Tài liệu CHAPTER XI Electromagnetic Induction pptx

In the previous chapter we know that: current =» ~ magnetic field In this chapter we will study currents produced by changing magentic fields... W Faraday's law: The emf € induced in a c

GENERAL PHYSICS II

Electromagnetism

7

Thermal Physics

CHAPTER XI

Electromagnetic Induction

§1 Induction experiment - Faraday’s and Lenz’s laws

§2 Induced electromotive force and induced electric fields

§3 Mutual inductance and self-inductance

§4 Magnetic field energy

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In the previous chapter we know that:

current =» ~ magnetic field

In this chapter we will study currents produced by changing magentic fields In other words,

changing magnetic field = current

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§1 Induction experiment - Faraday’s and Lenz’s laws:

1.1 Induction experiment:

+ Inside the shaded region, there is a S

Lorentz force (on the electrons in

(a) A Clockwise Current; (6b) A Counterclockwise Current; (c) No Current

+ Now the loop is pulled to the right at a velocity v

— The Lorentz force will now give rise to:

1831: Michael Faraday did the previous experiment, and a few others:

+ Move the magnet, not the loop Here

k there is no Lorentz force yx 8 but

there was still an identical current

(This phenomenon can be explained

by the relativity principle of motion)

+ Decrease the strength of B Now

nothing is moving, but M.F still saw vš

+ Instead of a magnet we use a loop

with current:

¢ Switch closed (or opened)

= current induced in coil b

¢ Steady state current in coil a

= no current induced in coil b

= What is the cause of the currents induced in the loops in the

mentioned experiment?

Conclusion:

A current is induced in a loop when:

« there is a change in magnetic field through it

¢ this can happen many different ways

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Note that we must fix the choice of the surface’s normal vector

Either choice is equally good, but once we make the choice we must stick with it

¢ Recall the analogous formula for the flux of the electric field Magnetic field lines is drawn by the same rule as electric field

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W Faraday's law:

The emf € induced in a circuit is determined by the time rate

of change of the magnetic flux through that circuit

2.3 Lenz's Law:

¢ The direction of induced currents is determined by the Lenz's Law:

“The induced current will appear in such a direction that it opposes the change in flux that produced It’

Conservation of energy considerations:

Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated

Why 22?

¢ {lf current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc

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te Example 1:

L

constant velocity v in the +x direction _ XX XXXXXXXX

through a region of constant magnetic field XX XXKXXXX

B in the -z direction as shown XXXXXXXXXXXX

= What is the direction of the induced x

current in the loop?

There is a non-zero flux ®g passing through the loop since B

is perpendicular to the area of the loop

since the velocity of the loop and the magnetic field are CONSTANT,

however, this flux DOES NOT CHANGE IN TIME

Therefore, there is NO emf induced in the loop; NO current will flow!!

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+ Example 2:

A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current/ flows tn the +x direction

induced current in the loop?

The flux ®, is decreasing — the induced current must create a

induced magnetic field which directs along the manetic field of the current I

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+ Example on the calculation of induced current:

| | | XXXXXX r7

resistance FA through a region of constant X x X X X |

« Direction of induced current?

Lenz’s Law > clockwise! XXXXXX

Appendix)

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§2 Induced electromotive force and induced electric fields:

2.1 Induced electric fields:

`

time

_ a a

A magnetic field, increasing in time, passes through the blue loop

+ What force makes the charges move around the loop?

- lt cant be a magnetic force because the loop isn't moving ina magnetic field and even isn't lying in a magnetic field

° [he force exerting on charges exists in any case while the loop

is large or small, even while the loop does not have to be a wire

[= We are forced to conclude that the curent is caused by a electric field

which is called induced electric field

Note: The wire isnt the causion of induced enff, it is only the device which

help us observe the current, the emf exists even in vacuum, and this emf

is related to the induced electric field!

+ The equation for the induced electric field: The work done by the induced

electric field E per unit charge is equal to the induced emf, so we have

(the integration path is stationary) + Between the electrostatic and induced electric fields there are radical differences:

¢ Recall that the electrostatic field is conservative (a work done by the field depends only the intial and final positions; over a closed path 1= 0) The induced electric field is nonconservative

— For induced electric fields we can not introduce the concept

of potential

¢ The electrostatic field is produced by a static charge distribution

The induced electric field can not be produced by any static charge distridution, it can be produced by a changing magnetic field

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2.2 Eddy currents:

Not only in conducting wires, induced currents appear in pices of metal moving in magnetic field or located in changing magnetic fields

