PHY222 Midterm #2 Study Guidance: Waves, Interference, and Thermodynamics

Q1A. Monochromatic light falls on two narrow slits that are 0.05 mm apart. On a screen 4.00 m away, the first order fringe (bright spot) is 6.00 cm from the middle of the interference pattern. What is the wavelength of the light?

Background

Topic: Double-Slit Interference

This question tests your understanding of the double-slit interference pattern and how to relate the position of bright fringes to the wavelength of light.

Key Terms and Formulas

• Double-slit equation for bright fringes: $d \sin\theta = m\lambda$

• Small angle approximation: $\sin\theta \approx \tan\theta = \frac{y}{L}$, where $y$ is the fringe position on the screen and $L$ is the distance to the screen.

• $d$ = slit separation, $m$ = order of fringe (first order means $m=1$), $\lambda$ = wavelength.

Step-by-Step Guidance

1. Identify the known values: $d = 0.05$ mm $= 5.0 \times 10^{-5}$ m, $L = 4.00$ m, $y = 6.00$ cm $= 0.060$ m, $m = 1$.

2. Write the double-slit equation for the first order bright fringe: $d \sin\theta = m\lambda$.

3. Use the small angle approximation: $\sin\theta \approx \frac{y}{L}$.

4. Substitute the values into the equation: $d \frac{y}{L} = \lambda$.

Try solving on your own before revealing the answer!

Q1B. Monochromatic light of wavelength 500 nm falls on two narrow slits that are 0.04 mm apart and this generates an interference pattern on a distant screen. At what angle (in degrees) does the first minima occur in the interference pattern?

Background

Topic: Double-Slit Interference Minima

This question tests your ability to find the angular position of the first dark fringe (minima) in a double-slit interference pattern.

Key Terms and Formulas

• Minima condition: $d \sin\theta = (m + 0.5)\lambda$ for minima, where $m=0$ for the first minima.

• $d$ = slit separation, $\lambda$ = wavelength.

Step-by-Step Guidance

1. Identify the known values: $d = 0.04$ mm $= 4.0 \times 10^{-5}$ m, $\lambda = 500$ nm $= 5.0 \times 10^{-7}$ m, $m = 0$ for first minima.

2. Write the minima condition: $d \sin\theta = (m + 0.5)\lambda$.

3. Plug in $m = 0$: $d \sin\theta = 0.5\lambda$.

4. Solve for $\sin\theta$: $\sin\theta = \frac{0.5\lambda}{d}$.

Try solving on your own before revealing the answer!

Q1C. A light beam traveling through air (n=1.00) reflects perpendicularly off the top and bottom surfaces of a thin layer of oil (n=1.50) floating on water (n=1.33). What is the thinnest layer of oil that will give rise to constructive interference at a wavelength of 700 nm?

Background

Topic: Thin Film Interference

This question tests your understanding of constructive interference in thin films, considering phase changes upon reflection.

Key Terms and Formulas

• Constructive interference for thin film (with phase change on both reflections): $2 n t = m \lambda$ (for $m=1$ for thinnest layer).

• $n$ = refractive index of oil, $t$ = thickness, $\lambda$ = wavelength in vacuum.

Step-by-Step Guidance

1. Identify the known values: $n = 1.50$, $\lambda = 700$ nm $= 7.0 \times 10^{-7}$ m, $m = 1$ for thinnest layer.

2. Write the constructive interference condition: $2 n t = m \lambda$.

3. Rearrange to solve for $t$: $t = \frac{m \lambda}{2 n}$.

Try solving on your own before revealing the answer!

Q1D. A soap film (n=1.35) of thickness 150 nm is surrounded by air (n=1.00) on both sides. If light of a certain wavelength is beamed at the soap film such that it hits the film perpendicular to its surface, what is the longest wavelength of light for which you’ll see no reflection?

Background

Topic: Thin Film Interference (Destructive)

This question tests your understanding of destructive interference in thin films, considering phase changes upon reflection.

Key Terms and Formulas

• Destructive interference for thin film (with one phase change): $2 n t = m \lambda$ (for $m=1$ for longest wavelength).

