Lecture 22 current loops. sources of magnetic field.

1. Lecture 22
Current loops.
Sources of magnetic field.

2. 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?
F
B
I
F
F
F
Magnitude of force is the same
on all four sides: F = ILB
Net force is zero.

3. Current loop in a uniform B field
r
B
(B in any direction, not necessarily in the
plane of the screen)
Ñ
∫ dl
r
F =
r
r
Ñ ×B
∫ Idl
=I
r
(Ñ )
∫ dl
r
×B
r
=0
=0
r
Constant B and I
The net force on a closed loop of current in a uniform
B field is always zero.

4. 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.
Net force = 0
τ
F =0
B
F
Net torque about the center of
the loop:
τ =F
= FL
F
= IL2B
I
F =0
L
L
+F
2
2
F = ILB

5. 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
FL
I
φ
z
FR
φ
Near side has force FN = ILB cosφ
out of the screen.
Far side has force FF = ILB cosφ
into of the screen.
⇒ These forces cancel out and don’t
do torque.
x
FL = FR = ILB
Net torque about the center of the loop:
ˆ
L
r  L
ˆ
τ =  FL sin φ + FR sin φ ÷j = IL2B sin φ j

2
2


6. Magnetic dipole
For a current loop of area A and current I:
r
r
µ = IA
Magnetic dipole moment of a current loop (=
magnetic dipole)
r
A = area vector, with direction given by the right-hand rule
Then, the torque by the uniform magnetic field is:
r
r r
2
ˆ
ˆ
τ = IL B sin φ j = µB sin φ j = µ × B
r r r
τ = µ ×B
B
φ
I
z
φ
x

7. Work by this torque as loop plane moves from φ1 to φ2:
W =
∫
φ2
φ1
φ2
φ2
r r
τ d φ = − ∫ τ d φ = − ∫ µB sin φd φ = + µB ( cos φ2 − cos φ1 )
φ1
φ1
r
τ is clockwise
r
φ is counterclockwise
r
µ
φ
r
τ
r
B
Motion of a magnetic dipole
(current loop) in a uniform B
field given by:
W = −∆U
r r
U = − µB cos φ = − µ ×B
Minimum (stable equilibrium) at φ = 0
r r r
τ = µ ×B
r r
U = − µ ×B
⇒ µ tends to align itself with B
Current loop in
magnetic field

8. 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 µ.
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. µ
B. µ/2
Rule of thumb: if there are N turns, count
area as NA (A = area of one loop)
C. 2µ
I
I
I
µ = I πR 2
equivalent to
2I
2
R 
1
1
µ ′ = 2I π  ÷ = I π R 2 = µ
2
2
2

9. 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 µ is aligned along B or against it.
– Quantum Mechanics: The spin is always ~aligned along B or against it
Aligned: U1 = − µB
Anti-aligned: U2
= µB
∆U ≡ U2 − U1 = 2µB = 2.82 × 10 −26 J
μproton = 1.41×10−26 Am2
B = 1 Tesla (= 104 Gauss) Big field!
In QM, you will learn that photon
energy = frequency • Planck’s constant
h ≡ 6.63 × 10-34 J s
2.82 × 10 −26 J
f =
= 42.5 MHz
6.63 × 10 −34J s

10. 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
nearby.
→ what molecules are
Finally, because f µ ∆U µ B , if we put a strong magnetic field gradient
across the sample, we can look at individual slices, with ~millimeter
spatial resolution.
B
Small B
low freq.
Bigger B
high freq.
Signal at the right frequency only from this slice!

11. Thanks to

12. What produces magnetic fields?
A moving charge experiences a force in a B-field
By symmetry, it is reasonable to think that B fields are also produced by
moving charges
generates B-field and
exerts a force on
moving charge 1
moving charge 2
generates B-field and
exerts a force on

13. Magnetic field by a moving charge:
experimental facts
• When q larger, and when v larger, larger B-field produced
1
• B-field decreases with 1/distance2 from the moving source B µ 2
r
• B-field is NOT directed away or towards moving charge
charge (q > 0) moving into screen
B-field line is circular
around moving charge

14. Magnetic field by a moving charge: equation
r µ vr × r
ˆ
B-field at point P B = 0 q
4π
r2
ˆ
r = unit vector from charge to P
µ0 permeability constant = 4π × 10 −7 T ×m/A
If q > 0 then
r
r
ˆ
B same direction as v × r