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The concept of magnetic moment and torque on current loops in a magnetic field. It covers the principles of net force being zero but net torque existing, the direction of magnetic moment given by the right-hand rule, and the dependence of torque on the angle between magnetic moment and magnetic field. The document also includes examples of calculating torque for a circular loop and the trajectory of a charge in a constant magnetic field.
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PHY2054: Chapter 19
21
a
a
b
b
Î
Consider rectangular current loop
Forces in left, right branches = 0
Forces in top/bottom branches cancel
No net force! (true for any shape)
Î
But there is a net torque!
Bottom side up, top side down (RHR)
Rotates around horizontal axis
Î
μ
= NiA
⇒
“magnetic moment” (N turns)
True for any shape!!
Direction of
μ
given by RHR
Fd
iBa b
iBab
iBA
Plane normal is
B here
PHY2054: Chapter 19
General Treatment of Magnetic Moment, Torque Î
μ
= NiA is magnetic moment (with N turns)
Direction of
μ
given by RHR
Î
Torque depends on angle
θ
between
μ
and B
τ
μ
θ
PHY2054: Chapter 19
x x x x x x x x x x x x x xx x x x x x x x x x x x x xx x x x x x x x x x x x x xx x x x x x x x x x x x x x
Î
A charge q enters B field with velocity v perpendicular to B. What path will q follow?
Force is always
⊥
velocity and
⊥
B
Path will be a circle.
F is the centripetal force needed to keep the
charge in its circular orbit. Let’s calculate radius R
F
F
v
R
v
B
q
F
v
PHY2054: Chapter 19
x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x
q
F
v
2
mv
qvB
R
=
mv
R
qB
=
PHY2054: Chapter 19
27
Î
Velocity of particle has 2 components
(parallel to B and perp. to B)
Only v
⊥
= v sin
φ
contributes to circular motion
v
||
= v cos
φ
is unchanged
Î
So the particle moves in a helical path
v
||
is the constant velocity along the B field
v
⊥
is the velocity around the circle
v
v
v
⊥
&
mv
qB
⊥
B
v
φ
v
⊥
v
||
PHY2054: Chapter 19
PHY2054: Chapter 19
Î
Two particles of the same charge enter a magnetic field with the same speed.
Which one has the bigger mass?
A
B
Both masses are equal
Cannot tell without more info
x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x
mv
qB
Bigger mass meansbigger radius
PHY2054: Chapter 19
PHY2054: Chapter 19
Î
A beam of deuterons travels right at v = 5 x 10
5
m/s
What value of B would make deuterons go undeflected through aregion where E = 100,000 V/m pointing up vertically?
If the electric field is suddenly turned off, what is the radius andfrequency of the circular orbit of the deuterons?
5
5
eE
evB
v
(
)(
)
5
6
2
10 Hz
v
f
−
(
)(
)
(
)(
)
27
5
2
2
19
m
mv
mv
evB
eB
−
−
−