• $n$ = refractive index of film, $t$ = thickness, $\lambda$ = wavelength in vacuum.

Step-by-Step Guidance

1. Identify the known values: $n = 1.35$, $t = 150$ nm $= 1.5 \times 10^{-7}$ m, $m = 1$ for longest wavelength.

2. Write the destructive interference condition: $2 n t = m \lambda$.

3. Rearrange to solve for $\lambda$: $\lambda = \frac{2 n t}{m}$.

Try solving on your own before revealing the answer!

Q2A. The displacement of a transverse wave on a string is given by $D(x,t)=4\sin(5\pi x-6\pi t)$, where all numbers are in SI units. What is the amplitude, wavenumber, angular frequency, and speed of the traveling wave?

Background

Topic: Wave Motion

This question tests your ability to identify wave parameters from the wave equation.

Key Terms and Formulas

• General wave equation: $D(x,t) = A \sin(kx - \omega t)$

• Amplitude $A$, wavenumber $k$, angular frequency $\omega$, wave speed $v = \frac{\omega}{k}$

Step-by-Step Guidance

1. Compare the given equation to the standard form to identify $A$, $k$, and $\omega$.

2. Write down the amplitude $A$ directly from the equation.

3. Identify the wavenumber $k$ (coefficient of $x$ inside the sine).

4. Identify the angular frequency $\omega$ (coefficient of $t$ inside the sine).

5. Write the formula for wave speed: $v = \frac{\omega}{k}$.

Try solving on your own before revealing the answer!

Q2B. What is the period, frequency, wavelength and direction of propagation of the traveling wave?

Background

Topic: Wave Properties

This question tests your understanding of how to extract period, frequency, and wavelength from the wave equation, and determine the direction of propagation.

Key Terms and Formulas

• Period: $T = \frac{2\pi}{\omega}$

• Frequency: $f = \frac{1}{T}$

• Wavelength: $\lambda = \frac{2\pi}{k}$

• Direction: If the argument is $(kx - \omega t)$, the wave moves in the $+x$ direction.

Step-by-Step Guidance

1. Use the identified $k$ and $\omega$ from the previous part.

2. Calculate the period $T$ using $T = \frac{2\pi}{\omega}$.

3. Calculate the frequency $f = \frac{1}{T}$.

4. Calculate the wavelength $\lambda = \frac{2\pi}{k}$.

5. State the direction of propagation based on the sign in the wave equation.

Try solving on your own before revealing the answer!

Q2C. What is the transverse velocity and acceleration of the string at a position $x=3$ m and time $t=2$ s?

Background

Topic: Wave Motion – Derivatives

This question tests your ability to find the velocity and acceleration by differentiating the displacement function with respect to time.

Key Terms and Formulas

• Transverse velocity: $v(x,t) = \frac{\partial D}{\partial t}$

• Transverse acceleration: $a(x,t) = \frac{\partial^2 D}{\partial t^2}$

Step-by-Step Guidance

1. Write the displacement function: $D(x,t) = 4\sin(5\pi x - 6\pi t)$.

2. Find the first derivative with respect to $t$ for velocity: $v(x,t) = \frac{\partial}{\partial t} D(x,t)$.

3. Find the second derivative with respect to $t$ for acceleration: $a(x,t) = \frac{\partial^2}{\partial t^2} D(x,t)$.

4. Substitute $x=3$ m and $t=2$ s into your expressions for $v(x,t)$ and $a(x,t)$.

Try solving on your own before revealing the answer!

Q2D. Make two plots: one of $D(x,0)$ vs $x$ (one full wavelength), and one of $D(0,t)$ vs $t$ (one full period). Label axes, maxima, minima, and zero crossings.

Background

Topic: Wave Visualization

This question tests your ability to interpret and sketch wave functions at fixed time and position, and to identify key features of the wave.

