Magnetic inductance Solenoids - folk.uio.nofolk.uio.no/ravi/cutn/elec_mag/15-eddy-displacement.pdf · Units of Magnetic Field • The SI unit of magnetic field is the tesla (T) Wb

Magnetic inductance & Solenoidsg

P.Ravindran, PHY041: Electricity & Magnetism 26 February 2013: Magnetic inductance, and Displacement current

A whole picture helps

Charge q as source Current I as source

Gauss’s Law Ampere’s LawFaraday’s Law

Electric field E Magnetic field B

Ampere‐Maxwell Law

Force on q in the E field

Ampere Maxwell Law qF E

qF v B

Force on q in the E fieldForce on or I in the B field

qv

Summarized inSummarized in

Maxwell equations


Units of Magnetic Field

• The SI unit of magnetic field is the tesla (T)• The SI unit of magnetic field is the tesla (T)

Wb N N2 ( / )

Wb N NTm C m s A m

– Wb (Weber) is the unit for magnetic field flux.

• A non‐SI commonly used unit is a gauss (G)– 1 T = 104 G


Poynting VectorPoynting Vector

• Electromagnetic waves carry energy• As they propagate through space, they canAs they propagate through space, they can transfer that energy to objects in their pathTh f fl f i• The rate of flow of energy in an electromagnetic wave is described by a vector called the Poynting vector


Poynting Vector contPoynting Vector, cont

• The Poynting Vector is defined as 1

S E B

• Its direction is the di i f i

oS

direction of propagation• This is time dependent

– Its magnitude varies inIts magnitude varies in time

– Its magnitude reaches a maximum at the same instant as the fields


Poynting Vector…Poynting Vector…

Th it d f th t t th• The magnitude of the vector represents the rate at which energy flows through a unit surface area perpendicular to the direction of the wave propagationp p g– This is the power per unit area

Th SI it f th P ti t J/ 2• The SI units of the Poynting vector are J/s.m2 = W/m2


ELECTRIC FIELD

An Electric field exists in the presence of a charged body

ELECTRIC FIELD INTENSITY (E)

A vector quantity: magnitude and direction (Volts/meter)

MAGNITUDE OF E: Proportional to the force acting on a unit positive charge at a point in the field

DIRECTION OF E: The direction that the force acts


The Electric Field (E) is represented by drawing the Electric Displacement Vector (D), which takes into account the characteristics of the medium within which the Electric Field existsField exists.

EmcoulD 2

, the Electric Conductive Capacity or Permittivity, is related to the ability of a medium, such as air to store electrical potential energy.

11212108508 j llV 112120 10850.8 mjoulecoulVacuum:

112121 10876.8 mjoulecoulAir:1 10876.8 mjoulecoul

Ratio: 003.11


0

The Electric Displacement Vector, D, is used to draw lines of force.

2mcoulUnits of D: mcoulUnits of D:


MAGNETIC FIELD

A Magnetic field exists in the presence of a current

MAGNETIC FIELD INTENSITY (H)

A vector quantity: magnitude and direction (amps/meter)

MAGNITUDE OF H: Proportional to the current

DIRECTION OF H: The direction that a compass needle points in aDIRECTION OF H: The direction that a compass needle points in a magnetic field


The Magnetic Field (H) is represented by drawing the Magnetic Induction Vector (B), which takes into account the characteristics of the medium within which the current flowswithin which the current flows.

HB , the Magnetic Inductive Capacity, or Permeability, is related to the ability of a medium, such as air, to store magnetic potential energy.

126102601 mampjouleVacuum:0 10260.1 mampjouleVacuum:

Air: 1261 10260.1 mampjoule

Ratio: 000.11


0

Magnetic Fields:

Magnetic fields associated with moving charges (electric currents)

BIForce

I: Current ampsorscoul 1

B: Magnetic Induction 21 mampjouleg pj

Magnetic Field Lines are closed loops surrounding the currents that produce themthat produce them


Electromagnetic Induction

The Most Important Point of Faraday’s LawA h i ti fi ld dA changing magnetic field produces

or creates an electric field.

Two types of electric fields. One is created by charge and the other is created by a


yp y g ychanging magnetic field.

Eddy currentsformed by induced emf in a rotating metal disk


formed by induced emf in a rotating metal disk.

Eddy currents

Eddy currents are small circular or swirling currents that arise in conductors like a sheet of metalconductors like a sheet of metal.


Eddy currents

d d d• Induced currents produce eddy currents that oppose the induced currents (and the induced currents (and the motion of the disk)

P.Ravindran, PHY041: Electricity & Magnetism 26 February 2013: Magnetic inductance, and Displacement current16

Metal detector

Metal detector – an alternating magnetic field Bo induces eddycurrents in a conducting object moved through the detector.Th dd ts i t d lt ti tiThe eddy currents in turn produce an alternating magneticfield B’ and this field induces a current in the detector’sreceiver coil


receiver coil.

