1

Discussion about the mid-term

4. A high voltage generator is made of a metal sphere with a radius of 6 cm sits on an insulating post. A wire connects to the sphere’s inner surface through a small hole. The wire carriers a current of 1 μA and flows into the sphere when the switch is closed. One (1) minute after the switch is closed, what is the electric potential at the surface of the surface assuming potential far away from the sphere is zero.

I = 1 μA

Q μC60sec60μA1 tIQ

(V)10998(V)106106010998 6

2

69

..rQkU

2

Discussion about the mid-term

7. Twisted wire pairs are used to reduce interference generated from signal transmission. In each twisted wire pair, the current is the same in magnitude but flows the opposite direction. Use Ampere’s Law to prove that the magnetic field generated by this twisted wire pair is zero.

Constructed an arbitrary Ampere loop around the twisted wire pair.

From Ampere’s Law because the current sum from the two wires I = 0.

Because the Ampere loop is arbitrary, only satisfies the condition that

00 IldB

0B

0ldB

3

Discussion about the mid-term

11. An electric pulse of 5V with duration of 1 second is applied at point A and B. calculate the potential difference of C and D (VCD) as a function of time.

4

Maxwell’s equations and the EM wave

Maxwell’s equations:

0enclosed

surfaceenclosed

qAdE

dtdldEemf B

loopenclosed

0surfaceenclosed

AdB

dtdIldB E

loopenclosed

000

0

E

tBE

0 B

tEIB

000

Gauss’s Law for magnetic flux

Integral form Differential form

Gauss’s Law

Faraday’s Law of induction

From

Ampere’s Law with Maxwell’s correction

5

Maxwell’s equations and the EM wave

Maxwell’s equations:

0enclosed

surfaceenclosed

qAdE

dtdldEemf B

loopenclosed

0surfaceenclosed

AdB

dtdIldB E

loopenclosed

000

Gauss’s Law for magnetic flux

Integral form picture

Gauss’s Law

Faraday’s Law of induction

From

Ampere’s Law with Maxwell’s correction

6

Maxwell’s equations and the EM wave

Gauss’s Law for magnetic flux:

0surfaceenclosed

AdB

Since magnetic field lines are loops, no matter how the Gaussian surface is constructed, there are always equal number of lines come into and leave the surface. So the magnetic flux on an enclosed surface is always 0:

This is also to say that magnetic monopole has not been discovered.

0 surfaceenclosedsurfaceenclosed

B AdB

7

Maxwell’s equations and the EM wave

Ampere’s Law with Maxwell’s correction:

dtdIldB E

loopenclosed

000

Ampere’s Law states that: IldBloopenclosed

0

Maxwell believes that there are two types of currents, the one we discussed already plus another one he calls “displacement current”:

8

Maxwell’s equations and the EM wave

Maxwell’s equations predicted electro-magnetic wave in vacuum:

dtdldE B

loopenclosed

dtdldB E

loopenclosed

00

Solve these two equations, one has:

Demo 35.2

9

Maxwell’s equations and the EM wave

The introduction of the physics constant: the speed of light in vacuum c.

Get a bit excitement of a theoretical physicist?

c.

.

BE

BE

max

max

m/sec1099863382

)m/N(C10858m/A)(T104

1

1

8

22127

00

cBE

BE

max

max 00

1

10

Maxwell’s equations and the EM wave

The electromagnetic spectrum: electromagnetic radiation ordered by frequency or wavelength

11

Maxwell’s equations and the EM wave

Electromagnetic wave carries energy with it and travel through vacuum (without a medium). This is why we get the energy from the Sun.

The Poynting vector, which is the instantaneous area power density of the EM wave:

Example problem:An electromagnetic wave in vacuum has an electric field amplitude of 417 V/m. What is the maximum energy density per unit area of this EM wave?

12

Maxwell’s equations and the EM wave

With the instantaneous area power density is known as

And the wave functions:

The average value of S over time gives the intensity:

Unit is the same as S: Watts/m2

This is also what you feel under the Sun.

2

2222

0

2

0

2

0

2

/EE

cBc

Ec

ESI

maxrms

maxrmsmaxaverage

What are the maximum field strengths E

max and Bmax for the light emanating from the white square above?