1 w13d1: displacement current, maxwell’s equations, wave equations today’s reading course notes:...
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W13D1:Displacement Current, Maxwell’s Equations,
Wave Equations
Today’s Reading Course Notes: Sections 13.1-13.4
AnnouncementsMath 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
Maxwell’s Equations
Displacement Current
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Maxwell’s Equations
Is there something missing?
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Maxwell’s EquationsOne Last Modification: Displacement Current
“Displacement Current”has nothing to do with
displacement and nothing to do with current
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Ampere’s Law: CapacitorConsider a charging capacitor:
Use Ampere’s Law to calculate the magnetic field just above the top plate
What’s Going On?1) Surface S1: Ienc= I
2) Surface S2: Ienc = 0
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Displacement CurrentWe don’t 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-Ampere’s Law
“flow of electric charge”
“changing electric flux”
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Concept Question: CapacitorIf 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
As we increase the radius of our Amperian loop we enclose more flux and hence the magnitude of the integral will increase.
Answer 2. The line integral of B is larger for larger r
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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. Positive3. Negative4. Can’t tell (need to know direction of E)
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Concept Q. Answer: CapacitorAnswer 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 time2. Constant in time.3. Decreasing in time.
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Concept Q. Answer: Capacitor
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.
Answer 1. The charge Q is increasing in time
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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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Maxwell’s Equations
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Electromagnetism ReviewE fields are associated with:
(1) electric charges (Gauss’s Law )
(2) time changing B fields (Faraday’s Law)
B fields are associated with(3a) moving electric charges (Ampere-Maxwell Law)
(3b) time changing E fields (Maxwell’s Addition (Ampere-Maxwell Law)
Conservation of magnetic flux
(4) No magnetic charge (Gauss’s 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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Maxwell’s Equationsin Vacua
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Maxwell’s Equations
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0
What about free space (no charge or current)?
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How Do Maxwell’s Equations Lead to EM Waves?
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Wave EquationStart with Ampere-Maxwell Eq and closed oriented loop
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Wave Equation
So in the limit that dx is very small:
Apply it to red rectangle:
Start with Ampere-Maxwell Eq:
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Group Problem: Wave EquationUse Faraday’s 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 Faraday’s Law:
So in the limit that dx is very small:
and apply it to red rectangle:
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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
