Phys 312- Assignment 9

Due Thursday, April 2

1.

Fourier integrals. (8 points)Suppose that an EM pulse is described by the Gaussian function  

$$f(t)={\frac{1}{\sqrt{2\pi \sigma ^{2}}}}e^{-t^{2}/2\sigma ^{2}}.$$ (1)

(a)

Calculate the Fourier transform $F(\omega )$ of the function f(t). If you use Maple, remember to say assume(sigma>0).

(b)

Define the moments of f(t) and $F(\omega )$ as  

$$\langle t^{n}\rangle ={\frac{\int_{-\infty }^{\infty }t^{n}f... ...\,d\omega }{\int_{-\infty }^{\infty }F(\omega )\,d\omega }}.$$ (2)

Calculate $\Delta t=\sqrt{\langle t^{2}\rangle -\langle t\rangle ^{2}}$ and $\Delta \omega =\sqrt{\langle \omega ^{2}\rangle -\langle \omega \rangle ^{2}}$, and the product $\Delta t\,\Delta \omega$. What happens to the bandwidth when you make the pulse sharper? Why?

2.

TV bandwidth. (4 points) In Melissinos, p. 90, it is stated that the bandwidth required to transmit a television signal is 6 MHz. Why? Try to understand this number by making an order of magnitude estimate. A television screen has 525 horizontal lines with about 700 dots per line; 30 images are produced per second. See Bloomfield, section 4.2, for details on image creation and getting the best images with the available bandwith.

3.

Phased antenna arrays and diffraction. (10 points)Obtain the radiation pattern shown in a "polar plot" by Melissinos in Fig 4.4(b) and the corresponding "straight" plot of the time-averaged $dP/d\Omega$ versus angle $\theta$, as shown in the appropriate frame of the movie scatdiff.mov.

(a)

You will need to generalize the equations for E₁ and E₂ at the top of page 124. Working with complex fields (to make life easier), show that for propagation at an angle $\theta$ to the y axis of the figure, when r tends to infinity

$$E_{1}=E_{0}\exp \left[ i\left( kr-\frac{k\lambda }{8}\cos \theta -\omega t-\phi _{1}\right) \right] .$$

Obtain the analogous equation for E₂. (Actually, E₁ and E₂ fall off like 1/r, but this factor is canceled when computing $dP/d\Omega ).$ Hence, find E.

(b)

The time-averaged $dP/d\Omega$ is proportional to the time-averaged $\left\vert E\right\vert ^{2};$ the other factors do not interest us, as indicated by Melissinos. Obtain a polar plot of $\left\langle \left\vert E\right\vert ^{2}\right\rangle$ as found in Melissinos, as well as a "straight" plot as shown on the appropriate frame of the movie (which frame?). Accurate plots are expected.

4.

(4 points) In a plane electromagnetic wave, what fraction of the energy is electric and what fraction is magnetic? A precise answer is expected, and should be derived using the simple plane wave described by Melissinos in eqs. (4.9) and (4.9$^{\prime }$), but the answer is true more generally.

5.

Applied diffraction. (10 points) One important physical limitation on the resolving power of an antenna is diffraction. Under ideal conditions:

(a)

From how high can an eagle see a mouse on the ground?

(b)

A diffraction-limited laser beam of diameter 1 cm is pointed at the moon. What is the diameter illuminated on the moon? Ignore atmospheric effects.

(c)

The world's largest single-dish radiotelescope is at Arecibo, Puerto Rico. It has a diameter of 305 m. What is the resolving power (the angular resolution, in degrees), when the telescope is operating at the famous 1420 MHz frequency of neutral hydrogen?

(d)

What is the angular resolution of an array of antennas spaced 3000 km, at the same frequency of 1420 MHz?

6.

(10 points) Briefly describe the characteristics of

• Light emitted from a sodium lamp (looks yellow)
• Natural light from the sun
• Synchrotron radiation
• Laser light

In each case, is the light broad-spectrum or narrow-spectrum, polarized (at least partly) or unpolarized, coherent (at least partly) or incoherent?