THE UNIVERSITY OF BRITISH COLUMBIA

*Physics 401 *
Assignment #
**6: **

** Electromagnetic Waves **

*SOLUTIONS:*

Wed. 8 Feb. 2006 - finish by Wed. 22 Feb.

Wed. 8 Feb. 2006 - finish by Wed. 22 Feb.

**CMBR:**Most of the electromagnetic energy in the universe is in the cosmic microwave background radiation (CMBR), sometimes referred to as the Kelvin background. Penzias and Wilson discovered the CMBR in 1965 using a radio telescope, and subsequently received the Nobel Prize for this discovery. This background radiation has wavelength mm. The energy density of the CMBR is about 4.0 x 10^{-14}J/m^{3}. What is the*rms*electric field strength of the CMBR?**ANSWER**:*If J/m*^{3}, then . (The wavelength, while interesting, is irrelevant to the question.)**STANDING WAVES:**Consider standing electromagnetic waves:

- Show that these satisfy the wave equation (9.2).
**ANSWER**:*When we're taking the spatial derivatives, the**t*-dependent factor is just part of the amplitude, and vice versa. Thus and ; and ; so and similarly for . But , since . Thus and similarly for . - Show that we must also have
and
*E*_{0}=*c B*_{0}.**ANSWER**:*Since is a universal property of all solutions of The Wave Equation (TWE), that's a given. Applying FARADAY'S LAW, , gives or . Dividing out the common factor gives or (since ) .* - Show that the time-averaged power flow
across
*any*area will be zero.**ANSWER**:

. Looking only at the*t*-dependence to get the time average, we note that which averages to zero. - Show that the Poynting vector will also be zero,
*i.e.*there is no net energy flow.**ANSWER**:*I must apologize for a defective question. [The hazards of using someone else's problem!] As explained above, . This is only zero where ,**i.e.*at*z*=0 and (where*n*is any integer). That is, for . At any other position, oscillates in the direction,__averaging__to zero.

- Show that these satisfy the wave equation (9.2).
- (p. 386, Problem
**9.14**) -**REFLECTED & TRANSMITTED POLARIZATION:**In Eqs. (9.76) and (9.77) it was tacitly assumed that the reflected and transmitted waves have the same*polarization*as the incident wave, namely along the direction. Prove that this*must*be so. [*Hint:*Let the polarization vectors of the reflected and transmitted waves be

and prove from the boundary conditions that .]**ANSWER**:*We must have continuous across the boundary. Since the normal direction is , is constituted of**x*and*y*components. Thus or [1] and [2]. Similarly, must be continuous across the boundary, and, as always, , giving [3] and [4]. If , Eq. [4] reads , which we can combine with Eq. [2] to conclude that , which can be true only if*E*_{T}= 0 (trivial case) or (mod ). Equation [2] then also requires (mod ). - (p. 392, Problem
**9.15**) -**COMPLEX ALGEBRA EXERCISE:**Suppose that we have six nonzero constants*A*,*B*,*C*,*a*,*b*,*c*such than*Ae*^{iax}+*Be*^{ibx}=*Ce*^{icx}for all*x*. Prove that*a*=*b*=*c*and*A*+*B*=*C*.**ANSWER**:*The first part is easy: if it were**not*true that*a*=*b*=*c*then even if the equation were satisfied at some position in*x*, it would*not*be satisfied at some nearby*x*. So*a*=*b*=*c*. The second part is even easier: at*x*=0,*A*+*B*=*C*. Done. - (p. 392, Problem
**9.17**) -**DIAMOND:**The index of refraction of diamond is 2.42. Construct the graph analogous to Figure 9.16 for the air/diamond interface. (Assume .)**ANSWER**:*FRESNEL'S EQUATIONS read*

*where and . In this case (we assume the light is**entering*the diamond rather than emerging) and . You can use your favourite spreadsheet or other plotting software to produce the graph below. (I used`http://musr.org/muview/`, a free*Java*spreadsheet applet we built at TRIUMF.)- the amplitudes at normal incidence;
**ANSWER**:*For , , giving or and or .* - Brewster's angle;
**ANSWER**:*or .* - and the "crossover" angle at which
the reflected and transmitted amplitudes are equal.
**ANSWER**:*Rather than try to read this off the graph, let's calculate it exactly: The condition is or or or or or or .*

- the amplitudes at normal incidence;
**PLANE WAVE STRESS TENSOR:**Find all the elements of the Maxwell stress tensor of a monochromatic plane wave traveling in the*z*-direction, polarized in the*x*-direction:

**ANSWER**:*Recall Eq. (8.19) on p. 352:*

*E*/c, so all off-diagonal elements are zero. We have or*T*_{11}= 0, or*T*_{22}= 0 and or (only nonzero element!) .In what direction does this EM wave transport momentum? Does this agree with the form of the Maxwell stress tensor you just deduced?

**ANSWER**:*If**T*_{ij}represents the force per unit area acting in the direction on a surface whose normal is in the direction, then the diagonal elements are*pressures*and*T*_{33}is the radiation pressure on a surface normal to . In the same way -*T*_{33}represents the the momentum current density transported by the fields, and is (as expected) in the same direction as and is, in fact, equal to .

**7.**- (p. 412, Problem
**9.33**) -**SPHERICAL WAVES:**Suppose that

with , as usual. [This is, incidentally, the simplest possible**spherical wave**. For notational convenience, let in your calculations.]- Show that obeys all four of Maxwell's equations,
in vacuum, and find the associated magnetic field.
**ANSWER**:*Since and**E*does not depend on , GAUSS' LAW reads (in spherical coordinates)

*In order to satisfy FARADAY'S LAW we must therefore have (within a constant of integration)*

*or where*

*This should satisfy GAUSS' LAW too:*

*It remains only to check AMPÈRE'S LAW: or*

*Now, if we're to get any joy from this, it had better be equal to*

*Thus the proposed function does satisfy all of MAXWELL'S EQUATIONS as advertised and is therefore also a valid solution of TWE (The Wave Equation). And this is the*__simplest possible__spherical wave! (Don't you just love curvilinear coordinates?) - Calculate the Poynting vector.
Average over a full cycle
to get the intensity vector .
Does point in the expected direction?
Does it fall off like
*r*^{-2}, as it should?**ANSWER**:

*The fact that has a non-radial component may seem alarming, but let's check the time average: all of , and oscillate in time, but only the first averages to zero; the other two average to , but their**difference*does average to zero. Thus

*which points radially outward and falls off like 1/**r*^{2}, as expected. - Integrate
over a spherical surface
to determine the total power radiated.

[You should get .]**ANSWER**:

*This was a tedious problem; it took me all day to get it right. I will be duly impressed if you managed to grind through it successfully. Now you know why we like our plane waves so much, Huygens' principle notwithstanding!* - Show that obeys all four of Maxwell's equations,
in vacuum, and find the associated magnetic field.

2006-02-20