Day 5: General Form of Faraday’s Law

• How a changing magnetic Flux Produces an Electric Field

• Example of an E-Field is produced by a changing B-Field

• The General form of Faraday’s Law

• The Electrostatic (Coulombic) Force vs.

the Induced Electric Force

A Changing Flux Produces an Electric Field• When a current flows through a wire, there is an electric

field in the wire that does the work of moving the electrons in the wire

• The current moving through the wire produces a magnetic field

• Conversely, a changing magnetic flux induces a current in the wire which implies there is an electric field induced in the wire by the magnetic flux

A changing Magnetic Flux Produces an Electric Field

E-Field moving the current

Induced E-Field a

The EMF induced in this circuit is equal dl

to the work done per unit charge by the

electric field around a closed path

b

b

a

ba dlEVVV

dt

dNdlEthen

dt

dNfrom

dlE

B

B

The General Form of Faraday’s Law

• The integral is taken around a closed path enclosing the area through which the magnetic flux ΦB is changing

• This is a more elegant statement of Faraday’s Law and is valid not only in conductors but in any region of space

dt

dNdlE B

E-Field Produced by a Changing B-Field

• Inside the magnet (r < r0)

• Outside the magnet (r > r0)

dt

dBrE 2

1

dt

dB

r

rE

2

20

E-Field Produced by a Changing B-Field

• Inside the magnet, the electric field increases linearly from zero (at the center) to

at the edge

• Outside the magnet, the electric field decreases inversely with the radial distance, beyond the edge of the magnetic field

dt

dBr02

1

dt

dB

r

r

2

20

The Electrostatic Force is a Conservative Force

• The general form of faraday’s law is a closed path integral, and the electric field produced by electric charges at rest (electrostatic field) yields:

• If the path is closed, then points a & b are the same points and: because these points are at

the same potential (ΔVa-a=0)

• This follows from the fact that the electrostatic (Coulombic) force is a conservative force and that the work done per unit charge around any closed path = 0 & is independent of the path taken

b

a

ab dlEVVV

0 dlEa

The Non-static Electric Force is a non-Conservative Force

• But in the non-electrostatic case when the electric field is produced (induced) by a changing magnetic field, then the closed integral is not zero.

• Therefore, the conclusion is that the forces resulting from the changing magnetic fields are non-conservative and the induced electric field is a non-conservative field !

dt

ddlE B