Bucknell Physics 212 Supplement

1.

Draw phasor diagrams depicting the oscillations described below at the noted times:

1. For the oscillation $x(t)=3\cos \left(\frac{\pi }{3}t\right)\text{,}$ draw phasor diagrams for $t=0\text{,}$ 1, 2, 3, 5, and $6\mathrm{s}\text{.}$

2. For the oscillation

   $$x(t)=3e^{i(\frac{\pi }{3}t+\frac{\pi }{3})}$$

   draw complex phasor diagrams for $t=0\text{,}$ 1, 2, 3, 5, and $6\mathrm{s}\text{.}$

2.

Consider water waves passing by a fixed point with a period of $6\mathrm{s}$ and a height $30\mathrm{cm}\text{.}$ Imagine that $t=0$ corresponds to the moment when the water surface returns to its equilibrium position just after a crest passes.

1. Draw phasor diagrams for the displacement of the water for times $t=0,1,2,3,5\text{,}$ and $6\mathrm{s}\text{.}$

2. Write an expression for this oscillation in the form

   $$x(t)=A{e}^{i(\omega t+{\varphi }_0)}$$

   specifying $A\text{,}$ $\omega \text{,}$ and ${\varphi }_0\text{.}$

3.

Draw a phasor diagram describing these oscillations:

1. ${\displaystyle {\displaystyle x(t)=4\cos (\frac{\pi }{4}t+\frac{\pi }{2})\text{at time }}}$

2. ${\displaystyle {\displaystyle x(t)=4e^{i(\frac{\pi }{2}-\frac{\pi }{4}t)}\text{at time }}}$

4.

Write the superposition of the following two oscillations

in the form

and determine $A_3$ and ${\varphi }_3\text{.}$ (Hint: Use phasor addition!)

5.

You're standing at the point $P$ on a line between two radio towers, A and B.

Both towers broadcast the same radio signal of wavelength $\lambda =12\mathrm{m}\text{.}$ With only tower A broadcasting, you measure a wave amplitude of 6 (in some unspecified unit). With only tower B broadcasting, you measure a wave amplitude of 3. What amplitude do you measure when both towers are broadcasting?

6.

You're sitting in the Weis Center listening to the “Sonorous Symphony in C,” which consists of a single 128 Hz tone played through two speakers separated by $5\mathrm{m}$ on stage. Both speakers emit sound waves in phase.

You happen to be sitting at the point $P$ in the above diagram, $5\mathrm{m}$ to the left of the leftmost speaker, and $15\mathrm{m}$ back from the stage.

If the wave amplitude at your location from each speaker individually is $A\text{,}$ what is the amplitude of the combined waves from both speakers at your location?

7.

Three radio towers, arrayed as indicated in the figure, each broadcast the same in-phase signal with wavelength $\lambda =5\mathrm{m}\text{.}$

If the amplitude from each tower individually at point $P$ is $A\text{,}$ what is the amplitude of the combined signal from the three towers?

(Hint: $d\sin \theta$ won't work here. Calculate the distances and figure out the path length differences directly.)

8.

Light of wavelength $\lambda =617\mathrm{nm}$ passes through a three-slit system where the slits are separated by a distance $d=1.4\times {10}^{-5}\mathrm{m}\text{,}$ and produces an interference pattern on a distant screen. Determine the amplitude of the light directed at an angle $\theta ={0.23}^{\circ }$ from the direction to the central maximum in the interference pattern. Assume that the amplitude of the light from each slit individually is $A\text{.}$

9.

Light of wavelength $\lambda =532\mathrm{nm}$ passes through a four-slit system where the slits are separated by a distance $d=1.7\times {10}^{-5}\mathrm{m}\text{,}$ and produces an interference pattern on a distant screen.

1. Draw phasor diagrams for the three minima between the central maximum and the first side maximum in the interference pattern, and determine the values of $\mathrm{\Delta }\varphi$ for each situation.

2. Calculate the angles (relative to the direction to the central maximum) where each of these minima occurs.

10.

Light of wavelength $600\mathrm{nm}$ is incident normally on a grating with $850$ lines per millimeter. The diffraction pattern is observed on a distant screen.

1. Sketch a phasor diagram for the light at the second-order maximum of the diffraction pattern. Be sure to indicate the phase difference between adjacent phasors on your diagram. (Don't worry about the fact that you can't draw all the thousands of phasors–just draw 5 or 6 representative phasors.)

2. Using your phasor diagram and the adjacent phase difference you determined in part (a), find the angle to the normal at which the second-order diffraction maximum occurs.

11.

Sodium has two emission lines with ${\lambda }_1=589.00\mathrm{nm}$and ${\lambda }_2=589.59\mathrm{nm}\text{.}$ You send light made up of both wavelengths through a transmission grating with $1000$ lines per millimeter and onto a screen one meter distant. By what distance will the first-order bright spots formed by the two wavelengths be separated?

12.

You wish to design a spectrometer, using a transmission diffraction grating to spread light of different wavelengths out horizontally for observation on a distant screen. Your design specifications state that the second-order diffracted light must all fit on a screen $1.3\mathrm{m}$ away and $1.7\mathrm{m}$ wide, with the zero-order light hitting the center of the screen. The wavelengths you intend to analyze with the spectrometer fall between $450\mathrm{nm}$ and $800\mathrm{nm}\text{.}$

1. Determine the maximum allowed number of lines per millimeter for the grating.

2. Using your answer for part (a), determine whether or not the first and second-order bands of diffracted light will overlap.

13.

Light of wavelength $500\mathrm{nm}$ illuminates a single slit of width $5\mu \mathrm{m}\text{.}$ At the center of the diffraction pattern on a screen $1\mathrm{m}$ away, the amplitude of the light is $A\text{.}$ Determine the amplitude of the light on the screen a distance of $2.5\mathrm{cm}$ away from the center of the central maximum.

14.

A telescope is being designed to detect distant binary-star systems. The telescope should be able to detect stars separated by ${10}^{-4}\mathrm{lt}-\mathrm{yr}$ at a distance of $300\mathrm{lt}-\mathrm{yr}\text{,}$ using near infrared light of wavelength $1\mu \mathrm{m}\text{.}$ Approximately what minimum diameter should the telescope have, if its performance is limited by diffraction?