Summary of electrostatics and magnetostatics

1
Summary of electrostatics and magnetostatics

• summarizing everything we have so far in the
  static case
• but what happens if something changes in time???

2
What happens if things are not constant in time
Faradays law

• Faradays law an electromotive force (emf) is
  induced in a loop if the magnetic flux through
  the loop changes in time
• this emf is an induced voltage
• recall we get a voltage by doing a path integral
  of an electric field
• magnitude of induced voltage is proportional to
  how fast the flux through the loop changes in
  time
• the sign of the induced voltage is such that it
  would induce a current in the loop that would
  produce a flux that opposes the change in the
  flux through the loop (Lenzs law)
• i.e., the induced current tries to keep the
  flux from changing
• in integral form Faradays law is
• the time variation could be due to either
  movement of the path or time variations in the
  magnetic field B

3
What happens if things are not constant in time
Faradays law

• Faradays law the sign of the induced voltage is
  such that it would induce a current in the loop
  that opposes the change in the flux through the
  loop (Lenzs law)
• the sign convention works in such a way that if
  you go around the path in a given direction, the
  surface normal direction is set by the right hand
  rule

4
Spark plugs and Faradays law

• example the firing of a spark plug
• you need about 40,000 V to set of the spark in a
  spark plug, but youre battery is only 12 V dc
• use a step-up transformer and a switch
• from http//hyperphysics.phy-astr.gsu.edu/hbase/m
  agnetic/ignition.htmlc1

5
Refs

• http//hyperphysics.phy-astr.gsu.edu/hbase/electri
  c/farlaw.html
• applet http//www.phas.ucalgary.ca/physlets/farad
  ay.htm

6
Differential form of Faradays law

• lets assume the path does not move in space, but
  the magnetic field is changing in time, so
• but recall Stokes theorem
• so now we have

7
Statics check of differential form of Faradays
law

• if this was an electrostatics problem, the
  voltage around a closed path is zero since the
  electric field is conservative
• which is consistent with the result from the
  differential form of Faradays law

8
Is there any problem with Amperes law under
non-static conditions?

• lets look at a simple circuit with a capacitor
  in it, and apply Amperes law
• here everything seems ok, since there could be
  conduction current in the wire

closed path C
surface bounded by C
9
Is there any problem with Amperes law under
non-static conditions?

• but what if we draw a different surface, with the
  same bounding path?
• the path integral of H should still be exactly
  the same, but where is the current???
• from circuits, we already know that there is
  current flow through a capacitor, but how is it
  represented using field concepts?

closed path C
surface bounded by C
10
Modifying Amperes law for non-static conditions

• Amperes law in differential form, statics
• lets take the divergence of both sides
• but by vector identity, div?curl is ALWAYS
  identically zero, so
• BUT we talked about the divergence of the current
  density before, and got the continuity equation
• this came from the idea that the spatial rate of
  change in current density had to balance the time
  rate of change in charge concentration

11
Amperes law under non-static conditions

• so under static conditions, combining, we have
• this is fine for statics, but what do we do under
  non-static conditions?
• lets try modifying Amperes law by adding an
  unknown vector G
• to try and find G to make this work, lets take
  the divergence of both sides, and use the fact
  that div?curl 0
• but we also know that the free charge is the
  divergence of D

12
Amperes law under non-static conditions

• so under non-static conditions we have
• or
• then combining we have
• the second term is called the displacement
  current density

13
Physical picture of displacement current

• we should take a closer look at the term we just
  added to the curl equation for H
• lets look at a simple circuit with a capacitor
  in it, and apply Amperes law

14
Displacement current
closed path C

• in this example, for the surface on the left,
  clearly the current is conduction current
• lets get the current from circuit analysis,
  assuming a voltage source driving the circuit
• source Vocos(wt)
• then the current, assuming we can ignore any
  inductance or resistance in the loop

15
Displacement current

• now for the surface on the right, the current
  must be from the displacement term
• so the displacement current is the same as the
  conduction current
• this also suggests that the current is the same
  everywhere in the circuit!!! CURRENT CONTINUITY!

16
Summary of electromagnetics Maxwells equations

• summarizing everything we have so far, valid even
  if things are changing in time
• plus material properties

Gausss law
Faradays law
Amperes law
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Maxwells equations in integral form

• we can also convert to integral form E gives
  Faradays law
• H gives Amperes law

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Maxwell in integral form

• Gausss law from

19
Maxwells equations in integral form

• Faradays law
• Amperes law
• Gausss law(s)

20
Solutions to Maxwells equations

• lets try a simple guess at a solution
• time harmonic solutions
• lets substitute into Maxwells equations, and
  see if this could work
• left hand side of equation

21
Solutions to Maxwells equations

• right hand side of curl E equation
• now substitute
• if the time independent forms Es and Hs satisfy
  this equation, they will satisfy Maxwell

22
Solutions to Maxwells equations

• lets keep going
• substitute into next Maxwell equation
• left hand side of equation
• left side

notice I took the material properties out of the
time derivative
23
Maxwells equations, time harmonic form

• so assuming that none of the materials change in
  time, time independent fields Es and Hs that
  satisfy
• when multiplied by exp(jwt), the product will
  satisfy the full time dependent set of Maxwell
  equations
• so now what???
• can we use these equations to predict/understand
  any other phenomena?
• lets try to solve this set of coupled partial
  differential equations