Early college-level class illustrating a simple application of Maxwell's equations: the induction motor.

1. Maxwell’s Equations, Part III -Maxwell’s Equations, Part III -
Faraday’s LawFaraday’s Law
Lecture 2:Application & Use –
The Induction Motor

2. OutlineOutline
Review from last time
Study an interesting application
systematically
Question period
09/24/13 PHYS-666 Pseudo-Physics Course 2

3. Review from last timeReview from last time
Faraday’s Law:
Integral form:
Differential form:
Force law for current-carrying conductors:
09/24/13 PHYS-666 Pseudo-Physics Course 3
dt
d
N BΦ
−=emf
∇×E=−
∂B
∂t
E⋅dl
C∫ =−
∂ΦB
∂t
=−
∂
∂t
B⋅ds
S
∫∫
F=Il
_
×B

4. The Induction MotorThe Induction Motor
Consider a square loop of wire (N turns) with a
current I running through it, that is fixed on an axis
so it can rotate around the x-axis
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x
y
z
L
I

5. The Induction MotorThe Induction Motor
Now put the wire in a magnetic field that is
rotating about the x-axis
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x
y
z
L
B=sin(ωt),ω=
dθ
dt

6. The Induction MotorThe Induction Motor
We know:
Since the magnetic field is rotating about
the x-axis, it can be written:
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F=Il
_
×B
B=By j
^
+Bz k
^
=Bcosθj
^
+Bsinθk
^
Bcosθ
θ
BsinθB

7. The Induction MotorThe Induction Motor
 Side 1:
 Since the loop can’t
move in x, there is no
motion caused by the
magnetic field on this
arm
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I
L
F = Il
_
×B, l
_
= L j
^
= ILBsin θ( )sin ωt( )i
^
+(stuff) j
^
× j(
^
=0)
x
y
z

8. The Induction MotorThe Induction Motor
 Side 2:
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I
L
F = Il
_
×B, l
_
= Li
^
= ILBcos θ( )sin ωt( )k
^
−ILBsin θ( )sin ωt( )j
^
x
y
z

9. The Induction MotorThe Induction Motor
 Side 3: Similar to Side 1
 Since the loop can’t move in
x, there is no motion caused
by the magnetic field on this
arm
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I
L
F = Il
_
×B, l
_
= −L j
^
= −ILBsin θ( )sin ωt( )i
^
−(stuff) j
^
× j(
^
=0)
x
y
z

10. The Induction MotorThe Induction Motor
 Side 4: Similar to Side 2
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I
L
F = Il
_
×B, l
_
= −Li
^
= −ILBcos θ( )sin ωt( )k
^
+ILBsin θ( )sin ωt( )j
^
x
y
z

11. The Induction MotorThe Induction Motor
Now sum up the forces on the loop:
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x
y
z L
I
F4z=−ILBcosθ()sinωt( )k
^
F4y=ILBsinθ()sinωt( )j
^
F2z=ILBcosθ()sinωt( )k
^
F2y=−ILBsinθ()sinωt( )j
^

12. The Induction MotorThe Induction Motor
The forces along the y-axis cancel, and only the
two torques in the z-direction remain
The torque on each arm of one loop is
And the overall torque (bearing in mind that
there are N turns, and same torque on arms 2
and 4 of each turn) is
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τ=r×F=
L
2
ILBcosθ()sinωt( )θ
^
τT=2τ=NIL2
Bcosθ()sinωt( )θ
^

13. The Induction MotorThe Induction Motor
The overall effect is that the rotating field pulls
the ring around with it at an angular frequency
equal to the angular frequency of the field
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14. The Induction MotorThe Induction Motor
Things to think about:
◦ N: # turns
◦ I: applied current
◦ B: magnetic field intensity
◦ A (=L2
): area enclosed by loop
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τT=NIABcosθ()sinωt( )θ
^

15. Review for midtermReview for midterm
Faraday’s Law of Induction
Right-hand rule/cross product
Faraday force law for current-carrying
conductors
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