**Homework set #10 (due 12/3/02)**

**Prove that the expressions in eq. (7.28) are identical to those given in eq. (7.27).****problem 7.1.****problem 7.2.**

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**Use of a pile of plane-parallel plates to produce polarized light:**

**Show that the ratio of the
transmission coefficient for
perpendicular vs the transmission coefficent for parallel light upon passing
the surface once is given by cos ^{2}(_{}), where the two angles are the incident and the refraction
angles, resepctively. Show that if such
a light passes thru a thin film, then the ratio above becomes cos^{4}(_{}).**

If you have five such thin films stack together, and the incident angle is at the Brewster angle, show that this ratio is about 0.2 if n=1.5. (n=1 for the air.)

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**In this exercise you calculate the phase change of the reflected wave under the total internal reflection condition.**

**Let a plane wave travelling in a
medium with index of refraction _{} is reflected from a
medium with index of refraction _{} where _{}. Show that for incident angle _{} greater than the
critical angle, the phase change for the an incident wave where the electric
field is parallel to the plane of incidence is**

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_{ ,}

_{ }

and for electric field perpendicular to the plane of incidence the phase change is

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** _{}**

_{ }

_{ } where n=_{}. Aslo show that

**
_{}**

**This shows that one can use total internal reflection to
change a plane wave into a circularly polarized wave.**

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**Homework set #9 (due 11/21/02)**

1.
In this exercise you will calculate the pressure due to a
plane wave when it is reflected from a planar surface. Consider the case
that the electric field of the plane wave is polarized parallel to the plane of
incidence. The incident angle is _{}.

(a) the elementary method.

Calculate the momentum of the
incident plane wave. Calculate the change of momentum when the incident wave is
reflected. From this calculate the pressure.

(b) Use the formula involving the Maxwell stress tensor.

Calculate the electric and
magnetic field near the planar surface. Then follow the definition to obtain
the pressure which is the diagonal component of the Maxwell stress tensor.

2.
Calculate the total force due to the radiation
pressure on a plane which is perpendicular to the direction of the
sunlight. The plane has an area of 1 km^{2} and the intensity of the sun at the surface
of Earth is 1350 W/m^{2}. Make sure that you use correct numbers
(depending on the units used) in the calculation.

3. A Ti.Sapphire
laser has a peak intensity of 10^{20} W/cm^{2}, calculate the
peak electric field in units of MV/m. What is the electric field on an electron
which is at 1 Bohr radius from the center of the hydrogen atom? The intensity of the sun at the surface of
Earth is 1350 W/m^{2}. What is the total power received on the surface
of Earth, and compare the number with the total power received in 1 cm^{2}
of the Ti:Sapphire laser.

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**Homework set #8 (due 11/14/02)**

** 1. **A current I
flows through a resistor R in the form of a long

straight wire. Show that the poynting vector flows radially
inward through

the surface of the wire with the correct magnitude to
produce the Joule heating.

2. This
problem is adopted from the book of Heald and Marion, 4-12 and 4-13. It

is a good exercise of manipulating the time-dependent Maxwell equations.

Consider a
parallel-plate capacitor consisting of two circular plates. The radius of

the plate is a and the plate separation is h and the medium is filled with a
dielectric which

has dielectric constant _{}. The capacitor is charged by connecting to a battery

with EMF V_{0} and a series
resistor R. If the circuit is closed at t=0 find the following

quantities within the capacitor as a function of time. Neglect the edge effect.

(a) The electric field.

(b) the magnetic field.

(c) the Poynting vector.

(d) the total field energy.

(f) the scalar potential.

(g) the vector potential

3. Continue problem 3, but now
assume that the material between the plates have

conductivity _{} in addition to being
a dielectric. Forget the resistor in problem

w2. This capacitor is charged to a potential V_{0} by a battery and
then disconnected at

t=0.

(a) Find the free charge on the capacitor as a function of time.

(b) Find the conduction current, the displacement current density.

(c) Find the magnetic field H within the capacitor,

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**Homework set #7 due (11/5/02)**

**5.3**

**5.6**

**5.13**

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**Homework set #6 due (10/10/02)**

**problem 4.2****problem 4.6****problem 4.8**

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**Homework set #5 due (10/1/02)**

**1.
****Problem 3.17**

**2.
****Problem 3.22**

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**Homework set #4 due (9/24/02)**

**Problem 3.2. Part (a)**

**Hint: treat the problem as a
uniformly charged sphere plus a negative charge on the northern cap. Then
consider the potential due to the northern cap using the results worked out
from the class. See example in Fig. 3.4.
You need to use eq. (3.28).**

**Problem 3.7.****problem 3.9—this is a simple exercise for you to set up the series solution in cyclindrical coordinates. Write down the general expression with the boundary conditions incorporated.**

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**Homework set #3 (due 9/17/02)**

**problem 2.2****problem 2.7**

3**. problem
2.13**

**Homework set #2 (due 9/10/02**)

- Problem 1.9. (consider only parallel plates case)
- Problem 1.10

The next three are just simple exercises.

- Derive eq. (2.5) of Jackson.
- Eq. (2.6) of Jackson shows that the force between charge q and the induced charges on the conducting sphere varies inversely with the cubic power of y. If the distance y of the charge q is now measured from the surface of the sphere, how is the force depends on y? Justify your result.
- Eq. (2.15) gives the induced surface charge, calculate the dipole moment from this surface charge and show that it is identical to the dipole moment due to the two image charges.

**Homework set 1:**

**Jackson pp50-52 (due
9/03/02)**

1.1; 1.3; 1.4; 1.5; 1.6a,b;