Today’s lecture
VII. Electrodynamics
•
•
•
•
Electromotive force
Motional emf
examples
Electromagnetic induction
Joule heating law
Electric field E does work on the moving charges,
The work does not increase the kinetic energy of the charges
(constant currents⇔constant velocity),
What is the energy used for?
Work of the electric field on the charges is converted into
heat in the resistor through collisions in the material.
The power delivered is given by the Joule heating law:
  
P = I ∫ E ( r ' ) ⋅ dl ' =I ⋅ ΔV = I 2 ⋅ R
high V
low V
P =V ⋅ I = I2 ⋅ R
Electromotive force, emf
Two fields (or forces) are involved in driving a current
through a wire: The known electrostatic field E and the
  
field (force) of the source fs :
f = fs + E
This could be a chemical force (battery), mechanical
pressure (piezo), light (photodiode), or a temperature
gradient (thermocouple).
   
Through battery: ε := ∫ f ⋅ dl = ∫ f s ⋅ dl
  
 
since ∫ E ⋅ dl = ∫∫ ∇ × E ⋅ da = 0
  P  P 
ε = ∫ f s ⋅ dl = ∫ f s ⋅ dl = − ∫ E ⋅ dl
(
N
)

E
N
=V = I ⋅ R
• An electrostatic field E cannot
maintain a steady current
  in a closed
circuit since: ∫ E ⋅ dl = I ⋅ R = 0

E
+  −
fs
• An emf-produced field is not conservative.
• According to Ohm’s law the emf results in a current.
Electromotive force, emf
Faraday discovered 1831 the following:
A short current is induced in a conducting loop when
•
The current in a neighboring loop is switched on or off,
•
The neighboring loop is moved relative to the 1st one,
•
A permanent magnet is brought/removed into/from the
loop
Faraday concluded this would be due to the change in
magnetic flux. This change induces an electric field (along
the loop) and its line-integral is called electromotive force
 
ε := ∫ f s ⋅ dl
The name is given for historical reasons only.
Remember: emf is not a force but a line integral over an
electric field!
Therefore the electromotive force is really a potential!
Battery
Example: (Problem 7.5)
A battery of emf ε and internal resistance r is hooked up to a
variable “load” resistance R.
If you want to deliver the maximum power to the load,
what resistance R should you choose? (ε and r are fixed)?
Motional emf
What happens when you move a wire through a magnetic
field?


ε = ∫ f mag ⋅ dl = vBh
 
φ = ∫∫ B ⋅ da = Bhx

B ×
dφ
dx
= Bh
= − Bhv = −ε
dt
dt
Flux rule for motional emf
h
x
ε =−
R

v
dφ
dt
The flux rule is a shortcut in order to calculate the emf.
It is applicable to arbitrary (non-rectangular!) loops in
arbitrary directions even if the magnetic field is not uniform.
Electromagnetic induction
Again Faraday’s experiments from 1831 :A current is
induced in a conducting loop when
•
The wire is moved
•
The magnet is moved
•
The magnetic field is changed
The 1st case is covered by the flux rule.
The 2nd case: How to describe the observation when the
loop does not move?
Faraday concluded the force could not be magnetic but had
to be an electric field:
A changing magnetic field induces an electric field.
It is this induced field that accounts for the emf in case 2.
In the 3rd case the magnetic field changes and induces an
emf (according to the flux rule.)



dφ
δB 
ε = − = ∫ E ⋅ dl = − ∫∫
⋅ da
dt
δt
Faraday’s law

 
δB
∇× E = −
δt
Historical facts
1785: Coulomb formulates the Coulomb law
Gauss (1777-1855) is the first physicist to measured electric
and magnetic quantities in absolute units.
1820: Oersted discovered (after 13 years) the link between
electric and magnetic fields: If the charges are moving with
constant speed, a static magnetic field is produced.
Ampère developed Oersted’s discovery and introduced the
concept of current element and the force between them.
Shortly thereafter, Baptiste and Savart carried out the
experiments and so did Ampère.
1831: Faraday: a time-varying field produces an induced
Voltage (Faraday’s law)
Lenz’s Law
Lenz’s law helps to keep track of the signs in Faraday’s law.
Lenz law states: Nature opposes a change in flux.
Problem 7.14:
As a lecture demonstration a short cylindrical bar magnet is
dropped down a vertical aluminum pipe of slightly larger
diameter, about 2 meters long. It takes several seconds to
emerge at the bottom, whereas an otherwise identical piece
of unmagnetized iron makes the trip in a fraction of a second.
Explain why the magnet falls more slowly.