Maxwells Equations

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Maxwells Equations

• We have been examining a variety of electrical
  and magnetic phenomena
• James Clerk Maxwell summarized all of electricity
  and magnetism in just four equations
• Remarkably, the equations predict the existence
  of electromagnetic waves

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Maxwells Equations
As you can see we need some vector calculus to
use these equations. That isnt going to happen
in this class, but I wanted you to see the
equations anyway.
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Maxwells Equations

• The first is Gausss Law which is an extended
  form of Coulombs Law
• The second is the equivalent for magnetic fields,
  except that we know that magnetic poles always
  occur in pairs (north south)

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Maxwells Equations

• The third is Faradays Law that a changing
  magnetic field produces an electric field
• The fourth is that a changing electric field
  produces a magnetic field
• The latter is a bit of a stretch. We knew that a
  current produces a magnetic field

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Maxwells Equations

• Start with Amperes Law

Earlier, we just went on a closed path enclosing
surface 1. But according to Amperes Law, we
could have considered surface 2. The current
enclosed is the same as for surface 1. We can
say that the current flowing into any volume must
equal that coming out.
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Maxwells Equations

• Suppose we have a charged capacitor and it begins
  to discharge

Surface 1 works but surface 2 has no current
passing through the surface yet there is a
magnetic field inside the surface.
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Maxwells Equations
Same problem here. Surface 1 works, but no
current passes through surface two which encloses
a magnetic field. What is happening???
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Maxwells Equations

• While the capacitor is discharging, a current
  flows
• The electric field between the plates of the
  capacitor is decreasing as current flows
• Maxwell said the changing electric field is
  equivalent to a current
• He called it the displacement current

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Maxwells Equations
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Electromagnetic Waves

• So, a magnetic field will be produced in space if
  there is a changing electric field
• But, this magnetic field is changing since the
  electric field is changing
• A changing magnetic field produces an electric
  field that is also changing
• We have a self-perpetuating system

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Electromagnetic Waves
Close switch and current flows briefly. Sets up
electric field. Current flow sets up magnetic
field as little circles around the wires. Fields
not instantaneous, but form in time. Energy is
stored in fields and cannot move infinitely fast.
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Electromagnetic Waves
Picture a shows first half cycle. When current
reverses in picture b, the fields reverse. See
the first disturbance moving outward. These are
the electromagnetic waves.
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Electromagnetic Waves
Notice that the electric and magnetic fields are
at right angles to one another! They are also
perpendicular to the direction of motion of the
wave.
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Speed of EM Waves

• Now that we have shown how the waves are formed
  from oscillating charges, we need to see if we
  can predict how fast the move
• We move far away from the source so that the wave
  fronts are essentially flat
• Just like dropping a rock in a pond and looking
  at the waves a few hundred feet away from the
  impact point

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Speed of EM Waves
This picture defines the coordinate system we
will use in our discussion. Wave propagates
along the x-axis. The electric field varies in
the y-direction and the magnetic field in the
z-direction.
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Speed of EM Waves
We are going to apply Faradays Law to the
imaginary moving rectangle abcd. Compute the
magnetic flux change
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Speed of EM Waves

• We can say the emf around the loop is the sum of
  the individual emfs going along each straight
  line segment in the loop
• We look at the work done in moving a test charge
  around the loop
• emf W/q Fd/q Ed
• emf Ey0 By0v
• E Bv

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Speed of EM Waves
Now we are going to look at the change in
electric flux. Set a new imaginary rectangle and
play the same game as before.
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Speed of EM Waves