Key Terms and Formulas

• At $t=0$, $D(x,0) = 4\sin(5\pi x)$

• At $x=0$, $D(0,t) = 4\sin(-6\pi t)$

• Wavelength $\lambda = \frac{2\pi}{k}$, period $T = \frac{2\pi}{\omega}$

Step-by-Step Guidance

1. For $D(x,0)$, plot $4\sin(5\pi x)$ over one wavelength ($x$ from $0$to$\lambda$).

2. Label the $x$-axis in meters and the $D$-axis in meters.

3. Mark the positions of maxima ($D=4$), minima ($D=-4$), and zero crossings ($D=0$).

4. For $D(0,t)$, plot $4\sin(-6\pi t)$ over one period ($t$ from $0$to$T$).

5. Label the $t$-axis in seconds and the $D$-axis in meters, and mark maxima, minima, and zero crossings.

Try sketching these plots before checking the answer!

Q3A. A standing sound wave is shown in a 30 cm long open tube at t=0. Sketch the standing wave pattern at a time T/2 (halfway through the period). Indicate where the nodes (N) and antinodes (A) occur along the tube.

Background

Topic: Standing Waves in Open Tubes

This question tests your understanding of standing wave patterns, nodes, and antinodes, and how the pattern evolves over time.

Key Terms and Formulas

• Standing wave: $D(x,t) = 2A\sin(kx)\cos(\omega t)$

• Nodes: points where displacement is always zero

• Antinodes: points where displacement reaches maximum amplitude

Step-by-Step Guidance

1. Recall that at $t=0$, the displacement pattern is at a maximum or minimum at antinodes.

2. At $t=T/2$, the cosine term changes sign, so the displacement pattern is inverted.

3. Nodes remain at the same positions, antinodes switch from maximum to minimum displacement.

4. Mark the positions of nodes and antinodes along the 30 cm tube.

Try sketching the pattern before checking the answer!

Q3B. Which of the following wave functions best describes the displacement for the above standing wave?

Background

Topic: Standing Wave Equations

This question tests your ability to recognize the mathematical form of a standing wave in an open tube.

Key Terms and Formulas

• Standing wave: $D(x,t) = 2A\sin(kx)\cos(\omega t)$ or $D(x,t) = A\sin(kx)\cos(\omega t)$

Step-by-Step Guidance

1. Recall the general form of a standing wave in an open tube.

2. Compare each option to the standard form.

3. Identify which option matches the expected form for a standing wave.

Try reasoning through the options before checking the answer!

Q3C. What is the wavelength of the standing wave shown above?

Background

Topic: Standing Waves in Open Tubes

This question tests your ability to relate the length of the tube to the wavelength of the standing wave.

Key Terms and Formulas

• For an open tube: $L = n\frac{\lambda}{2}$, where $n$ is the harmonic number.

Step-by-Step Guidance

1. Identify the length of the tube: $L = 30$ cm $= 0.30$ m.

2. Determine the harmonic number based on the standing wave pattern shown.

3. Use the formula $L = n\frac{\lambda}{2}$ to solve for $\lambda$.

Try solving on your own before revealing the answer!

Q3D. If you run toward your friend at 10 m/s while your friend plays the open pipe, what frequency will you hear? (Speed of sound in air is 343 m/s.)

Background

Topic: Doppler Effect

This question tests your understanding of the Doppler effect for sound when the observer is moving toward the source.

Key Terms and Formulas

• Doppler effect (observer moving toward stationary source): $f' = f \left(\frac{v + v_o}{v}\right)$

• $v$ = speed of sound, $v_o$ = observer speed, $f$ = source frequency

Step-by-Step Guidance

1. Calculate the frequency of the sound produced by the pipe using $f = \frac{v}{\lambda}$, where $\lambda$ is from Q3C.

2. Plug values into the Doppler effect formula for a moving observer.

3. Solve for the observed frequency $f'$.

Try solving on your own before revealing the answer!

Q3E. If you and your friend play wind pipes with frequencies differing by 3 Hz, what is the wavelength that corresponds to the beat frequency?

Background

Topic: Beats and Sound Waves

This question tests your understanding of beat frequency and how to relate it to wavelength using the speed of sound.

