Faraday’s Law

By: Ahmed M. AttiyaBy: Ahmed M. Attiya

King Saud University

Electrical Engineering Dept.

Fundamental Laws of Electrostatics

• Integral form • Differential form

∫ =⋅ ldE 0 E =×∇ 0

∫∫

∫=⋅

C

dvqsdD evqDE

=⋅∇×∇ 0

∫∫V

evS

dvqsdD evq

ED ε=

Fundamental Laws of Magnetostatics

• Integral form • Differential form

⋅=⋅ ∫∫ sdJldH =×∇ JH

0=⋅∫

∫∫SC

sdB 0=⋅∇=×∇

BJH

0∫S

sdB

HB μ=

Electrostatic, Magnetostatic, and , g ,Electromagnetostatic Fields

• In the static case (no time variation), the electric field (specified by E and D) and the ( p y )magnetic field (specified by B and H) are described by separate and independent setsdescribed by separate and independent sets of equations.

b h l• In a conducting medium, both electrostatic and magnetostatic fields can exist, and are coupled through the Ohm’s law (J = σE). Such a situation is called electromagnetostatic.Such a situation is called electromagnetostatic.

Electromagnetostatic Fieldsg

• In an electromagnetostatic field, the electric field is completely determined by the p y ystationary charges present in the system, and the magnetic field is completely determinedthe magnetic field is completely determined by the current.

h f l h• The magnetic field does not enter into the calculation of the electric field, nor does the electric field enter into the calculation of the magnetic field.magnetic field.

Faraday’s Experimenty p

83 i h l dIn 1831, Michael Faraday discovers that a changing

magnetic flux can induce an l t ti felectromotive force.

dΦ ∫ ⋅= AdBrr

Φdt

dVemfΦ−= ∫

area

∫ ∫∫ dBdldE0=⋅∫C

ldE ∫∫ ⋅−=⋅SC

sdBdt

ldE

Move a bar magnet toward

Conducting loop

the loop, a current suddenly appears in the circuit

Sensitive current meter

The current disappears when the bar magnet stops

Since there is no battery or other source of emf included,

If we then move the bar magnet away, a current again suddenly appears, but now in the opposite directionthere is no current in the circuit the opposite direction

Discovering of the First Experiments

1. A current appears only if there is relative motion bet een the loop and the magnetmotion between the loop and the magnet

2 Faster motion produces a greater current

3 If moving the magnet’s N-pole towards the

2. Faster motion produces a greater current

3. If moving the magnet s N pole towards the loop causes clockwise current, then moving the N-pole away causes counterclockwise.

Magnetic Field can produce an electric current in a closedloop, if the magnetic flux linking the surface area of the loopchanges with time.

This mechanism is called “Electromagnetic Induction”

dΦdNVemfΦ−=

dtemf

LENZ’S LAWLENZ S LAW

The polarity of the emf induced by aThe polarity of the emf induced by a changing flux will produce a current that

t ti fi ld i thgenerates a magnetic field opposing the flux change that produced it

Conclusions from the experiment

• Current induced in the closed loop when magnetic flux changes,and direction of current depends on whether flux is increasing ordecreasingg

• If the loop is turned or moved closer or away from the coil, theph sical mo ement changes the magnetic fl linking its s rfacephysical movement changes the magnetic flux linking its surface,produces a current in the loop, even though B has not changed

In Technical TermsTime ar ing magnetic field prod ces an electromoti eTime-varying magnetic field produces an electromotive

force (emf) which establish a current in the closed circuitcircuit

Electromotive force (emf) can be obtained through thefollowing ways:following ways:

1. A time-varying flux linking a stationary closed path. (i.e.Transformer)

2. Relative motion between a steady flux and a close path.(i.e. D.C. Generator)

3. A combination of the two above, both flux changing andd i i l l l d hconductor moving simultaneously. A closed path may

consists of a conductor, a capacitor or an imaginary line inspace etcspace, etc.

Faraday’s LawFaraday’s LawThe magnitude of the induced emf in a closed

i ig

circuitis equal to the time rate of change of the linked

magnetic flux Φ.

( l )(volts)

Minus Sign Lenz’s Law

I di t th t th f i d d i i h di ti tIndicates that the emf induced is in such a direction as to produces a current whose flux is opposite to the original

fluxflux.

If the closed path is taken by an N-turn filamentary conductors

Magnetic flux Φ?Magnetic flux Φ?

The magnetic flux Φ linking a surface S is defined as the total magnetic flux density B passing through S:total magnetic flux density B passing through S:

(Wb)(Wb)

⇒

For a closed loop with contour C, the emf is defined by:

Take N = 1

⇒

In Electrostatics – an electric field intensity E due to static charge distribution must lead to zero potential difference b t l ed thabout a closed path.

Here – the line integral leads to a potential difference

15

g pwith time-varying magnetic fields, the results is an emf or a voltage

Stationary Loop in a Time-Varying Magnetic Field

A i l t (N 1) d ti l iA single-turn (N =1), conducting loop is placed in a time-varying magnetic field B(t).

i h l i i d/d ( ) lSince the loop is stationary, d/dt operates on B(t) only

⇒⇒

Applying Stoke’s theorem to the closed line integral

⇒

∴∴

If B is not time varying i eIf B is not time-varying, i.e.

