The surface of a metal is illuminated alternately with photons of energies E₁ = 4 eV and E₂ = 2.5 eV respectively. The ratio of maximum speeds of the photoelectrons emitted in the two cases is 2. The work function of the metal in (eV) is _____.

Solution

Physics sir · 4 steps

Speed ratio 2 means kinetic energy ratio 4, because KE goes as the square of speed.

1. Einstein's equation. For each photon, the fastest photoelectron gets 1/2mvₘₐₓ² = E - φ, where φ is the work function.
2. Square the ratio. Kinetic energy depends on v². So a speed ratio of 2 means a kinetic energy ratio of 2² = 4.

   (4 - φ)/(2.5 - φ) = (v₁/v₂)² = 4

3. Solve for φ. 4 - φ = 10 - 4φ, so 3φ = 6 and φ = 2 eV. Check: the kinetic energies are 2 eV and 0.5 eV, a ratio of 4.
4. So. The work function of the metal is 2 eV.

The common slip: 1. Using the speed ratio 2 directly as the kinetic energy ratio gives φ = 1 eV. Square it first to get 4.

The idea to keep. When a question gives a ratio of speeds, square it before comparing kinetic energies. Einstein's equation is about energy, not speed.

Answer from the NTA final answer key. Solution by PrepWiser.