field is always zero.

0

0dl

F I dl B

I dl B

Constant and B I

Torque on a square loop of current

The square loop below has side length L and carries a current I. The magnetic field B is uniform.

B

I

F = 0

F = 0

FF

Net torque about the center of the loop:

2 2L L

F F

FL

2I L B

F = ILB

Net force = 0

Torque on a square loop of current (2)

Now the loop is in the xy plane and B is parallel to the xz plane.

B

I

Near side has force FN = ILB cos out of the screen.Far side has force FF = ILB cos into of the screen. These forces cancel out and don’t do torque.

FL

z

x

FR

Net torque about the center of the loop:

L Rˆsin sin

2 2L L

F F j

L RF F I LB

2 ˆsinI L B j

Magnetic dipole

Magnetic dipole moment of a current loop (= magnetic dipole)

For a current loop of area A and current I:

I A

area vector, with direction given by the right-hand ruleA

Then, the torque by the uniform magnetic field is:

2 ˆ ˆsin sinI L B j B j B

B

B

Iz

x

Work by this torque as loop plane moves from 1 to 2:

2 2 2

1 1 12 1sin cos cosW d d B d B

is clockwise

is counterclockwise

B

W U

cosU B B

Minimum (stable equilibrium) at = 0

Motion of a magnetic dipole (current loop) in a uniform B field given by: U B

B

tends to align itself with B Current loop in magnetic

field

ACT: Two turns

A cable forms a circular circuit of radius R. When connected to a battery, current flows through it and we can assign it a magnetic moment .

If we use the same cable to make a circular circuit with two turns of radius R/2, and use the same battery, the magnetic moment is:

A.

B. /2

C. 2

I

Iequivalent to

2I

I 2I R

2

21 12

2 2 2R

I I R

Rule of thumb: if there are N turns, count area as NA (A = area of one loop)

MRI (Magnetic Resonance Imaging) and NMR (Nuclear Magnetic Resonance)

A single proton (like the one in every hydrogen atom) has a charge (+|e|) and an intrinsic angular momentum (“spin”). If we (naively) imagine the charge circulating in a loop magnetic dipole moment μ.

In an external B-field:– Classically: there will be torque unless is aligned along B or against

it.– Quantum Mechanics: The spin is always ~aligned along B or against

itAligned:1U B

Anti-aligned:2U B

262 1 2 2.82 10 JU U U B

26

34

2.82 10 J42.5 MHz

6.63 10 J sf

In QM, you will learn that photonenergy = frequency • Planck’s constant

h ≡ 6.63 10-34 J s

μproton = 1.4110−26 Am2

B = 1 Tesla (= 104 Gauss) Big field!

If we “bathe” the protons in radio waves at this frequency, the protons can flip back and forth. If we detect this flipping hydrogen!

The presence of other molecules can partially shield the applied B, thus changing the resonant frequency (“chemical shift”).

Looking at what the resonant frequency is what molecules are nearby.

Finally, because , if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution.

f U B

B

Small Blow freq.

Bigger Bhigh freq.

Signal at the right frequency only from this slice!

Thanks to

What produces magnetic fields?

By symmetry, it is reasonable to think that B fields are also produced by moving charges

A moving charge experiences a force in a B-field

moving charge 1

generates B-field and exerts a force

on

moving charge 2

generates B-field and exerts a force

on

Magnetic field by a moving charge: experimental facts

• When q larger, and when v larger, larger B-field produced• B-field decreases with 1/distance2 from the moving source• B-field is NOT directed away or towards moving charge

2

1B

r

charge (q > 0) moving into screen

B-field line is circular around moving charge

Magnetic field by a moving charge: equation

B-field at point P 02

ˆˆ unit vector f rom charge to P

4v r

B q rr

I f 0 then

ˆB same direction as

q

v r

70 permeability constant 4 10 T m/ A