Introduction:
The ratio of the wavelengths of two materials can be determined using the work functions of those materials. Work function is the minimum amount of energy required to remove an electron from the surface of a material. The energy of a photon is directly related to its wavelength through the equation E = hc/λ, where E is the energy, h is the Planck's constant, c is the speed of light, and λ is the wavelength.

Given:
Work function of sodium (ϕ₁) = 2.3 eV
Work function of copper (ϕ₂) = 4.5 eV

Calculating Energy:
To find the ratio of the wavelengths, we first need to calculate the energies associated with the work functions of sodium and copper. We know that 1 eV is equal to 1.6 x 10^-19 Joules.

Energy of sodium (E₁) = ϕ₁ x 1.6 x 10^-19 J
Energy of copper (E₂) = ϕ₂ x 1.6 x 10^-19 J

Calculating Wavelength:
The energy of a photon can be related to its wavelength using the equation E = hc/λ. Rearranging the equation, we get λ = hc/E.

Wavelength of sodium (λ₁) = (6.63 x 10^-34 Js x 3 x 10⁸ m/s) / E₁
Wavelength of copper (λ₂) = (6.63 x 10^-34 Js x 3 x 10⁸ m/s) / E₂

Calculating the Ratio:
To find the ratio of the wavelengths, we can divide the wavelength of copper by the wavelength of sodium.

Ratio = λ₂ / λ₁

Substituting Values:
Now let's substitute the given values and calculate the ratio of the wavelengths.

Ratio = [(6.63 x 10^-34 Js x 3 x 10⁸ m/s) / E₂] / [(6.63 x 10^-34 Js x 3 x 10⁸ m/s) / E₁]
= E₁ / E₂

Substituting Energy Values:
Substituting the energy values calculated earlier, we can find the ratio of the wavelengths.

Ratio = (ϕ₁ x 1.6 x 10^-19 J) / (ϕ₂ x 1.6 x 10^-19 J)
= ϕ₁