Question

When a light of wavelength \( \lambda \) falls on the emitter of a photocell, maximum speed of emitted photoelectrons is \( v \). If the incident wavelength is changed to \( \frac{2}{3} \lambda \), the maximum speed of emitted photoelectrons will be:

• A. \( \sqrt{3}v \)
• B. \( \frac{v}{\sqrt{2}} \)
• C. \( v \)
• D. \( \sqrt{\frac{3}{2}} v \)

Hint:

The maximum speed of the emitted photoelectrons is proportional to the square root of the photon energy. When the wavelength is reduced, the energy increases, and so does the speed.

Answer 1

The Correct Option is A

Step 1: Use Einstein’s photoelectric equation.
Einstein’s photoelectric equation relates the energy of the incident photon to the maximum kinetic energy of the emitted photoelectron:
\[ E = h \nu = K_{\text{max}} + \phi, \]
where:
- \( E = h \nu \) is the energy of the incident photon,
- \( K_{\text{max}} = \frac{1}{2} m v^2 \) is the maximum kinetic energy of the emitted photoelectron,
- \( \phi \) is the work function of the material (the minimum energy required to emit an electron).

Step 2: Relate wavelength to photon energy.
The energy of a photon is also related to its wavelength \( \lambda \) by the equation:
\[ E = \frac{hc}{\lambda}, \]
where:
- \( h \) is Planck’s constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength.

Step 3: Effect of wavelength on maximum speed of photoelectron.
If the wavelength is reduced to \( \frac{2}{3} \lambda \), the energy of the incident photon increases. Since the energy is proportional to \( \frac{1}{\lambda} \), the energy of the photon changes as:
\[ E' = \frac{hc}{\frac{2}{3} \lambda} = \frac{3}{2} E. \]
Thus, the maximum kinetic energy of the photoelectron increases by a factor of \( \frac{3}{2} \).

Step 4: Relate kinetic energy to speed.
Since kinetic energy is related to the speed of the photoelectron by \( K = \frac{1}{2} m v^2 \), the maximum speed of the emitted photoelectron increases by a factor of \( \sqrt{\frac{3}{2}} \). Final Answer:
Thus, the new maximum speed of the photoelectron is:
\[ \boxed{\sqrt{3} v}. \]