Key Terms and Formulas

• Beat frequency: $f_{beat} = |f_1 - f_2|$

• Wavelength: $\lambda = \frac{v}{f}$, where $v$ is the speed of sound and $f$ is the beat frequency.

Step-by-Step Guidance

1. Identify the beat frequency: $f_{beat} = 3$ Hz.

2. Use the speed of sound $v = 343$ m/s.

3. Calculate the wavelength: $\lambda = \frac{v}{f_{beat}}$.

Try solving on your own before revealing the answer!

Q4A. Sketch the adiabatic compression process from C to A and determine the ratio $P_2/P_1$ for the engine cycle described.

Background

Topic: Thermodynamics – Adiabatic Processes

This question tests your understanding of adiabatic processes and how to relate pressures and volumes using the adiabatic condition for an ideal gas.

Key Terms and Formulas

• Adiabatic process: $P V^\gamma = \text{constant}$, where $\gamma = \frac{C_P}{C_V}$

• For a monoatomic ideal gas: $C_P = \frac{5R}{2}$, $C_V = \frac{3R}{2}$, so $\gamma = \frac{5}{3}$

Step-by-Step Guidance

1. Write the adiabatic condition: $P_2 V_2^\gamma = P_1 V_1^\gamma$.

2. Identify the volumes at points C and A from the cycle description.

3. Express $P_2/P_1$ in terms of the volume ratio and $\gamma$.

Try solving on your own before revealing the answer!

Q4B. What is the heat for each step in the process, $Q_{AB}$, $Q_{BC}$, and $Q_{CA}$?

Background

Topic: Thermodynamics – Heat in Processes

This question tests your ability to calculate heat transfer for isobaric, isochoric, and adiabatic processes in an ideal gas cycle.

Key Terms and Formulas

• Isobaric: $Q = n C_P \Delta T$

• Isochoric: $Q = n C_V \Delta T$

• Adiabatic: $Q = 0$

• Use the ideal gas law to relate $T$, $P$, and $V$ at each state.

Step-by-Step Guidance

1. Calculate the temperature at each state using $T = \frac{PV}{nR}$.

2. For AB (isobaric): $Q_{AB} = n C_P (T_B - T_A)$.

3. For BC (isochoric): $Q_{BC} = n C_V (T_C - T_B)$.

4. For CA (adiabatic): $Q_{CA} = 0$.

Try solving on your own before revealing the answer!

Q4C. Which of the above heats equals $Q_H$ (heat absorbed) and $Q_L$ (heat dumped)? Use these to compute the efficiency.

Background

Topic: Thermodynamic Cycles and Efficiency

This question tests your understanding of heat flow in a cycle and how to compute the efficiency of a heat engine.

Key Terms and Formulas

• Efficiency: $e = 1 - \frac{Q_L}{Q_H}$

• $Q_H$ is the heat absorbed (positive $Q$), $Q_L$ is the heat expelled (negative $Q$)

Step-by-Step Guidance

1. Identify which process corresponds to $Q_H$ and $Q_L$ based on the sign of $Q$.

2. Plug the values into the efficiency formula: $e = 1 - \frac{Q_L}{Q_H}$.

Try reasoning through the steps before checking the answer!

Q4D. Compute the change in entropy for each process, $\Delta S_{AB}$, $\Delta S_{BC}$, and $\Delta S_{CA}$.

Background

Topic: Entropy Changes in Thermodynamic Processes

This question tests your ability to calculate entropy changes for isobaric, isochoric, and adiabatic processes.

Key Terms and Formulas

• Entropy change: $\Delta S = n C \ln\left(\frac{T_f}{T_i}\right)$ (for constant $C$ process)

• For adiabatic, reversible process: $\Delta S = 0$

Step-by-Step Guidance

1. For AB (isobaric): $\Delta S_{AB} = n C_P \ln\left(\frac{T_B}{T_A}\right)$.

2. For BC (isochoric): $\Delta S_{BC} = n C_V \ln\left(\frac{T_C}{T_B}\right)$.

3. For CA (adiabatic): $\Delta S_{CA} = 0$.

Try solving on your own before revealing the answer!