OR

M ll’ E f El i

OR

Maxwell’s Eqn of Electrostatic

Moving Conductor in a Static Magnetic Field

A wire with length l moving across a static magnetic fieldat a constant velocity u (points to x).

The conducting wire contains free electron.

Magnetic force Fm acting on any charged particle “q”any charged particle q

moving with velocity u is:

This Fm is equivalent to the electrical force that would be exerted o the particle by an electric field Em given by:

The electric field Emgenerated by the motion of

Em is in a direction perpendicular to the plane generated by the motion of

the charged particle is called a motional electric field

perpendicular to the plane containing u and B

For the wire, Em is along -y

Magnetic force acting on the electrons in the wire causes them to move in the direction of -Em

i.e. towards the end labeled 1

Induces a voltage differenceInduces a voltage difference between ends 1 and 2

End 2 being at the higher potentialVoltage induced: motional emf, End 2 being at the higher potential

For the conducting wire:g

⇒

∴

In general, if any segment of a closed circuit with contour Cmoves with a velocity u across a static magnetic field B, then the induced motional emf is:

O l th t f th i it th t tiOnly those segments of the circuit that cross magnetic field lines contribute to

Fleming Right Hand RuleFleming Right Hand Ruleg gg gDirection of Induced e.m.f, Magnetic Flux, Conductor Motion

Fore FingerDi ti fDirection of Field Flux

ThumbDirection of

Conductor Motion

Middle FingerDirection of Induced emf or Current Flowemf or Current Flow

Fleming's right hand rule (for generators)

Fleming's right hand rule shows the direction ofFleming s right hand rule shows the direction of induced current flow when a conductor moves in a magnetic field.

The right hand is held with the thumb, first fingerand second finger mutually at right angles, as shown in the diagramshown in the diagram

The Thumb represents the direction of Motion of the conductorThe Thumb represents the direction of Motion of the conductor. The First finger represents the direction of the Field. The Second finger represents the direction of the induced or generated Current (i h l i l di i f i i i )(in the classical direction, from positive to negative).

Fleming's left hand rule (for electric motors)

Fl i ' l f h d l h h di i f

Fleming s left hand rule (for electric motors)

Fleming's left hand rule shows the direction of the thrust on a conductor carrying a current in a magnetic field.

The left hand is held with the thumb, index fingerand middle finger mutually at right angles.g y g g

The First finger represents the direction of the Field. The Second finger represents the direction of the Current (in the classical direction, from positive to negative).direction, from positive to negative). The Thumb represents the direction of the Thrust or resultant Motion.

Application of Faraday’s LawApplication of Faraday’s Law

The rectangular loop shown in the

Example 1:

g pfigure is situated in the x-y planeand moves away from the originat a velocity (m/s) in amagnetic field given by:

(T)

If R = 5Ω , find the current I at the instant that the loop sides are at y1 = 2m and y2= 2.5m .

The loop resistance may be ignored.The loop resistance may be ignored.

h i d d l i i bThe induced voltage V12 is given by:

Since is along g

Voltage are induced across only the sides oriented along

i.e. sides (1-2) and (3-4)

xB decreases exponentially with yB decreases exponentially with y

The induced voltage V12 is given by:The induced voltage V12 is given by:

B decreases exponentially with yB decreases exponentially with yAt y1 = 2 m

Induced voltage V12 is:

At y2 = 2 5 mAt y2 2.5 m Induced voltage V43 is:

Current in the circuit is:

Example Example 22: AC Generator: AC Generator

The Faraday’s Law is the principle at work in an electric generator. The essential design is a conducting coil rotating in the magnetic field of a fixed magnet.g

G tGenerators

Si l Ph G tSingle-Phase Generator

l l i h iFor constant angular velocity, the magnetic flux through the coil area A is:

Conducting CoilConducting Coil

B

⇒

Some Other Applications of M ti I d tiMagnetic Induction

The Magnetic Playback Head of a Tape DeckDeck

• Tape / Hard Drive etc– Tiny coil responds to change in flux as the magnetic– Tiny coil responds to change in flux as the magnetic

domains go by (encoding 0’s or 1’s).

– Credit Card ReaderM t i d– Must swipe card

generates changing fluxb l– Faster swipe bigger signal

TransformersA transformer is a device for increasing or decreasing an ac voltagevoltage.

The changing magnetic flux produced by the current in the i il i d f i h d ilprimary coil induces an emf in the secondary coil.

At the far right is the symbol for a transformer.

Transformer EquationsUsing Faraday’s law we can write expressions for the primary and secondary voltages as follows:primary and secondary voltages as follows:

NV SSΔΦ−= NV ΔΦ−=.

tNV SS Δ

.t

NV PP Δ=

Dividing the above equations we get,

.SS NV=

Dividing the above equations we get,

PP NVAssuming that there is no power loss, we can write,

.PPSS IVIV =

NIV.

P

S

S

P

P

S

NN

II

VV

==

Power Loss in Transmission LinesTransformers play a key role in the transmission of electric power.

RIP 2RIPLoss2=

Example 3Example 3

Example 4Example 4

Example 5Example 5