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
b
V  Va  Vb   E  dl
a
The EMF induced in this circuit is equal
to the work done per unit charge by the
electric field around a closed path
   E  dl
d B
from    N
dt
d B
then  E  dl   N
dt
dl
b
The General Form of Faraday’s Law
d B
 E  dl   N dt
• 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
E-Field Produced by a Changing B-Field
dB
• Inside the magnet (r < r0) E  r
dt
r02 dB
• Outside the magnet (r > r0) E 
2r dt
1
2
E-Field Produced by a Changing BField
• Inside the magnet, the electric field increases linearly
from zero (at the center) to
at the edge
1
2
dB
r0
dt
• Outside the magnet, the electric field decreases inversely
with the radial distance, beyond the edge of the magnetic
field
r02 dB
2r dt
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:
b
V  Vb  Va    E  dl
a
• If the path is closed, then points a & b are the same
points and:  E  dl  0 because these points are at
a
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
The Non-static Electric Force is a nonConservative 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.
d B
 E  dl   dt
• 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 !