(a) Well, well, well, let's figure out the work function of this metal surface. So, we know that the energy of a photon is given by Planck's equation, E = hf, where h is the famous Planck's constant. Now, the energy of the ejected electron is given by E = KE + W, where KE is the kinetic energy of the ejected electron and W is our desired work function. Hold on tight, we're about to unleash some math!
We can express the energy of the photon in terms of its wavelength using the equation E = hc/λ, where c is the speed of light, and λ is the wavelength. This photon energy must be equal to the sum of the work function and the kinetic energy of the electron, so we have hc/λ = KE + W.
Since we're dealing with light and electrons here, we know that the kinetic energy of the ejected electron can be expressed as KE = (1/2)mv², where m is the mass of the electron and v is its velocity. We know the velocity, which is 4.50x10^5 m/s, but we don't need to bother with the mass of the electron since it cancels out in our calculations. Fancy, isn't it?
Alright, let's plug in the values and solve for the work function. hc/λ = (1/2)m(4.50x10^5)² + W. Given that λ is 615 nm, convert it to meters by dividing by 10^9: λ = 615x10^-9 m. Now, rearrange the formula to solve for W: W = hc/λ - (1/2)m(4.50x10^5)².
Crunch all those numbers, and you'll finally find the work function of the metal surface. Ta-da!
(b) Now that we've got the work function sorted, let's find the cutoff frequency! The cutoff frequency is the minimum frequency at which electrons can escape from the metal surface.
We can bring in Einstein's photoelectric equation here: E = hf = Φ + KE, where E is the energy of the photon, f is the frequency of the light, Φ is the work function, and KE is the kinetic energy of the ejected electron.
We know the energy of a photon can be expressed as E = hc/λ, where λ is the wavelength. Now, the cutoff frequency is the frequency at which the kinetic energy becomes zero, so we get 0 = Φ + KE.
Rearranging our equation, we have Φ = -KE. But since KE = (1/2)m(v_cutoff)², where v_cutoff is the cutoff velocity, we can rewrite Φ = -m(v_cutoff)²/2.
Now we can introduce the equation v_cutoff = fλ, where f is the cutoff frequency. Substituting this into our equation, we have Φ = -m(fλ)²/2.
Now, to find the cutoff frequency, let's solve for f: f = sqrt(2Φ/mλ²).
Plug in the value of the work function you calculated earlier, the mass of the electron, and the wavelength, and you'll have your cutoff frequency. Good luck with those number-crunching acrobatics!