# W13D1: Displacement Current, Maxwell’s Equations, Wave Equations

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W13D1: Displacement Current, Maxwell’s Equations, Wave Equations
Today’s Reading Course Notes: Sections Class 18

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 or Next Reading Assignment W13D2 Course Notes: Sections Class 28

Outline Maxwell’s Equations Displacement Current Class 28

Maxwell’s Equations Is there something missing? Week 13, Day 1 4
Class 30 4

Maxwell’s Equations One Last Modification: Displacement Current “Displacement Current” has nothing to do with displacement and nothing to do with current Class 28

Ampere’s Law: Capacitor
Consider a charging capacitor: Use Ampere’s Law to calculate the magnetic field just above the top plate 1) Surface S1: Ienc= I 2) Surface S2: Ienc = 0 What’s Going On? Class 28

Displacement Current We 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 Class 28

Displacement Current:
If surface S2 encloses all of the electric flux due to the charged plate then Idis = I Class 28

Maxwell-Ampere’s Law “flow of electric charge”
“changing electric flux” Class 28

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 the same. larger. smaller. Class 28

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. Class 28

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. Class 20

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. Class 20

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 Zero (No current through loop) Positive Negative Can’t tell (need to know direction of E) Class 28

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. Class 28

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: Increasing in time Constant in time. Decreasing in time. Class 28

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. Class 28

Group Problem: Capacitor
Week 13, Day 2 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 Find the electric field E(t) at P vs. time t (mag. & dir.) Find the magnetic field B(t) at P Class 31

Week 13, Day 1 Maxwell’s Equations Class 30

Electromagnetism Review
E 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) Class 30

Electromagnetism Review
Conservation of charge: E and B fields exert forces on (moving) electric charges: Energy stored in electric and magnetic fields Class 30

Maxwell’s Equations in Vacua
Class 30

Maxwell’s Equations What about free space (no charge or current)?
What about free space (no charge or current)? Class 30

How Do Maxwell’s Equations Lead to EM Waves?
Class 30

Class 30

So in the limit that dx is very small: Class 30

Group Problem: Wave Equation
Use Faraday’s Law and apply it to red rectangle to find the partial differential equation in order to find a relationship between Class 30

Group Problem: Wave Equation Sol.
Use Faraday’s Law: and apply it to red rectangle: So in the limit that dx is very small: Class 30

1D Wave Equation for Electric Field
Take x-derivative of Eq.(1) and use the Eq. (2) Class 30

1D Wave Equation for E This is an equation for a wave. Let Class 30

Definition of Constants and Wave Speed
Week 13, Day 2 Definition of Constants and Wave Speed Recall exact definitions of The permittivity of free space is exactly defined by Class 31

Group Problem: 1D Wave Eq. for B
Take appropriate derivatives of the above equations and show that Class 30

Wave Equations: Summary
Both electric & magnetic fields travel like waves: with speed But there are strict relations between them: Class 30

Electromagnetic Waves
Class 30