Lenzs LawLenzs Law: There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic fi ld th h i th flfield opposes the change in the flux.

Lenzs LawLenzs Law: There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic fi ld th h i th flfield opposes the change in the flux.

Lenzs LawLenzs Law: There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic fi ld th h i th flfield opposes the change in the flux.

Lenzs LawTACTICS BOX. Using Lenzs law

1 Determine the direction of the applied magnetic field. The field must pass through the loop.field. The field must pass through the loop.

2 Determine how the flux is changing. Is it increasing, decreasing, or staying the same?

3 Determine the direction of an induced magnetic field that will oppose the change in the flux..-Increasing flux: the induced magnetic field points opposite the applied magnetic field.. -Decreasing flux: the induced magnetic field points in the same direction as the applied magnetic field.. -Steady flux: there is no induced magnetic field.

4 Determine the direction of the induced current4 Determine the direction of the induced current.Use the right-hand rule to determine the current direction in the loop that generates the induced magnetic field you found in step 3.

Faradeys LawA f i i d d i d ti l if th ti fl th h th lAn emf is induced in a conducting loop if the magnetic flux through the loop changes. The magnitude of the emf is given by the formula below and the direction of the emf is such as to drive an induced current in the direction so that the induced magnetic field opposes the change in the flux.

d m=g pp g

dt

dd vlBxlBdtd

dtd m === )(

Faradeys Law1 Th L d t t ti1. The Loop can move or expand or rotate, creating a

motional emf.

2 The magnetic field can change

BdAAdBd m +==

2. The magnetic field can change

dtA

dtB

dt+==

Faradays Law

Remember B is the magnetic flux through the circuit and is found bythe circuit and is found by

If the circuit consists of N loops all of theB d = B A

If the circuit consists of N loops, all of the same area, and if B is the flux through one loop an emf is induced in every loop andloop, an emf is induced in every loop and Faradays law becomes

dBd Ndt=

Induced Currents. Generators.BAm =

tABABBAm coscos ===

ABNtdABNdN m icos tABNdt

tdABNdt

dN mcoil sincos ===

Induced emf and Electric Fields

The emf for any closed path can be The emf for any closed path can be expressed as the line integral of E.ds over the pathp

Faradays law can be written in a general form:form:

Bdddt = E s dt

Two ways to create an electric field

Faradays Law RevisitedCalculating the emf

WqW=

sdEqdW =

b =a

sdEqW

Calculating the emf

= sdEqW curveclosed dEW curveclosed == sdEq curveclosed

Establishing the Sign

dsdE m== dt[ ]AdBdd [ ]dt AdBddtdsdE m ==

Transformers An AC transformer consists of two

coils of wire wound around a core of soft iron

The side connected to the input AC The side connected to the input AC voltage source is called the primaryand has N1 turns

The other side, called the secondary, is connected to a resistor and has Nis connected to a resistor and has N2turns

The core is used to increase the magnetic flux and to provide a

di f th fl t fmedium for the flux to pass from one coil to the other Eddy current losses are

minimized by using a laminated y gcore

Iron is used as the core material because it is a soft ferromagnetic substance and reducessubstance and reduces hysteresis losses

Transformers

Assume an ideal transformer One in which the energy losses in the windings One in which the energy losses in the windings

and the core are zero Typical transformers have power efficiencies of 90% to

99%

The rate of change of the flux is the same for both coils

1 1BdV N

dt =

Transformers The voltage across the

secondary is

d The voltages are related by

2 2BdV N

dt =

The voltages are related by

22 1

1

NV VN

= When N2 > N1, the

transformer is referred to as a step-up transformer

1N

p p When N2 < N1, the

transformer is referred to as a step-down transformer

Transformers

The power input into the primary equals the The power input into the primary equals the power output at the secondary I1V1 = I2V2 I1V1 I2V2

The equivalent resistance of the load resistance when viewed from the primary isresistance when viewed from the primary is

2

1eq L

NR R = eq

2LN

Metal Detectors

Inductors. The Potential Difference across an Inductor

Self-Inductance As the current increases with time, the magnetic flux through the

circuit loop due to this current also increases with time This corresponding flux due to this current also increases This corresponding flux due to this current also increases This increasing flux creates an induced emf in the circuit The direction of the induced emf is such that it would cause an

induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field

The direction of the induced emf is opposite the direction of the emf of the battery

This results in a gradual increase in the current to its final This results in a gradual increase in the current to its final equilibrium value

This effect is called self-inductance Because the changing flux through the circuit and the resultant Because the changing flux through the circuit and the resultant

induced emf arise from the circuit itself The emf L is called a self-induced emf

Self-Inductance, Coil Example

A current in the coil produces a magnetic field directed toward the left (a)

If the current increases, the increasing flux creates an inducedIf the current increases, the increasing flux creates an induced emf of the polarity shown (b)

The polarity of the induced emf reverses if the current decreases (c)decreases (c)

Self-Inductance

A induced emf is always proportional to theA induced emf is always proportional to the time rate of change of the current

Id L=

L is a constant of proportionality called the

L L dt=

p p yinductance of the coil and it depends on the geometry of the coil and other physical h t i ticharacteristics

Inductance of a Coil

A closely spaced coil of N turns carrying A closely spaced coil of N turns carrying current I has an inductance of

B LN L

Th i d t i f th

I IB LN L

d dt= =

The inductance is a measure of the opposition to a change in current

Inductance Units

The SI unit ofThe SI unit of inductance is the henry(H)

VA

sV1H1 = Named for Joseph

Henry (pictured here)

Inductance of a Solenoid

Assume a uniformly wound solenoid having N Assume a uniformly wound solenoid having Nturns and length Assume is much greater than the radius of the Assume is much greater than the radius of the

solenoid The interior magnetic field is g

I Io oNB n = = ll

Inductance of a Solenoid, cont The magnetic flux through each turn is

NA IB oNABA = = l

Therefore, the inductance is2

B oN N AL = = This shows that L depends on the

IL = = l

pgeometry of the object

Energy in Inductors and Magnetic Fields

dtdILIVIP Lelec ==

2I

The electric power:

2

2

0

LIIdILUI

L == 2

2LIUL =2

0

20

0

22

02

21

21

22AlB

lNIAlI

lANLIUL

=

===

21

2

2

021 AlBUL =

1 202

1 BuB =