= We call these eddy currents

Turn on the magnet —

EP ReNTs the pendulum motion

is arrested The eddy teva act to oppose the change in magnetic

2.3 Meissner effect for superconductors:

¢ Superconducitivity: The property of some materials that the resistance becomes zero

at temperatures under a critical one T < /,,

(Tc ~ some °K)

¢ Superconductors have not only this property, they also have extraordinary magnetic

properties An important property is the

C) Consider a magnet on a sample of superconducting material:

¢ Above the critical temperature (7 > 7), the motion of magnet

produces induced currents in the sample, but these currents die away due the resistance

- Cool the sample below 7, — the magnet lifts off and hovers above the superconductor

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@ Why does the magnet hover ? This phenomenon can not be explained by Faraday’s law Itis a property called the Meissner effect

lt is caused by the fact that superconductors exclude magnetic fields just like ordinary conductors exclude electric fields

§3 Mutual inductance and self-inductance:

in a neighboring coil

We can describe this effect quantitatively

in terms of the concept of mutual inductance

3.1 Mutual inductance:

A current is induced in one coil eee

when the current is changed

3.1.1 Definition of mutual inductance:

¢ The induced emf in a coil with

N, turns producing by the change

of the current /, :

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Tu),

This equation can be used as the

definition of mutual inductance M,,

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3.1.2 Properties of mutual inductance:

¢ M,, depends only on the shapes and the

relative positions of the coils, not on the

current in coil 1 or on time

¢ The induced emf in coil 1 producing by

the change of the current /, in coil 2 has

the analogous formula:

at two coefficients always

3.1.3 Applications of mutual inductance:

=» Transformers (still to come)

- Change one AC voltage into another

- Pulsed current > pulsed magnetic field

> Induces emf in metal Ferromagnetic metals “draw in” more B

> larger mutual inductance > larger emf Emf > current (how much, how long it lasts, depends on the resistivity of the material)

Decaying current produces decaying magnetic field

> induces current in receiver coils Magnitude & duration of signal depends

on the composition and geometry of the metal object

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(secondary)

» Pacemakers

- It's not easy to change the battery!

- Instead, use an external AC supply

- Alternating current

> alternating B

> alternating ©, inside “wearer”

> induces AC current to power pacemaker

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3.2.1 Definition of inductance: XXXX

XXXXXX Consider the loop with f„„= 0 loop ® yyxy

switch closed => current starts to flow in the Fs) b

loop (is it infinite??) 1=/0 !!! No “inductance” mm

limits dI / dt

A magnetic field produced in the area enclosed by the loop (B proportional to J ) The flux through loop increases as the current increases The emf induced in loop opposing initial direction of current flow because it opposes increasing flux (Faraday’s Law)

this emf acts like a battery to oppose the real battery — reduce current flow]

¢ Fact: the current turns on at such a rate:

¬_.h

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= Self-induction: A change current through a loop induces an opposing

emf in that same loop Then an inductor (a set of coils in some geometry; e.g., solenoid) is characterized by a quantity called self-inductance, or simply,

inductance It can be calculated from its geometry alone if the device is constructed from conductors and air (similar to the Capacitance of a capacitor)

3.2.2 Calculation of inductance:

Consider a long solenoid:

N turns total, radius r, length I

N ane

3.2.0 Inductors in Circuits:

General rule: inductors resist change in current

¢ Hooked to current source

— Initially, the inductor behaves like an open switch

— After a long time, the inductor behaves like an ideal wire

« Disconnected from current source

— Initially, the inductor behaves like a current source

— After a long time, the inductor behaves like an open switch

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+ Change of current in AL circuit:

¢ At rt=0, the switch is closed and the current / starts to flow

To find the current | as a function of time t, we need to choose an

exponential solution which satisfies the boundary condition:

6

eae [(t=0)=—

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=> For RL circuit with s on:

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After the switch has been in position

a for a long time, redefined to be f=0,

¢ The appropriate initial condition is:

¢ The solution then must have the form:

= For RL circuit with ¢ off

37% Max at t=L/R

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§4 Magnetic field energy:

ba STATE Cop energy is stored in an inductor when a current is flowing

W Where is the Energy Stored?

Claim: (without proof) energy is stored in the magnetic field itself

(just as in the capacitor / electric field case)

To calculate this energy density, consider the uniform field

Summary

I Self-Inductance F a: |

2

@ Self-Inductance for solenoid [ "xỉ : mr |

Inductors in series add a rae

¢ Inductors in parallel add reciprocally

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Devices in a circuit

Resistor A disperses energy in the heat form

Capacitor C can store energy in the electric field form, and then liberate energy as a source of current

Inductor L can store energy in the magnetic field form, and then liberate energy as a source of current