W13D1Displacement Current, Maxwells

Equations,Wave Equations

Todays Reading Course Notes Sections 13.1-13.4

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

Outline

- Maxwells Equations
- Displacement Current

Maxwells Equations

Is there something missing?

4

Maxwells EquationsOne Last Modification

Displacement Current Displacement

Currenthas nothing to do with displacement and

nothing to do with current

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?

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

Displacement Current

If surface S2 encloses all of the electric flux

due to the charged plate then Idis I

Maxwell-Amperes Law

flow of electric charge

changing electric flux

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.

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.

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.

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.

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
- Cant tell (need to know direction of E)

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.

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.

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.

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

Maxwells Equations

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)

Electromagnetism Review

- Conservation of charge
- E and B fields exert forces on (moving) electric

charges - Energy stored in electric and magnetic fields

Maxwells Equationsin Vacua

Maxwells Equations

What about free space (no charge or current)?

How Do Maxwells Equations Lead to EM Waves?

Wave Equation

Start with Ampere-Maxwell Eq and closed oriented

loop

Wave Equation

Start with Ampere-Maxwell Eq

Apply it to red rectangle

So in the limit that dx is very small

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

Group Problem Wave Equation Sol.

Use Faradays Law

and apply it to red rectangle

So in the limit that dx is very small

1D Wave Equation for Electric Field

Take x-derivative of Eq.(1) and use the Eq. (2)

1D Wave Equation for E

This is an equation for a wave. Let

Definition of Constants and Wave Speed

Recall exact definitions of

The permittivity of free space is exactly

defined by

Group Problem 1D Wave Eq. for B

Take appropriate derivatives of the above

equations and show that

Wave Equations Summary

Both electric magnetic fields travel like waves

with speed

But there are strict relations between them

Electromagnetic Waves

Electromagnetic Radiation Plane Waves

http//youtu.be/3IvZF_LXzcc