Vardaan Institute - Physics Worksheet (Level 2)

Topic 11.1: Photoelectric Effect & Einstein’s Equation

When light of frequency $2\nu_0$ (where $\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When frequency of incident radiation is increased to $5\nu_0$, the maximum velocity of electrons emitted is $v_2$. Find the ratio $v_1/v_2$.

Radiation of frequency $10^{15} \text{ Hz}$ is incident on two photosensitive surfaces A and B. Surface A has a work function of $2.0 \text{ eV}$ and Surface B has $5.0 \text{ eV}$. For which surface will photoelectric emission not occur? Justify your answer.

A metal of work function $3.3 \text{ eV}$ is illuminated by light of wavelength $300 \text{ nm}$. Calculate (i) the maximum kinetic energy of the photoelectrons, and (ii) the stopping potential.

The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength $400 \text{ nm}$ is $1.1 \text{ V}$. If the illuminating light is replaced by a source of $300 \text{ nm}$, find the new stopping potential.

AI Image Prompt: A clean, high-quality, mathematically correct landscape graph showing the variation of Photoelectric Current (y-axis) with Collector Plate Potential (x-axis) for three different FREQUENCIES (nu1, nu2, nu3, where nu3 > nu2 > nu1) of incident radiation, keeping INTENSITY constant. The graph must show three distinct negative Stopping Potentials (-V01, -V02, -V03) merging into a single saturation current level on the positive potential side. The background of the whole image should be fully white.

Filename: Level2_Q5_Current_vs_Potential_Frequency.png

Based on the graph, explain the physical relation between the frequency of incident light and the stopping potential.

In a photoelectric effect experiment, the slope of the graph between stopping potential and incident frequency is $4.12 \times 10^{-15} \text{ V}\cdot\text{s}$. Calculate the value of Planck’s constant $h$.

The threshold wavelength for a metal is $500 \text{ nm}$. Light of wavelength $400 \text{ nm}$ is incident on it. Find the maximum velocity of photoelectrons. ($h = 6.6 \times 10^{-34} \text{ Js}, m = 9 \times 10^{-31} \text{ kg}$)

Explain conceptually why the photoelectric current reaches a "saturation" value as the collector potential is increased. Does this saturation current depend on the frequency of light?

A monochromatic source of light operating at $200 \text{ W}$ emits $4 \times 10^{20}$ photons per second. Find the wavelength of the light.

10.

Calculate the force exerted by a monochromatic light beam of intensity $I$ falling normally on a perfectly reflecting surface of area $A$.

11.

Light of frequency $7.21 \times 10^{14} \text{ Hz}$ is incident on a metal surface. Electrons with a maximum speed of $6.0 \times 10^5 \text{ m/s}$ are ejected from the surface. What is the threshold frequency for photoemission of electrons?

12.

The work functions for Silver and Sodium are $4.73 \text{ eV}$ and $2.3 \text{ eV}$ respectively. Which of these will have a longer threshold wavelength? Prove with calculation.

13.

Using Einstein's photoelectric equation, show that the stopping potential $V_0$ varies linearly with the frequency of incident radiation. What is the physical significance of the intercept on the potential axis?

14.

Find the number of photons emitted per second by a $25 \text{ W}$ source of monochromatic light of wavelength $6000 \text{ \AA}$.

15.

Why is the maximum kinetic energy of photoelectrons independent of the intensity of light but the photoelectric current depends on it? Explain using the particle nature of light.

Topic 11.2: Matter Waves (de Broglie Hypothesis)

16.

Find the de Broglie wavelength of an electron moving with speed $v = c/100$. Compare it with the wavelength of a $100 \text{ eV}$ photon.

17.

Calculate the ratio of the de Broglie wavelength of a proton and an $\alpha$-particle accelerated through the same potential difference $V$.

18.

An electron and a photon each have a wavelength of $1.00 \text{ nm}$. Find (i) their momenta, and (ii) the ratio of their kinetic energies.

19.

What is the de Broglie wavelength of a nitrogen molecule in air at $300 \text{ K}$? (Mass of nitrogen molecule $= 4.6 \times 10^{-26} \text{ kg}, k_B = 1.38 \times 10^{-23} \text{ J/K}$)

20.

