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Introduction to Electromagnetism - PHYS2189

Assignment 1

Submission instructions:

1. Upload your assignment as a single pdf file to Canvas by the due date (1/9/23). Other file formats will not be accepted.

2. Please use a different piece of paper for each of the problems. Make sure they are in the correct order in your submission.

Assignment instructions:

1. This assignments relates to the content you have studied in weeks 1-6.

2. Make sure you show all your working. No marks will be awarded for results - even if they are correct - if no working is shown.

3. Include a diagram for every problem, even if it is not asked for explicitly.

4. Remember to include units throughout your calculations, not only in the final answers (numbers without units do not mean anything for dimensional quantities).

Problem 1. [4+3+3+3+3=16 points]

A 50.0 N force stretches a vertical spring 0.350 m.

(a) What mass m must be suspended from the spring so that the system will oscillate with a period of 1.00 s?

(b) If the amplitude of the motion is 0.050 m, write the position, velocity and acceleration of mass m as a function of time. Assume the mass is released from rest at the equilibrium position at t = 0

(c) Where is the object and in what direction is it moving 0.35 s after it has passed the equilibrium position?

(d) What force (magnitude and direction) does the spring exert on the object when it is 0.010 m below the equilibrium position, moving upward?

(e) What is the total energy of the oscillations? Write equations for the potential and kinetic energy as a function of time.

Problem 2. [4+4+4+4=16 points]

A transverse travelling wave can be represented by the equation

y(x, t) = A sin (kx − ωt + ϕ) (1)

For this wave, determine:

(a) Sketch the wave as a function of x at t = 0 for ϕ = 0, ϕ = π/2 and ϕ = 3π/4.

(b) Find an equation for the transverse velocity of a particle on the string.

(c) At t = 0, a particle on the string at x = 0 has displacement y = A/√ 2. Is this enough information to determine the value of ϕ?In addition, if you are told that a particle at x = 0 is moving towards y = 0 at t = 0, what is the value of ϕ?

(d) Explain, in general, what you must know about the wave’s behaviour at a given instant to determine the value of ϕ

Problem 3. [2+4+2=8 points]

Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode oscillates in SHM with amplitude 0.850 cm and period 0.0750 s. The string lies along the x axis and is fixed at x = 0

(a) How far apart are the adjacent nodes?

(b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern?

(c) At a particular instant, what is the phase difference (in radians) between two points 4.4 cm apart?

Problem 4. [4+4+4=12 points]

A person leaning over a 150-m-deep well accidentally drops a siren emitting sound of frequency 3000 Hz. Just before this siren hits the bottom of the well, find the frequency and wavelength of the sound the person hears

(a) coming directly from the siren; and

(b) reflected off the bottom of the well.

(c) What beat frequency does this person perceive?

Problem 5. [16 points]

Red light (λ = 700 nm in air) is passed through a double slit apparatus. At the same time, monochromatic visible light of a different wavelength is passed through the same apparatus. As a result, most of the pattern that appears on the screen is a mixture of two colours. However, the center of the third bright fringe (m = 3) of the red light appears pure red, with none of the other colour. What are the possible wavelengths of this second colour of visible light? Do you need to know the slit spacing to answer this question? Why (or why not)?

Problem 6. [4+3+3=10 points]

10.0 W of unpolarised light falls on two consecutive linear polarisers whose axes make a right angle.

(a) How much power will be transmitted through the system?

(b) How much power will be transmitted if an additional linear polariser, with its axis making a 45º angle with the axis of the first polariser, is inserted between the two?

(c) What if the additional polariser is placed in front of the other two?

Problem 7. [6+3+3=12 points]

A diverging lens with a focal length of -14 cm is placed 12 cm to the right of a converging lens with a focal length of 18 cm. An object is placed 33 cm to the left of the converging lens.

(a) Draw a ray diagram for this system.

(b) What is the total magnification of the system?

(c) Where will the image be located?