Maxwell’s Equations, Part Maxwell’s Equations, Part III - Faraday’s LawIII - Faraday’s LawLecture 2: Application & Use –

The Induction Motor

OutlineOutlineReview from last timeStudy an interesting application

systematicallyQuestion period

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Review from last timeReview from last timeFaraday’s Law:

Integral form:

Differential form:

Force law for current-carrying conductors:

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dt

dN B

= emf

E Bt

EdlC

B

t

t

BdsS

F I l_

B

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

The Induction MotorThe Induction MotorNow put the wire in a magnetic field

that is rotating about the x-axis

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x

y

z

L

B sin(t), ddt

The Induction MotorThe Induction MotorWe know:Since the magnetic field is

rotating about the x-axis, it can be written:

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F I l_

B

B By j^

Bz k^

Bcos j^

Bsin k^

Bcos

BsinB

The Induction MotorThe Induction MotorSide 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 I l_

B, l_

L j^

ILBsin sin t i^

(stuff) j^

j(^

0)

x

y

z

The Induction MotorThe Induction MotorSide 2:

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I

L

F I l_

B, l_

Li^

ILBcos sin t k^

ILBsin sin t j^

x

y

z

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 I l_

B, l_

L j^

ILBsin sin t i^

(stuff) j^

j(^

0)

x

y

z

The Induction MotorThe Induction MotorSide 4: Similar to

Side 2

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I

L

F I l_

B, l_

Li^

ILBcos sin t k^

ILBsin sin t j^

x

y

z

The Induction MotorThe Induction Motor

Now sum up the forces on the loop:

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x

y

z L

I

F4 z ILBcos sin t k^

F4y ILBsin sin t j^

F2z ILBcos sin t k^

F2y ILBsin sin t j^

The Induction MotorThe Induction MotorThe 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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rF L

2ILBcos sin t

^

T 2 NIL2Bcos sin t ^

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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The Induction MotorThe Induction MotorThings to think about:

◦N: # turns◦I: applied current◦B: magnetic field intensity◦A (=L2): area enclosed by loop

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T NIABcos sin t ^

Review for midtermReview for midtermFaraday’s Law of InductionRight-hand rule/cross productFaraday force law for current-

carrying conductors

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