AI Image Prompt: A clean, high-quality, mathematically correct landscape graph plotting the de Broglie wavelength (lambda) on the y-axis against (1/p) on the x-axis for a particle. The graph must show a straight line passing through the origin. On the same plot, show another line with a different slope representing a particle with a different Planck's constant (hypothetically) or for comparison of two different masses against sqrt(K). Label the axes clearly. The background of the whole image should be fully white.

Filename: Level2_Q20_Lambda_vs_Reciprocal_Momentum.png

21.

A particle A of mass $m$ and charge $q$ is accelerated through a potential difference of $V$. Another particle B of mass $2m$ and charge $q$ is also accelerated through the same potential $V$. Find the ratio of their de Broglie wavelengths.

22.

Show mathematically that for a non-relativistic particle, the de Broglie wavelength $\lambda$ is related to its kinetic energy $K$ by $\lambda = h / \sqrt{2mK}$. How does this relation change for a photon?

23.

An electron, a proton and an alpha particle have the same momentum. Compare their de Broglie wavelengths. Which one has the highest kinetic energy?

24.

For what kinetic energy of a neutron will the associated de Broglie wavelength be $1.40 \times 10^{-10} \text{ m}$? ($m_n = 1.675 \times 10^{-27} \text{ kg}$)

25.

An electron is accelerated through a potential difference of $V$ volts. If its de Broglie wavelength is $\lambda_1$, and when the potential is increased to $4V$, the wavelength becomes $\lambda_2$, find the ratio $\lambda_1/\lambda_2$.

26.

Determine the de Broglie wavelength of an electron in the $n=2$ state of a hydrogen atom. (Radius of $n=1$ is $0.53 \text{ \AA}$).

27.

A proton and an electron have the same de Broglie wavelength. Which one moves faster? Prove using the mass-velocity relationship.

28.

In the Bohr model of the hydrogen atom, the circumference of the $n^{th}$ electron orbit is an integral multiple of its de Broglie wavelength. Use this to derive the quantization condition for angular momentum.

29.

What is the de Broglie wavelength of a thermal neutron at temperature $T$? Write the formula and explain why neutrons are used for crystal structure analysis.

30.

If the uncertainty in the position of an electron is $10^{-10} \text{ m}$, find the minimum uncertainty in its de Broglie wavelength. (Use Heisenberg's Uncertainty Principle).

Topic 11.3: Davisson-Germer Experiment

31.

Briefly describe the Davisson-Germer experiment and how it confirmed the wave nature of electrons. State the role of the Nickel crystal.

32.

AI Image Prompt: A clean, high-quality, mathematically correct landscape diagram showing the geometry of electron diffraction in the Davisson-Germer experiment. Show the incident beam, the Nickel crystal planes with spacing d, the scattering angle phi, and the glancing angle theta. Clearly show the relationship theta = 90 - (phi/2). The background of the whole image should be fully white.

Filename: Level2_Q32_Diffraction_Geometry.png

In the setup above, if the scattering angle is observed to be $50^\circ$, calculate the glancing angle $\theta$ relative to the Bragg planes.

33.

For an electron beam accelerated by $54 \text{ V}$, the scattering maximum was found at $50^\circ$. Calculate the de Broglie wavelength and the Bragg wavelength (for $d = 0.91 \text{ \AA}$) and check for consistency.

34.

Why did Davisson and Germer observe a "peak" in the intensity of scattered electrons only at specific voltages and angles? Relate this to the concept of constructive interference.

35.

If the accelerating voltage in the Davisson-Germer experiment is increased from $54 \text{ V}$ to $64 \text{ V}$, what happens to the position (angle) of the intensity peak?

36.

Explain the function of the "Velocity Selector" (or collimator) in the electron gun assembly of the Davisson-Germer experiment.

37.

What is a Faraday cylinder in this experiment, and why was its position made movable?

38.

Draw a qualitative graph showing the variation of electron intensity with scattering angle for $40 \text{ V}, 48 \text{ V}, 54 \text{ V},$ and $64 \text{ V}$. (Mention only the features).

39.

Compare the de Broglie wavelength of an electron accelerated through $54 \text{ V}$ with the interatomic spacing in most crystals. Why was Nickel a suitable choice for this experiment?

40.

Does the Davisson-Germer experiment prove that electrons are "waves" or that they "behave as waves" under specific conditions? Discuss the dual nature conclusion.

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