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Sunday, March 22, 2009

THE MAGNETIC FIELD

1. The magnetic field exerts a force upon any charge that moves
in the field. The greater the size of the charge, and the faster
it moves, the larger the force. The direction of the force is
perpendicular to both the direction of motion and the magnetic
field. If is the angle between v and , then v sin is the
magnitude of the force. This vanishes when v and B are
para
θ B F=qB θ
􀁇 􀁇
􀁇 􀁇
llel (θ = 0), and is maximum when they are perpendicular.
2. The unit of magnetic field that is used most commonly is the A charge of one
coulomb moving at 1 metre per second perpendicularly to a field of one tesla experiences
a force of 1 newt
tesla.
4
on. Equivalently,
1 tesla 1 newton 1 newton 10 gauss (CGS unit)
coulomb meter/second ampere meter
In order to have an appreciation for how much a tesla is, here are some typical value
= = =
⋅ ⋅
-4
s of
the magnetic field in these units:
Earth's surface 10 T
Bar magnet 10-2 T
Powerful electromagnet 1 T
Superconducting magnet 5 T
3. When both magnetic and electric fields are present at a point, the total force acting upon
a charge is the vector sum of the electric and magnetic forces, v . This is
F=qE+q ×B
􀁇 􀁇 􀁇 􀁇
known as the . Note that the electric force and magnetic force are very
different. The electric force is non-zero even if the charge is stationary, and it is in the
same directi
Lorentz Force
on as .
4. The Lorentz Force can be used to select charged particles of whichever velocity we want.
In the diagram below, particles enter from the left with velocity v. They experience a force
E 􀁇
due to the perpendicular magnetic field, as well as force downwards because of an electric
field. Only particles with speed v =E/B are undeflected and keep going straight
5. A magnetic field can be strong enough to lift an elephant, but it can never increase or
decrease the energy of a particle. Proof: suppose the magnetic force moves a particle
through a di
F
􀁇
splacement . Then the small amount of work done is,
(v B)
(v B)
dr
dW F dr q dr
q drdt
dt
= ⋅ = × ⋅
= × ⋅
􀁇
􀁇 􀁇 􀁇 􀁇 􀁇
􀁇 􀁇 􀁇
(v B) v 0.
Basically the force and direction of force are orthogonal, and hence there can be no work
done on the particle or an increase in its energy
=q × ⋅ dt =
􀁇 􀁇 􀁇
.
6. A magnetic field bends a charged particle into a circular orbit because the particle feels
a force that is directed perpendicular to the magnetic field. As we saw above, the particle
cannot change its speed, but it certainly does change direction! So it keeps bending and
bending until it makes a full circle. The radius of orbit can be easily calculated: the magnetic
and centrifugal
forces must balance each other for equilibrium. So, qv v2 and we
find that v . A strong B forces the particle into a tighter orbit, as you can see. We
can also calculate the angular fre
B m
r
r m
qB
=
=
quency, v . This shows that a strong B makes
the particle go around many times in unit time. There are a very large number of applications
of these facts.
qB
r m
ω = =
7. The fact that a magnetic field bends charged particles is responsible for shielding the
earth from harmful effects of the "solar wind". A large number of charged particles are
released from the sun and reach the earth. These can destroy life. Fortunately the earth's
magnetic field deflects these particles, which are then trapped in the "Van Allen" belt
around the earth.
F
􀁇
F
􀁇
F
􀁇
v 􀁇
v 􀁇
v 􀁇





















qvB mv2 r mv.
r qB
= ⇒ =
r mv
qB
=
+
+
+ v























• +
μ 􀁇
I
3 i
2 i
1 i
θ
ds􀁇 B
􀁇
0 enclosed ∫B⋅ds =μ I
􀁇 􀁇


• Amperian loop
8. The mass spectrometer is an extremely important equipment
that works on the above principle. Ions are made from atoms
by stripping away one electron. Then they pass through a
velocity selector so that they all have the same speed. In a
beam of many different ions, the heavier ones bend less, and
lighter ones more, when they are passed through a B field.
9. A wire carries current, and current is flowing charges. Since each charge experiences a
force when placed in a magnetic field, you might expect the same for the current. Indeed,
that is exactly the case, and we can easily calculate the force on a wire from the force on
individual charges. Suppose N is the total number of charges and they are moving at the
average (or drift) velocit d
d
y v . Then the total force is . Now suppose that the
wire has length , crossectional area , and it has charges per unit volume. Then clearly
, and so v . Remembe
d F Nev B
L A n
N nAL F nALe B
= ×
= = ×
􀁇 􀁇 􀁇 􀁇
􀁇 􀁇 􀁇
d
r that the current is the charge that flows
through the wire per unit time, and so v . We get the important result that the
force per unit length on the wire is .
nAe I
F I B
=
= ×
􀁇 􀁇
􀁇 􀁇 􀁇
10. A current that goes around a loop (any shape) produces a
magnetic field. We define the as the
product of current and area, ˆ. Here A is the area of
the loop and I th
magnetic moment
μ􀁇 = IAz
e current flowing around it. The direction
is perpendicular to the plane of the loop, as shown.
11. Magnetic fields are produced by currents. Every small bit of current produces a small
amount of the field. Ampere's Law, illustrated below, says that if one goes around a
loop (of any sh
B
1 2 3
ape or size) then the integral of the B field around the loop is equal to the
enclosed current. In the loop below I=I +I . Here I is excluded as it lies outside.
R
r
ds􀁇 B
􀁇
B
B
r=R
inside outside
( )
2
0
0 2 2 2
2
B r I r B Ir
R R
π μ
π μ
π π
⎛ ⎞
= ⎜ ⎟⇒ =
⎝ ⎠
12. Let us apply Ampere's Law to a circular loop of radius outside an infinitely long wire
carrying current through it. The magnetic field goes around in circles, and so and
are bot
r
I B ds
􀁇 􀁇
( ) 0
0
h in the same direction. Hence, 2 . We get the
important result that .
2
13. Assuming that the current flows uniformly over the crossection, we can use Ampere's
Law
B ds B ds B r I
B I
r
π μ
μ
π
⋅ = = =
=
∫ ∫
􀁇 􀁇
to calculate the magnetic field at distance r, where r now lies inside the wire.
Here is a sketch of the B field inside and outside the wire as a function of distance r.
r
N S
θ
B 􀁇
A

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