Please See Homework Below Each Problem Must Follow The Struc ✓ Solved

Please See Homework Below Each Problem Must Follow The Structured Pro

Please see homework below, each problem must follow the structured problem solving format (see attached document) and be complete in Microsoft Word format.

Sample Paper For Above instruction

Introduction

This paper addresses two physics problems related to the photoelectric effect and Planck's constant. The first problem involves calculating the photocurrent generated by a specified laser illuminating a sodium target, and the second involves determining Planck's constant from experimental data, as well as deriving the threshold frequency and work function of the metal. Both problems are approached systematically, following a structured problem-solving methodology which includes analyzing the given data, applying relevant physics formulas, performing calculations, and interpreting the results within the context of physical principles.

Problem 1: Calculating Photocurrent from Laser Illumination

The first problem provides data about a laser with a power of 1.0 milliwatt (mW) and a wavelength of 405 nm illuminating a sodium target. The task is to compute the resulting photocurrent, given that one in 100,000 incident photons produces a photoelectron. The problem requires understanding the relationship between photon flux, the number of photoelectrons generated, and the resulting electrical current.

Step 1: Analyze the given data

The key parameters are:

- Laser power, P = 1.0 mW = 1.0 x 10^-3 W

- Wavelength, λ = 405 nm = 405 x 10^-9 m

- Quantum efficiency: 1 in 10^5 photons produces a photoelectron → efficiency = 1 / 10^5

- The goal: Calculate the photocurrent, which is the electric current due to the movement of photoelectrons.

Step 2: Calculate the energy of a single photon

Using Planck's equation:

\[ E_{photon} = \frac{hc}{\lambda} \]

where h = 6.626 x 10^-34 J·s, c = 3.0 x 10^8 m/s.

\[ E_{photon} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^{8}}{405 \times 10^{-9}} \]

\[ E_{photon} \approx 4.91 \times 10^{-19} \text{ J} \]

Step 3: Determine the incident photon flux per second

Total power divided by photon energy:

\[ \Phi_{total} = \frac{P}{E_{photon}} \]

\[ \Phi_{total} = \frac{1.0 \times 10^{-3}}{4.91 \times 10^{-19}} \approx 2.04 \times 10^{15} \text{ photons/sec} \]

Step 4: Calculate the number of photoelectrons generated per second

Since 1 in 10^5 photons produces a photoelectron:

\[ \Phi_{photo} = \frac{\Phi_{total}}{10^5} \]

\[ \Phi_{photo} = \frac{2.04 \times 10^{15}}{10^{5}} = 2.04 \times 10^{10} \text{ electrons/sec} \]

Step 5: Calculate the photocurrent

Current due to moving electrons:

\[ I = e \times \Phi_{photo} \]

where e = 1.602 x 10^-19 C.

\[ I = 1.602 \times 10^{-19} \times 2.04 \times 10^{10} \]

\[ I \approx 3.27 \times 10^{-9} \text{ A} \]

Final Result:

The photocurrent generated is approximately 3.27 nA.

Problem 2: Determining Planck's Constant from Experimental Data

The second problem involves an experiment where ultraviolet light illuminates a metal target. Measurements with two different wavelengths provide information needed to approximate Planck's constant and to derive the work function and threshold frequency of the metal.

Given Data

- Wavelength 1, λ₁ = 300 nm, stopping potential V₁ = 1.10 V

- Wavelength 2, λ₂ = 200 nm, stopping potential V₂ = 3.06 V

Step 1: Convert wavelengths to frequencies

Using:

\[ f = \frac{c}{\lambda} \]

- \(f_{1} = \frac{3.0 \times 10^{8}}{300 \times 10^{-9}} = 1.0 \times 10^{15} \text{ Hz}\)

- \(f_{2} = \frac{3.0 \times 10^{8}}{200 \times 10^{-9}} = 1.5 \times 10^{15} \text{ Hz}\)

Step 2: Use photoelectric equation to find h

The maximum kinetic energy of emitted electrons:

\[ KE_{max} = eV \]

Corresponds to:

\[ KE_{max} = hf - \phi \]

where \(\phi\) is the work function.

For the two wavelengths:

\[ eV_{1} = hf_{1} - \phi \]

\[ eV_{2} = hf_{2} - \phi \]

Subtracting:

\[ e(V_{2} - V_{1}) = h (f_{2} - f_{1}) \]

\[

h = \frac{e (V_{2} - V_{1})}{f_{2} - f_{1}}

\]

Substituting known values:

\[ h = \frac{1.602 \times 10^{-19} \times (3.06 - 1.10)}{1.5 \times 10^{15} - 1.0 \times 10^{15}} \]

\[ h = \frac{1.602 \times 10^{-19} \times 1.96}{0.5 \times 10^{15}} \]

\[ h \approx \frac{3.14 \times 10^{-19}}{0.5 \times 10^{15}} \]

\[ h \approx 6.28 \times 10^{-34} \text{ J·s} \]

This value closely matches the accepted Planck's constant \(6.626 \times 10^{-34}\) J·s, with a percent error:

\[

\text{Percent error} = \frac{|6.626 \times 10^{-34} - 6.28 \times 10^{-34}|}{6.626 \times 10^{-34}} \times 100\% \approx 5.2\%

\]

Step 3: Find threshold frequency and work function

Using data from the lowest energy photons:

\[ \phi = hf_{1} - eV_{1} \]

\[ \phi = (6.626 \times 10^{-34}) \times 1.0 \times 10^{15} - 1.602 \times 10^{-19} \times 1.10 \]

\[ \phi \approx 6.626 \times 10^{-19} - 1.762 \times 10^{-19} = 4.864 \times 10^{-19} \text{ J} \]

Threshold frequency:

\[ f_{th} = \frac{\phi}{h} \]

\[ f_{th} = \frac{4.864 \times 10^{-19}}{6.626 \times 10^{-34}} \approx 7.34 \times 10^{14} \text{ Hz} \]

This is the minimum frequency needed to eject electrons.

Conclusion:

The experiment approximates Planck's constant as approximately \(6.28 \times 10^{-34}\) J·s, with a 5.2% error relative to the accepted value. The threshold frequency of the metal is approximately \(7.34 \times 10^{14}\) Hz, and the work function is about \(4.86 \times 10^{-19}\) J.

Conclusion

These calculations reinforce the fundamental principles of quantum physics, especially the photoelectric effect, which directly supports the quantization of energy emitted by electromagnetic radiation. The first problem demonstrated how the incident laser power translates into measurable current due to photoelectron generation, while the second used spectral data to approximate Planck’s constant and derive properties intrinsic to the metal's electronic structure. Such problem-solving approaches exemplify the application of theoretical physics to experimental data, bridging abstract principles with practical measurement.

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