physics 54 lecture march 1, 2012 - duke universitygoshaw/lecture_em_induction_march_1... · 2012....

Physics 54 March 1, 2012 1

OUTLINE �� Micro-quiz problems (magnetic fields and forces)

�� Magnetic dipoles and their interaction with magnetic fields

�� Electromagnetic induction Introduction to electromagnetic induction Production of emf’s by a time varying magnetic field Lenz’s Law Motional emf’s Eddy currents

Physics 54 Lecture March 1, 2012


A couple of Micro-quiz problems


Micro Quiz 1


Micro Quiz 1

�� An electron is traveling horizontally east in the magnetic field of the earth near the equator near the equator. The direction of the force on the electron is:

N

E W

B

v

-e

N

S

E W

The electron is deflected downward: Fe = - e v x B

S


Micro Quiz 2


Micro Quiz 2

v

B Fm = q v x B up

Fe = q E down

q v B = q E balance

E/B = v independent of q or m


Micro Quiz 3


Micro Quiz 3

I into page B

Electron velocity out of page

Fe = - e v x B

The electrons move to the right and accumulate on surface 2

2


Magnetic dipoles


�� The simplest electric object in Nature is a point charge q or electric monopole. An electron is an example of an ideal point charge.

�� The next simplest electric object is an electric dipole. Atoms and molecules can carry permanent or induced electric dipole moments p.

Introduction: monopoles and dipoles


�� No single magnetic single poles have been observed in Nature. There are apparently no magnetic monopoles!

�� The simplest magnetic objects in Nature are magnetic dipoles given the symbol µ .

�� Magnetic dipoles are created by current loops or the rotation of charged objects. Elementary particles, such as electrons, carry a magnetic dipole moment.



�� A simple iron bar magnet is a magnetic dipole.

µ

p

+

-

break up

break up

+

-

electric monopoles = single charges

N

S

N

N

S

S There are

no isolated magnetic monopoles


�� The source of “current loops” in this case are from the atomic structure of the atoms in the magnet. �� Electric dipoles can be broken apart:

�� In contrast magnetic dipoles can not:


�� A magnetic dipole is created by a closed current loop.

Definition of a magnetic dipole

µ = I N A n

�� Consider a loop with N turns of wire.

�� Let n be a unit vector perpendicular to the wire loop in a direction determined by the current and the right hand rule.

v

�� Define the magnetic moment µ of this loop to be:

v


�� Recall that for an electric dipole in a uniform electric field: The net force is zero The torque τ = p x E The potential energy is U = - p E

Torque on a magnetic dipole

. �� For a magnetic dipole in a uniform magnetic field:

The net force is zero

The torque is τ = µ x B (See derivation in text)

The potential energy is U = - µ B .


Example 1 (torque on a wire loop)

Find the torque on the current loop.

current I a

b

magnetic field B


Electromagnetic Induction (a new topic: Chapter 34)


�� We will now extend electromagnetic theory by introducing a new phenomenon called electromagnetic induction.

Introduction

�� This is the observation that a time varying magnetic field creates an electromotive force (emf) , and at a more basic level an electric field.

�� The applications of electromagnetic induction are far reaching. For example electric motors and generators operate based upon the the principle of electromagnetic induction.

�� Electromagnetic induction is described by two of the simplest laws of electromagnetic theory:

Faraday’s Law and Lenz’s Law


Faraday’s Law and induced emf’s

�� We will first introduce Faraday’s Law in a very practical way.

S Surface bounded

by the loop Normal defined by curve direction and the right hand rule

�� We will use it to relate a time varying magnetic flux to an electromotive force (emf) induced in a wire.

�� Let ΦB = the magnetic flux through the area bounded by a single closed wire loop.

ΦB


�� With the definition of magnetic flux given on the previous page, Faraday’s Law predicts that the emf induced in the wire loop will be:

emf induced in wire loop

magnetic flux through the loop

As stated above the emf is for one loop of wire. If there are N overlapping loops in series then the emf is: = N | dΦB/dt | where ΦB = the magnetic flux through one loop

= | dΦB/dt |

Faraday’s Law and induced emf’s


�� An easy way to keep track of the sign of the induced emf is formulated by a rule called Lenz’s Law.

