Exercise 9.11:

Question: A metal surface is illuminated by light of wavelength 400 nm. The stopping potential is measured to be 2.5 V. If the work function of the metal is 2.3 eV, calculate the threshold frequency and the photoelectric current.

Solution:

Step 1: Understanding the problem
We are given the wavelength of the incident light (λ = 400 nm), the stopping potential (V = 2.5 V), and the work function of the metal (ϕ = 2.3 eV). We need to calculate the threshold frequency and the photoelectric current.

Step 2: Convert the given wavelength to frequency
The speed of light (c) is given by the equation c = λν, where c is the speed of light, λ is the wavelength, and ν is the frequency. Rearranging the equation, we get ν = c/λ.

Plugging in the values, we have:
ν = (3 × 10^8 m/s) / (400 × 10^(-9) m)
ν = 7.5 × 10^14 Hz

Step 3: Calculate the threshold frequency
The threshold frequency (ν₀) is the minimum frequency required for the photoelectric effect to occur. It can be calculated by dividing the work function (ϕ) by Planck's constant (h).

The value of Planck's constant (h) is 6.63 × 10^(-34) J s.
The value of work function (ϕ) is given in electron volts (eV), so we need to convert it to joules (J) by multiplying by 1.6 × 10^(-19) J/eV.

ϕ = 2.3 eV = 2.3 × (1.6 × 10^(-19)) J
ϕ = 3.68 × 10^(-19) J

Now, we can calculate the threshold frequency:
ν₀ = ϕ/h
ν₀ = (3.68 × 10^(-19) J) / (6.63 × 10^(-34) J s)
ν₀ ≈ 5.55 × 10^14 Hz

Step 4: Calculate the photoelectric current
The photoelectric current (I) can be calculated using the equation I = ne, where n is the number of photons incident on the metal per second and e is the charge of an electron. Since the number of photons per second is directly proportional to the intensity of light, we can assume n as a constant.

The charge of an electron (e) is approximately 1.6 × 10^(-19) C.

Step 5: Conclusion
The threshold frequency is approximately 5.55 × 10^14 Hz, and the photoelectric current can be calculated using the equation I = ne, where n is the number of photons incident on the metal per second and e is the charge of an electron.