welcome to physics 2112 · electricity & magnetism lecture 22, slide 37 consider a point (x,y,z) at...

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2112

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Electricity & Magnetism Lecture 22, Slide 1

Physics 2112

Unit 22

Outline

➢ Displacement Current

➢ Maxwell’s Equations (Final Form)

➢ E&M Waves


Your Comments


“

AmazingThis pre-lecture was a lot more fun then a lot of the stuff we have learned over the last week of schoolI really liked how the checkpoint questions were not well explained by the prelecture. In seriousness, the derivation of the DE for the wave solution was done at breakneck speeds and was impossible for me to follow.Pretty fascinating prelecture... if you just make yourself believe everything that's going on and accept the last conclusion. Couldn't really understand much in the middle after the displacement current section.I think that the after doing a few problems it will become clearer but the relationship between magnetic fields and electric fields is hard to visualize.Can you explain the graph for electromagnetic waves.why are integrals everywhere in this course ?!

Where we are now

=•surface

AdB 0

ENCLo

loop

IldB =•

•−=• AdBdtd

ldEloop

o

enc

surface

QAdE

=•

Our equations so far…..

Unit 22, Slide 4

Gauss’ Law

Gauss’ Law for B fieldAmpere’s Law

Faraday’s Law

dt

d B−=

3 Points, all a distance r from axis of a current c

carrying wire connected to capacitor

Displacement Current


●

r●

I1

1 3●

2

rIBB o 2/|||| 31 ==

??0|| 2 =B

)( disENCLoloop

IIldB +=• Define

“displacement current”

such that:

Modify Ampere’s Law


A

QE

00

==

0

QEAE ==

EQ = 0

dt

d

td

dQ E= 0

DItd

dQDefine:

Real Current: Charge Q passes through area A in time t:

Displacement Current: Electric flux through area A changes in time

Displacement Current


dt

dI ED

= 0

td

dQI =

)(dt

dIldB EoENCLo

loop

+=•

A parallel plate capacitor has plates that are 2cm in diameter and 1mm apart. If the current into the capacitor is 0.5A, what is the magnetic field between the plates 0.5cm from the axis of the center of the plates?

R

r●

I1

Q1

d

Conceptual PlanUse modified Ampere’s Law

Strategic PlanFind electric flux contained within circle with radius of 0.5cmFind time rate of change of that flux

Example 22.1


1cm

0.5cm

R

r●

I1

Q1

d

To calculate the magnetic field using the displacement current, I’m going to have the find the E flux over a certain area and the circumference of a particular circle. To find the B field at r = 0.5cm case, what is the area and what is that circle?

a) area of a circle with r=0.5cm, circumference of a circle with r = 0.5cm

b) area of a circle with r=1.0cm, circumference of a circle with r = 0.5cm

c) area of a circle with r=0.5cm, circumference of a circle with r = 1.0cm

Question


1cm

0.5cm

A parallel plate capacitor has plates that are 2cm in diameter and 1mm apart. If the current into the capacitor is 0.5A, what is the magnetic field between the plates 3.0 cm from the axis of the center of the plates?

R

r

●

I1

Q1

Conceptual PlanUse modified Ampere’s Law

Strategic PlanFind electric flux contained within circle with radius of 0.5cmFind time rate of change of that flux

Example 22.1(b)


1cm

3cm

To calculate the magnetic field using the displacement current, I’m going to have the find the E flux over a certain area and the circumference of a particular circle. To find the B field at r = 3.0cm case, what is the area and what is that circle?

a) area of a circle with r=3cm, circumference of a circle with r = 3cm

b) area of a circle with r=1.0cm, circumference of a circle with r = 3cm

c) area of a circle with r=3.0cm, circumference of a circle with r = 1.0cm

Question


CheckPoint 1(A)


At time t = 0 the switch in the circuit shown below is closed.

Points A and B lie inside the capacitor; A is at the center and

B is at the outer edge..

A

After the switch is closed, there will

be a magnetic field at point A which

is proportional to the current in the

circuit.

A. True

B. False

CheckPoint 1(A)


B is proportional to Ibut

At A, B = 0 !!

2

10

2 R

rIB

=

At time t = 0 the switch in the circuit shown below is closed.

Points A and B lie inside the capacitor; A is at the center and

B is at the outer edge..

A

After the switch is closed, there will

be a magnetic field at point A which

is proportional to the current in the

circuit.

A. True

B. False

Switch S has been open a long time when at t =0, it is closed. Capacitor C has circular plates of radius R. At time t = t1, a current I1 flows in the circuit and the capacitor carries charge Q1.

What is the time dependence of the magnetic field B at a radius r between the plates of the capacitor?

0 11 22

I rB

R

=

(A) (B) (C)

Follow-Up


S

Ra

V C

A B C

Final form

=•surface

AdB 0

•+=• AdEdtd

IldB oENCLoloop

•−=• AdBdtd

ldEloop

o

enc

surface

QAdE

=•

Ta da!..... Maxwell’s Equations

Unit 22, Slide 19

Gauss’ Law

Gauss’ Law for B fieldAmpere’s Law

Faraday’s Law

Wave Equation


Remember this guy? Not the spring constant!

Remember from 2111??


++

--

•−=• AdBdtd

ldEloop

Bhdxdt

dhEd )*(* −=

dx

Bd

dt

d

dx

Ed

−=2

2

•=• AdEdtd

ldB ooloop

o o

dB dE

dx dt = −

Some Calculations

dt

dB

dx

Ed−=

2 2

2 2o o

d E d E

dx dt =

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see PHYS 2115


2 2

2 2o o

d E d E

dx dt =

A wave equation????

oo

v

1=

With a velocity of …?

Some Calculations


The Sun

Keep us warm Keep us safe

High Energy, Low Wavelength Low Energy, Long Wavelength

What makes it through the atmosphere?

Unit 15 Climate Science - Slide 28


Example 22.2

An electromagnetic plane wave has a wavelength of

0.100nm.

a) What is its wave number, k?

b) What is its frequency?

c) What portion of the electro-magnetic spectrum

does it fall in?


Past Confusion

Nothing is moving here.

Arrows only represent strength of field.

Actually a plane wave.

CheckPoint 2(A)


Ex = E0sin(kz − wt)

E = E0 sin (kz − wt):E depends only on z coordinate for constant t. z coordinate is same for A, B, C.

An electromagnetic plane wave is traveling in the +z direction. The

illustration below shows this wave an some instant in time. Points A, B, and

C have the same z coordinate.

Compare the magnitudes of the electric field at points A and B.

A. Ea < EbB. Ea = EbC. Ea > Eb

CheckPoint 2(A)







Compare the magnitudes of the electric field at points A and B.

A. Ea < EbB. Ea = EbC. Ea > Eb

CheckPoint 2(B)







Compare the magnitudes of the electric field at points A and C.

A. Ea < EcB. Ea = EcC. Ea > Ec

Question


Consider a point (x,y,z) at time t when Ex is negative and has its maximum value.

At (x,y,z) at time t, what is By?

A) By is positive and has its maximum valueB) By is negative and has its maximum valueC) By is zeroD) We do not have enough information


welcome to physics 2112 · electricity & magnetism lecture 22, slide 37 consider a point (x,y,z) at...

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