Faraday’s Law (continued)

What causes current to flow in wire?Answer: an E field in the wire.

A changing magnetic flux not only causesan EMF around a loop but an inducedelectric field.Can write Faraday’s Law:

dtdAdB

dtdldE BΦ−=∫ ⋅−=∫ ⋅=

rrrrε

Note: For electric fields from static charges, the voltage differencearound a closed path is always zero. Not true here! There are two sources for electric fields!

Remember for a long straight wire of length l, V = El.

N

S

Suppose we have electromagneticthat has an increasing magnetic fieldUsing Faraday’s Law we predict,

If we take a circular path inside andcentered on the magnet center axis, the electric field will be tangent to thecircle. (E field lines are circles.) NOTEsuch an E field can never be made bystatic charges

dtdAdB

dtdldE BΦ−=∫ ⋅−=∫ ⋅

rrrrB

E

E field lines will look like an onion slice (w/o smell!!)

Induced Electric Fields

NOTE there are no wire loops, the E fields can appear w/o loopsIf we place a loop there, a current would flow in the loop

Cylindrical symmetry here gives circularfield lines, but Faraday’s Law general.

B Field Inside a Long Wire• Suppose a total current I flows through

the wire of radius a into the screen as shown.

• Calculate B field as a function of r, the distance from the center of the wire.

x x x x xx x x x x x x x

x x x x x x x x xx x x x x x x x

x x x x x a

r

• B field is only a function of r ⇒ take path to be circle of radius r: ⇒ )π(2 rBldB =•∫

rr

• Current passing through circle: IarI 2

2

enclosed =

• Ampere's Law: enclosedoIμldB =•∫rr

⇒ 20

π2 arIμB =

B

• B field direction tangent to circles.

dl

Induced Electric Fields; example If we have a solenoid coil with changing current there will be circularelectric fields created outside the solenoid. It looks very much like themag. field around a current carrying wire, but it is an E field and there are no wires or loops.

E

dtdBArE

dtd

AdBdtdldE B

−=

Φ−=⋅−=⋅ ∫∫

)2( π

rrrr

Note the E fields are predicted by Faraday’s Law.

Eddy CurrentsChanging magnetic fields in metal induce eddy currents.

Example: Energy loss in transformers.To reduce, use laminations.

But eddy currents often useful.

Name Equation DescriptionGauss’ Law for

ElectricityCharge and

electric fields

Gauss’ Law for Magnetism

Magnetic fields

Faraday’s Law

Electrical effects from changing B

fieldAmpere’s Law Magnetic effects

from current

0=∫ ⋅ AdBrr

0εQAdE =∫ ⋅

rr

dtdldE BΦ−=∫ ⋅

rr

Maxwell’s Equations (integral form)

idlB 0μ=⋅∫r

There is a serious asymmetry.

Needs to be modified. + ?

cylinder square

00

=∫ ⋅

=∫ ⋅

AdB

QAdE enclosed

rr

rr

εGauss Law’s works for ANY CLOSED SURFACE

bagel

sphere

Remarks on Gauss Law’s with different closed surfaces

Surfaces forintegration of E flux

Fish bowl

dtAdBdldE ∫ ⋅

−=∫ ⋅rr

rr Faraday’s Law works for any closed Loop and ANYattached surface area

disk

Line integraldefines the Closed loop

This is proven in Vector Calculus with Stoke’s Theorem

cylinder

Remarks on Faraday’s Law with different attached surfaces

Surface areaintegration for B flux

Generalized Ampere’s Law and displacement current

Ampere’s original law, , is incomplete.Consider the parallel plate capacitor and suppose a current ic isflowing charging up the plate. If Ampere’s law is applied for thegiven path in either the plane surface or the bulging surface wewe should get the same results, but the bulging surface has ic=0,so something is missing.

encloseIdlB 0μ=∫ ⋅r

Generalized Ampere’s Law and displacement current

Maxwell solved dilemma by adding an addition term called displacement current, iD = ε dΦE/dt, in analogy to Faraday’s Law.

( ) ⎟⎠⎞

⎜⎝⎛ Φ

+=+=⋅∫ dtd

iiidlB EcDc 000 εμμ

r

( )

EEA

EddACVq

Φ==

=

=

εε

ε

dtdi

dtdq E

cΦ== ε

Current is once more continuous: iD between the plates = iC in the wire.

Summary of Faraday’s Law

dtdldE BΦ−=∫ ⋅

rr

If we form any closed loop, theline integral of the electric field equals the time rate change of magnetic flux through the surfaceenclosed by the loop.

If there is a changing magnetic field, then there will beelectric fields induced in closed paths. The electric fieldsdirection will tend to reduce the changing B field.

E

B

Note; it does not matter if there is a wire loop or an imaginaryclosed path, an E field will be induced. Potential has no meaning in this non-conservative E field.

Summary of Ampere’s Generalized Law

If we form any closed loop, the line integral of the B field is nonzero ifthere is (constant or changing) current through the loop.

If there is a changing electric fieldthrough the loop, then there will be magnetic fields induced about a closed loop path. B

E

⎟⎠⎞

⎜⎝⎛ Φ

+=∫ ⋅dtdidlB E

c 00 εμr

B

Current ic

James Clerk Maxwell (1831-1879)• generalized Ampere’s Law• made equations symmetric:

– a changing magnetic field produces an electric field– a changing electric field produces a magnetic field

• Showed that Maxwell’s equations predicted electromagnetic waves and c =1/√ε0μ0

• Unified electricity and magnetism and light.

Maxwell’s Equations

All of electricity and magnetism can be summarized by Maxwell’s Equations.