Download Magnetic Force

Magnetic Force
Read: Chapter 21
Force on Moving Charge
~
(−e)~v × B
The magnetic force on a moving charge is given by
~.
F~B = q~v × B
~v
−e
~
~v × B
~
B
The direction of F~ can be determined using the right hand rule.
Magnetic forces are typically very small compared to electric force, and are proportional to q/m.
~ is always perpendicular to displacement, so magnetic forces do no work.
Note that F~B = q~v × B
1
Magnetic Force
Force on Moving Charge
Moving charged particles in uniform magnetic fields exhibit uniform circular motion. Non-relativistically, their angular speed
ωB (also called the cyclotron frequency) is given by
ωB =
~
|q||B|
.
m
Question: Why does a uniform magnetic field cause uniform circular motion for a moving charged particle?
Exercise: Use the principles of uniform circular motion to find expressions for both the period T and the momentum |~
p| of
a moving chaged particle in a uniform magnetic field.
2
Magnetic Force
Force on Current-Carrying Wire
Using the principle of superposition, we can show that the magnetic force on a short length ∆l of current-carrying wire is
given by
~.
∆F~B = I∆~l × B
Exercise: Determine the direction and magnitude of the magnetic force on the wire.
I
h
L
What happens to the force if the battery terminals are reversed?
3
Magnetic Force
Parallel Wires
The magnetic field due to wire 1 at the location of wire 2 is
L
F~21
d
~ 1 = µ0 2I1
B
4π d
I1
I2
and into the page by the RHR.
~
B
Exercise: Determine F~21 , the force on wire 2 due to wire 1.
~ 2 at the location of wire 1.
Exercise: Now determine F~12 . You will need B
Question: What happens if the direction of one of the currents is reversed?
4
Magnetic Force
~ and B
~ combined
E
A moving charged particle can experience electric and magnetic forces simultaneously. Combining Fe and Fb gives us the
Lorentz Force Law:
~ + q~v × B
~.
F~ = q E
Exercise: Determine the direction of the Lorentz force for each situation below.
y
y
~
B
y
~
B
+
~
B
+
~v
~
E
~
B
+
~
E
x
z
y
~v
+
~
E
~v
x
~
E
x
z
z
x
z
Exercise: Repeat the exercise, but with a negative instead of a positive charge.
y
y
~
B
~
B
−
~v
~
B
−
~
E
~v
x
x
y
~v
~
B
−
~
E
z
y
−
~
E
~
E
x
z
z
5
x
z
Magnetic Force
~ and B
~ combined
E
~ and B
~ fields can be configured to select charged particles of a particular velocity.
E
~ and B
~ that could exert zero net foce on the given
Exercise: Draw in a configuration of E
particle.
y
+
~v
x
z
Question: If the magnetic field has magnitude B, what magnitude of E will produce no net force?
Question: If the same particle comes in at a slower speed, why won’t it move in a straight line?
6
Magnetic Force
~ and B
~ combined
E
~ earth , what are the magExample: Given the details in the figure and neglecting B
nitude and direction of I?
2 turns, radius R
~v
I
+Q
−e
3R
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Magnetic Force
Motional EMF
A conducting bar of length L moving with velocity ~v through a uniform magnetic
~ polarizes due to magnetic forces, inducing an electric field E.
~
field B
+
L
~
E
−
~v
~
B
~ will the polarization process conclude?
Question: If the bar is kept at a constant speed |~v |, at what value of |E|
Question: In the steady state, what is ∆V between the top and bottom of the bar? Is an external force needed to keep the
bar moving at |~v |? Why or why not?
Hard Question: If the bar is released (i.e., not maintained at a constant speed) before the steady state is reached, what
happens?
8
Magnetic Force
Motional EMF
The conducting bar on rails acts like a battery, where the source of emf is the magnetic
force on the charge carriers: emf = vBL. This is known as motional emf.
+
R
~ve
~v
~
B
−
An external force F~ext is required to maintain a constant bar speed |~v | because the
induced current I generates an opposing force F~B = ILB.