The sign of the induced emf is determined by Lenz’s Law

The induced emf produces a current, I, that creates a magnetic field that opposes the change in ΦB .

See the next example.


Example 2

A magnet is moved into a wire loop as shown. In which direction will the current flow?


The magnetic field B1 caused by the moving magnet causes a magnetic flux increase through the loop.

This induces an emf which causes a current I that produces a magnetic field B2 opposing B1.

B1

B1

B2

B2

current counterclockwise

Example 2 (solution using Lenz’s Law)


Example 3

A wire loop is a circle of radius r = 50 cm. It has a resistance R = 0.1 Ω . A magnetic field is applied perpendicular to the loop.

B(t)

t

B(t)

T

Bo a) Find the direction of the current b) Find I(t) c) Find the heat generated in the coil.

T = 10s Bo = 1 Tesla


Example 4

�� A loop of wire with resistance R falls under the influence of gravity.

A. zero B. non-zero clockwise C. non-zero counterclockwise

Io

The current induced in the wire is:


Example 5

�� A magnetic field is constant in space but varies with time: B(t) = [0.3 + 0.5 t2 ] Tesla (perpendicular to page) .

A. a b t B. a b t2 C. ab(0.3 + t)

D. ab( 0.3 + a b t2)

The magnitude of the emf induced in the above loop is:

a

b


Example 6

�� A bar magnet is dropped through a wire loop as shown

A. zero B. clockwise C. counter clockwise

D. clockwise and counter clockwise

The current induced in the loop will be:

N

S


Motional emf

�� A time-varying magnetic field through a stationary conducting loop generates an emf via Faraday’s Law.

�� Faraday’s Law also predicts that a conductor moving through a constant magnetic field will generate an emf.

stationary loop

time varying magnetic fux ΦB

= | d ΦB/dt |

constant magnetic field Bo

= Bo L v L v

“motional” emf


Motional emf (con.)

�� The form of the motional emf can be derived as follows.

L v

�� Consider a metal rod sliding on two fixed conducting bars

constant magnetic field Bo

out of page

x �� Magnetic flux in loop = ΦB = Bo L x and dΦB/dt = Bo L dx/dt = Bo L v

�� Using Faraday’s Law the magnitude of the emf = dΦB/dt and the motional emf = = Bo L v


Example 7

�� The magnetic field Bo is out of the page

The polarity across the resistor will be:

A. top + bottom - B. top - bottom + C. no potential difference


Currents generated in moving conducting plates

�� When a metal plate or conductor is pulled through a non-uniform magnetic field, currents in the form of “eddys”,i.e. eddy currents, are formed.

�� The time-varying magnetic flux through loops in the metal generate emf’s and therefore currents.

�� The currents react back on the magnetic field causing a retarding force.


Currents generated in moving conducting plates

�� Eddy currents therefore can be used to generate a magnetic brake in which the force increases as the speed of the moving conductor increases.

�� In some cases eddy currents are undesirable as they transform energy into heat. (for example in an AC transformer).

�� The eddy currents can be suppressed by cutting or laminating the metal plates.


Example 8

�� A metal plate falls into a region of magnetic field as shown below.

A. clockwise B. counter clockwise C. zero

The current when the plate enters the field will be:


Example 8 (solution)

The induced eddy currents act as a break to stop the motion of the metal plate


Example 9 (eddy currents)

Find the direction of the force on the copper ring due to the induced eddy currents.


End Lecture: Next one on March 13:

Will cover the remaining material in Chapter 34


Example 8

A constant current Iwire = Io flows through the long straight wire. The loop of wire moves upward with constant velocity vo so that r(t) = ro + vo t

Find the direction and magnitude of the current I(t) in the loop of wire if it has resistance R.

physics 54 lecture march 1, 2012 - duke universitygoshaw/lecture_em_induction_march_1... · 2012....

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