Did you know?
During dark nights at high latitudes you may be able to see an aurora borealis
(northern light) a ghostly “curtain” of light that hangs down from the sky.
The aurora may be several hundred km high, several thousand km long, stretching around
Earth in an arc.
However, it is less than 1km thick!
What causes Aurora Borealis, and what makes it so thin?
Answer by the end of this class
29.3 Torque on a Current Loop in a Uniform
Magnetic Field
29.4 Motion of a Charged Particle in a
Uniform Magnetic Field
Suggested HW problems: Chp.29: 23-25,30,31,36
1

Torque on a Current Loop in B = const
• There is no net force on a current loop in a uniform magnetic field
F =
I
( )
× =
• There is a finite torque on a current loop in a uniform magnetic field
τ =
!
× F µ
µ =
I
A
B current loop area current intensity magnetic moment
Torque on a Current Loop in B = const
F
1
=
F
2
=
IlB
τ =
=
F b
1
( / 2)
+
F b
2
( / 2)
( )
=
IAB
= µ
B
τ
F
1
=
F
2
=
IlB
= θ +
F b
=
B sin
θ
θ
2

Potential energy of a magnetic dipole in magnetic field
θ dU
= d
θ τ µ
B sin
θ θ
U
=
dU
= µ
B
U
= − µ
B cos
θ
sin
Force exerted by a non-uniform magnetic field on a magnetic dipole:
F
U
(
µ B
) ( )
B
How does the galvanometer work?
What is the orientation of the current in the coil ???
3

Motion of a charged particle in a uniform magnetic field uniform electric field
F
q
E trajectory is an arc of parabola if v
0
" E uniform magnetic field
F
q v B trajectory is a(n arc of) circle if v
0
⊥ B otherwise is a helix
Cyclotron radius and frequency
A magnetic field cannot change the kinetic energy of a charged particle!
dE c F v q
( ) dt
⇒
E c
= mv
2
2
0
= con st
⇒ v
= c on st
R assume:
F
= qvB
= m v 2
R
⇒
R
= mv qB
⇒
T
1
ν
2
π v
R
=
2
π m qB
4

How does a mass spectrograph work?
R
= mv qB
0 qvB
= qE
⇒ v
=
E qB m q
=
RBB
0
E
Helical trajectory of a charged particle in a uniform magnetic field assume: v
0
⊥ B =
B ˆ in the y-z plane the motion is uniform circular with
R
= mv
⊥ qB
, T
=
2
π qB m along the x-direction the motion is uniform
( )
= x (0)
+ the pitch of the helix: p
= v T
5

Motion in a non-uniform magnetic field the spiraling motion of a charged particle in a nonuniform magnetic field may end in a “magnetic bottle”, a region with sufficiently strong B
Electrons and protons are trapped in magnetic bottles of the terrestrian magnetic field, located at the two poles. The trapped particles form the
Van Allen radiation belts which loop above Earth’s atmosphere between the north and south geomagnetic poles. The particles bounce back and forth, between the poles, within a few seconds!
Aurora Borealis (Northen Lights)
When a large solar flare shoots additional high energy electrons & protons into the radiation belts, an electric field is produced at the magnetic bottles opening them up. The electrons are driven down into the atmosphere where intense collisions with atoms causes the latter to emit light!
6