W13D1: Displacement Current, Maxwell

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W13D1Displacement Current, Maxwells
Equations,Wave Equations
Todays Reading Course Notes Sections 13.1-13.4
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Announcements
Math Review Week 13 Tuesday 9pm-11 pm in
32-082 PS 9 due Week 13 Tuesday April 30 at 9 pm
in boxes outside 32-082 or 26-152 Next Reading
Assignment W13D2 Course Notes Sections 13.5-13.7

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Outline

• Maxwells Equations
• Displacement Current

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Maxwells Equations
Is there something missing?
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Maxwells EquationsOne Last Modification
Displacement Current Displacement
Currenthas nothing to do with displacement and
nothing to do with current
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Amperes Law Capacitor

• Consider a charging capacitor

Use Amperes Law to calculate the magnetic field
just above the top plate
1) Surface S1 Ienc I 2) Surface S2 Ienc 0
Whats Going On?
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Displacement Current
We dont have current between the capacitor
plates but we do have a changing E field. Can we
make a current out of that?
This is called the displacement current. It is
not a flow of charge but proportional to changing
electric flux
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Displacement Current
If surface S2 encloses all of the electric flux
due to the charged plate then Idis I
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Maxwell-Amperes Law
flow of electric charge
changing electric flux
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Concept Question Capacitor
If instead of integrating the magnetic field
around the pictured Amperian circular loop of
radius r we were to integrate around an Amperian
loop of the same radius R as the plates (b) then
the integral of the magnetic field around the
closed path would be

1. the same.
2. larger.
3. smaller.

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Concept Q. Answer Capacitor

Answer 2. The line integral of B is larger for
larger r
As we increase the radius of our Amperian loop we
enclose more flux and hence the magnitude of the
integral will increase.
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Sign Conventions Right Hand Rule
Integration direction clockwise for line integral
requires that unit normal points into page for
surface integral. Current positive into the page.
Negative out of page. Electric flux positive into
page, negative out of page.
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Sign Conventions Right Hand Rule
Integration direction counter clockwise for line
integral requires that unit normal points out
page for surface integral. Current positive out
of page. Negative into page. Electric flux
positive out of page, negative into page.
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Concept Question Capacitor
Consider a circular capacitor, with an Amperian
circular loop (radius r) in the plane midway
between the plates. When the capacitor is
charging, the line integral of the magnetic field
around the circle (in direction shown) is

1. Zero (No current through loop)
2. Positive
3. Negative
4. Cant tell (need to know direction of E)

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Concept Q. Answer Capacitor

Answer 2. The line integral of B is
positive. There is no enclosed current through
the disk. When integrating in the direction
shown, the electric flux is positive. Because the
plates are charging, the electric flux is
increasing. Therefore the line line integral is
positive.
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Concept Question Capacitor
The figures above show a side and top view of a
capacitor with charge Q and electric and magnetic
fields E and B at time t. At this time the
charge Q is

1. Increasing in time
2. Constant in time.
3. Decreasing in time.

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Concept Q. Answer Capacitor
Answer 1. The charge Q is increasing in time
The B field is counterclockwise, which means that
the if we choose counterclockwise circulation
direction, the electric flux must be increasing
in time. So positive charge is increasing on the
bottom plate.
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Group Problem Capacitor
A circular capacitor of spacing d and radius R is
in a circuit carrying the steady current i shown.
At time t 0 , the plates are uncharged

1. Find the electric field E(t) at P vs. time t
   (mag. dir.)
2. Find the magnetic field B(t) at P

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Maxwells Equations
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Electromagnetism Review
E fields are associated with (1) electric
charges (Gausss Law ) (2) time changing B
fields (Faradays Law) B fields are associated
with (3a) moving electric charges
(Ampere-Maxwell Law) (3b) time changing E fields
(Maxwells Addition (Ampere-Maxwell
Law) Conservation of magnetic flux (4) No
magnetic charge (Gausss Law for Magnetism)
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Electromagnetism Review

• Conservation of charge
• E and B fields exert forces on (moving) electric
  charges
• Energy stored in electric and magnetic fields

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Maxwells Equationsin Vacua
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Maxwells Equations
What about free space (no charge or current)?
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How Do Maxwells Equations Lead to EM Waves?
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Wave Equation
Start with Ampere-Maxwell Eq and closed oriented
loop

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Wave Equation
Start with Ampere-Maxwell Eq
Apply it to red rectangle
So in the limit that dx is very small
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Group Problem Wave Equation
Use Faradays Law and apply it to red rectangle
to find the partial differential equation in
order to find a relationship between

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Group Problem Wave Equation Sol.

Use Faradays Law
and apply it to red rectangle
So in the limit that dx is very small
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1D Wave Equation for Electric Field
Take x-derivative of Eq.(1) and use the Eq. (2)
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1D Wave Equation for E
This is an equation for a wave. Let
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Definition of Constants and Wave Speed
Recall exact definitions of
The permittivity of free space is exactly
defined by
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Group Problem 1D Wave Eq. for B
Take appropriate derivatives of the above
equations and show that
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Wave Equations Summary
Both electric magnetic fields travel like waves
with speed
But there are strict relations between them
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Electromagnetic Waves
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Electromagnetic Radiation Plane Waves

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