+
I
R
F~ext
~v
L
~
B
−
Exercise: Use the energy principle to show that the power generated by F~ext is equal to the power dissipated by the resistor.
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Magnetic Force
Motional EMF
Example: As shown, a bar on rails is moving downward with speed |~v | in a uniform
~ What is the reading on the voltmeter? The + and − refer to the positive
magnetic field B.
and negative terminals of the voltmeter, respectively.
w
~
B
−
+
~v
− V +
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Magnetic Force
~ and Inertial Reference Frames
B
An inertial reference frame is a coordinate system that is moving with respect to other coordinate systems. For example:
the inertial reference frame inside an airplane is moving with respect to the earth.
A charged object observed from within its own inertial frame (i.e., at rest) will be seen to produce only an electric field. Seen
from an inertial frame moving relative to it, the charged object will produce both electric and magnetic fields.
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Magnetic Force
~ and Inertial Reference Frames
B
Lab Frame
+e1
~1
B
v
r̂
F~21
Given protons moving in parallel, the net force on e2 due to e1 (in the Lab frame) is
v2
1 e2
1
−
(−ŷ) ,
F~net =
4π0 r2
c2
B
v
+e2
where µ0 0 = 1/c2 and the ratio FB /FE = v 2 /c2 .
~
E
~
F21E
Particle Frame
+e1
Question: If the protons are moving through a chamber, in which inertial frame (particle
or lab) do the protons strike the chamber walls “first?” How is this possible?
r̂
+e2
~
E
F~21
E
Question: Why do we put “first” in quotes?
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Magnetic Force
Magnetic Torque
Wire Loop Viewed Edge On
~
µ
~ = IA
~
B
~
B
~ =IwB
F
~ = Iwh(Â).
A wire loop of area A = wh and current I has magnetic moment µ
~ = IA
~ the wire loop experiences a torque
In a uniform magnetic field B,
I
axle
h
I
~.
~τ = µ
~ ×B
~
The torque will act to align µ
~ and B.
~ =IwB
F
Exercise: Draw in ~τ and the direction of rotation on the figure. Can you think of a RHR relating ~τ to the direction of
rotation?
~ determined?
Question: How is the direction of A
~ in which direction must µ
Question: Relative to B,
~ point for the loop to be in unstable equilibrium?
13
Magnetic Force
Potential Energy of Magnetic Dipole
Since a magnetic dipole in a uniform magnetic field experiences a torque, and that torque acts through some angle (i.e., work
~ the dipole must have some magnetic potential energy.
is done) until µ
~ k B,
Exercise: Calculate the minimum work required to rotate the wire loop through an
angle θ to verify that the potential energy of a magnetic dipole in a magnetic field is
given by
~.
Um = −~
µ•B
Wire Loop Viewed Edge On
µ
~
~
B
F~ext
I
θ
F~B
I
F~B
h
~
Question: What is the maximum potential energy of the loop for a given value of B?
14
Magnetic Force
Force on a Magnetic Dipole
F~B
Exercise: Using Biot-Savart, integrate the force F~B about the loop to obtain the net
~ is known.
force on the loop dipole due to the bar magnet. Assume B
~
B
θ
~
B
S
N
µ
~
R
F~B
Using the energy principle, we can show that the force on a magnetic dipole µ
~ is given by
~ .
F~B = −∇UB = ∇(~
µ • B)
Exercise: Prove the statement above for the case where the loop dipole is moved a
distance ∆x by an external force against the force of the magnetic field
S
N
~1
B
µ
~
R
∆x
15
~2
B
Magnetic Force
Motors and Generators
A simple generator consists of a wire loop in a magnetic field.
~
B
~
B
~v =
θ
hω
2
Exercise: Show that a loop of area A = hw turning at angular speed ω will generate an
emf around the loop equal to
emf = ωBA sin(ωt) .
ω
h
Exercise: An external force must be exerted in order to make the generator turn, resulting
in an input power equal to
dW
= IBAω sin θ = IV ,
P =
dt
F~ext
as required.
~
B
F~B
F~ext
θ
ω
h
16
F~B = IωB