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2/20/12 1
Bar Magnets
N
S
N
S
Attraction
S
N
N
S
Repulsion
- Bar magnet ... two poles: N and S
Like poles repel; Unlike poles
attract.
- Magnetic Field lines: (defined in
same way as electric field lines,
direction and density)
A m
N
1
C/s m
N
1
C m/ s
N
1 T 1
⋅
=
⋅
=
⋅
=
A common unit gauss (G): 1 G = 10
-
T
~Earth’s surface
field!
The SI unit for magnetic field is Tesla (T):
2/20/12 2
Lorentz
F = q
E + q
v ×
B
F
B
= q
v ×
B
F = q vB sin φ
2/20/12 3
Right Hand Rule
Direction of F B
is perpendicular to plane containing v & B.
If q is positive, F B
has the same sign as v x B.
If q is negative, F B
has the opposite sign of v x B.
F B
is never parallel to v.
F B
can only change the direction of the particle not the speed.
2/20/12 4
v
v
v
x x x x x x
x x x x x x
x x x x x x
v
F
B
q
x x x x x x
x x x x x x
x x x x x x
F
v
x x x x x x
x x x x x x
x x x x x x
B
+q
B
+q
B
+q
The direction of the force is:
The Magnetic Force
F = q
v ×
B
2/20/12 5
Magnetic Force on a Current-Carrying Wire
2/20/12 6
I
motion by reversing
Direction of I, by reversing V
B
I
F
I
V
Note that this example assumes that the magnetic field
caused by the currents in the rails is negligible compared to
the external magnetic field B shown.
The length L is the distance between the rails, where B is and
where the current I flows in the green bar.
THIS IS A FORM
OF ELECTRIC
MOTOR, TURNING
ELECTRICAL INTO
MECHANICAL
ENERGY
Top view of Current-Carrying Bar Sliding on two
current carrying frictionless rails in a magnetic
field.
DEMO
2/20/12 7
glow of ionized gas
⇒
B cannot change the kinetic energy of a charged particle.
B can only change the direction of a particle.
Motion of a Point Charge in a Magnetic Field
B cannot change |v| of a charged particle.
2/20/12 8
x x x x x x
x x x x x x
x x x x x x
q
- Suppose charge q enters B field with velocity v as
shown below. What will be the path q follows?
v
B
x x x x x x
x x x x x x
x x x x x x
F F
R
v
Trajectory in Constant Magnetic Field
2/20/12 13
Workingwithbothequations :
Firstsolvefor thevelocityon thefirst one,
Thensubstituteiton thekineticenergy equation
A mass spectrometer can be improved if instead of having
ions with the same kinetic energy entering the B field we
have ions with the same velocity.
Mass Spectrometer (Ions with same KE)
2/20/12 14
- If we shoot charged particles into a region of space
that has both an electric and a magnetic field, we
would end up with a net electro-magnetic force that is
equal to the vector sum of the electric and magnetic
forces acting on the charge:
F F F qE qv B
E B
= + = + ×
- A very interesting effect can be achieved when we apply
an electric and a magnetic force to a charged particle in
such a way that these forces balance.
E B
F F F
Combine an Electric Field and a Magnetic Field
2/20/12 15
CONCLUSION:
There is only one
particular velocity of a
will balance the
magnetic and electric
forces
This device is called a Velocity Selector.
Crossed E and B Fields
2/20/12 16
- We first have to define an unambiguous
direction of the loop, perpendicular to the
plane of the loop.
- We do this with our right hand (again)
n
your right hand in the
direction of the
current, then your
thumb should point in
the direction of
n
Torque on a Current Loop
2/20/12 17
n ˆ wants to align with
B : τ = NIabB sin θ = NIAB sin θ
whereA=ab and the formula does NOT depend on
the shape of the loop, only on the area A
N counts the number of turns of wire in this loop, each turn contributes.
Rectangular Current Loop in a B Field
2/20/12 18
Flat current loop
of arbitrary shape
area of loop
number of turns in loop
an fieldwhere.
rememberanelectricdipole in
E p E
B
= ×
= ×
τ
τ μ
Torque on a Magnetic Dipole
2/20/12 19
τ = μ× B ⇒τ= μ B sin θ
When = 0
o
or 180
o
However, = 180
o
is unstable.
U U B B
U dU B B U
dU dW B d
dW d B d
∫ ∫
Choose ( 90 ) 0 cos
sin cos
sin
sin
0
0
0
When a torque is exerted through an angle, work is
done. When a dipole is rotated through an angle d "
Magnetic Dipole in a Uniform B Field
2/20/12 20
U B B
U B
μ μ
μ
cos 0
U B
U B
o
μ
μ
= − cos 180
Potential Energy of Magnetic Dipole