Eddy currents lead to heat being generated in the conductor

This is the basic principle behind induction stoves. Safe to touch p punless you are metallic. Eddy currents are established in cookware causing metal to heat up.


Maxwell’s Displacement Current• Consider applying Ampere’s Law to the current shown in the diagram.

• If the surface is chosen as 1, 2 or 4, the l d t I

circuit

enclosed current = I

• If the surface is chosen as 3, the enclosed current = 0! (i e there is noenclosed current = 0! (i.e., there is no current between the plates of the capacitor)

Big Idea: In order to haveCurrent produced by displacement ofBig Idea: In order to have

for surface 1 = for surface 3Maxwell proposed there was an extra “displacement current” in the

dB

dB

displacement ofcharges in a dielectricmedia by polarization.

)( IIdB

Maxwell proposed there was an extra displacement current in the region between the plates, equal to the current in the wire→

Modified Ampere’s law:


)(0 DIIdB p

Maxwell’s Displacement Current• But where does the “displacement current” come from?!p

Although there is no actual charge moving between the plates, nevertheless, something is changing – the electric field between them!nevertheless, something is changing the electric field between them!

• The Electric Field E between the plates of the capacitor is determined by the charge Q on the plate of area A:

E = Q/(A0) → Q = E A0• Because there is current flowing through the wire, there must be a

Recall def’n of flux:

change in the charge on the plates:

( ) EdQ d d EA dI EA I

dI EQdSEE

0

1

Recall def n of flux: 0 0 0 DI EA I

dt dt dt dt

dtI E

D0

Modified Ampere’s Law: 0 0 0EdB d I

˜


Modified Ampere s Law: 0 0 0 dt

Displacement current (J ):Displacement current (Jd):

M ll t l t d th t it i t l th t iMaxwell postulated that it is not only the current in aconductor that produces a magnetic field, but a changingelectric field in ac m or in a dielectric also prod ces aelectric field in vacuum or in a dielectric also produces amagnetic field. i.e. Changing electric field is equivalentto a current which flows as long as the electric field isto a current which flows as long as the electric field ischanging.

D

This equivalent current produces the same magnetic effect

Dt

This equivalent current produces the same magnetic effectas an ordinary current in a conductor. This equivalentcurrent is known as displacement current


current is known as displacement current.

Maxwell’s Displacement CurrentMaxwell’s Displacement Current

If we are charging a capacitor, there is a current left and right of the capacitor

EEBB BB

left and right of the capacitor.

Thus, there is the same magnetic field right andThus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires.

But no magnetic field inside the capacitor?

With a compass, we can verify there is indeed a The missing magnetic field, equal to the field elsewhere.

B t th i t d i it! ?

The missingMaxwellEquation!


But there is no current producing it! ?

Maxwell’s FixEE

We calculate the magnetic field produced by i d/dt g p ythe currents at left and at right using Ampere’s law :

idsB

id=0d/dt

C

idsB 0

We can write the current as:

dEAdEddAdVCCVddqi E 0 )()()(

dtdtdtddtC

dtdti 00

CV


q=CV V=EdC=0A/d E=E•dA=EA

Displacement CurrentDisplacement CurrentMaxwell proposed it based on symmetry and math — no experiment! dsB 0

dAEddB

and math no experiment!C

SC

dAEdt

dsB 00

B ! BB

E

ii

E


Ampere’s Law (constant currents):

0. encB dl I

Ampere’s Law for constant currents.

What about currents which are not continuous? Displacement current in a capacitorDisplacement current in a capacitor

The capacitor holds a charge Q over the two plates. How can h b f

E‐field increasing as Q increases!there be a current emerging from the capacitor plates?

I I

+Q charge deposits on the plate

‐Q charge induced by E‐field


Problems with Ampere’s Law

o enclCB d I ?

B 2 I

oB 2 r I

oIB

B

2 r


But what ifBut what if…..

o enclCB d I ?

B 2 0

oB 2 r 0

B 0

?????


Maxwell’s correction to Ampere’s Law

Q CV

oAC V Ed

d

A oo o E

AQ Ed AEd

d E

odQ dI

Called “displacement current”, Id


oIdt dt

p , d

Documents

Magnetic inductance Solenoids - folk.uio.nofolk.uio.no/ravi/cutn/elec_mag/15-eddy-displacement.pdf · Units of Magnetic Field • The SI unit of magnetic field is the tesla (T) Wb