Question

When electromagnetic radiation of wavelength 310 nm falls on the surface of a metal having work function 3.55 eV, the velocity of photoelectrons emitted is \( x \times 10^5 \text{ms}^{-1} \). The value of \( x \) is (Nearest integer) (\( m_e = 9 \times 10^{-31} \text{kg} \))

• A. 2
• B. 4
• C. 5
• D. 6

Hint:

Using \( 1240 \) or \( 12400 \) (for \AA) is a great time-saver. Remember \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

Using Einstein's photoelectric equation: \[ E = W + K.E. \] where \( E \) is the energy of the incident photon, \( W \) is the work function, and \( K.E. \) is the maximum kinetic energy of the emitted electrons. \( K.E. = \frac{1}{2}mv^2 \).
Step 2: Key Formula or Approach:

1. Photon Energy: \( E = \frac{1240}{\lambda (\text{nm})} \) eV (shortcut formula). 2. Kinetic Energy: \( K.E. = E - W \). 3. Velocity: \( v = \sqrt{\frac{2 \times K.E.}{m}} \). Note: Convert K.E. to Joules first.
Step 3: Detailed Explanation:

Calculate energy of incident radiation: \[ E = \frac{1240}{310} = 4 \text{ eV} \] Given Work Function \( W = 3.55 \text{ eV} \). Calculate Kinetic Energy: \[ K.E. = 4 - 3.55 = 0.45 \text{ eV} \] Convert K.E. to Joules: \[ K.E. = 0.45 \times 1.6 \times 10^{-19} \text{ J} = 0.72 \times 10^{-19} \text{ J} \] Calculate velocity \( v \): \[ \frac{1}{2} m v^2 = 0.72 \times 10^{-19} \] \[ v^2 = \frac{2 \times 0.72 \times 10^{-19}}{9 \times 10^{-31}} = \frac{1.44 \times 10^{-19}}{9 \times 10^{-31}} \] \[ v^2 = 0.16 \times 10^{12} = 16 \times 10^{10} \] \[ v = \sqrt{16 \times 10^{10}} = 4 \times 10^5 \text{ ms}^{-1} \] Given velocity is \( x \times 10^5 \text{ ms}^{-1} \). Thus, \( x = 4 \). Correction based on Answer Key: The provided Answer Key marks Option 4 (\(x=6\)) as correct. Let's recheck the calculation. If \( E = \frac{hc}{\lambda} \). \( h = 6.626 \times 10^{-34} \), \( c = 3 \times 10^8 \). \( E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{310 \times 10^{-9}} = \frac{19.878 \times 10^{-26}}{310 \times 10^{-9}} \approx 0.0641 \times 10^{-17} = 6.41 \times 10^{-19} \text{ J} \). Convert to eV: \( 6.41 \times 10^{-19} / 1.6 \times 10^{-19} \approx 4.006 \text{ eV} \). So \( E = 4 \) eV is correct. \( KE = 0.45 \) eV is correct. \( v = 4 \times 10^5 \) m/s is correct. There seems to be a discrepancy with the key provided (Option 4 is 6). Mathematically, the result is 4. However, adhering to the provided answer key, the answer is 6. This might be due to slightly different constants or a typo in the question's wavelength/work function values (e.g., if wavelength was smaller, energy would be higher). But based on standard calculation, the answer is 4 (Option 2). I will list the correct option based on calculation but note the key indicates Option 4. Given strict instructions to follow the key: Correct Answer:
(D) 6 (Note: Calculation yields 4, discrepancy in key). Wait, let's re-read the options. 1. 2 2. 4 3. 5 4. 6 The green tick is on Option 2 (4). Ah, looking at the crop images again. Image 1, Question 122: Option 1: 2 Option 2: 4 (Green Tick) Option 3: 5 Option 4: 6 Okay, the correct answer IS 4. My manual calculation matches the key. The confusion came from the text prompt saying "Option 4: 6" might be correct in similar contexts or misreading the tick position. The image clearly marks Option 2.
Step 4: Final Answer:

The value of \( x \) is 4.