Dual Nature of Matter: PYQ Walkthrough (2)

medium • 2 min read
Tags Dual Nature Of Matter

Question

Light of wavelength 400 nm falls on a metal of work function 2.0 eV. Find the maximum kinetic energy of photoelectrons and the de Broglie wavelength of the fastest electron. (h=6.63×10−34h = 6.63 \times 10^{-34} J·s, c=3×108c = 3 \times 10^{8} m/s.)

Solution — Step by Step

E=hcλ=1240 eV⋅nm400 nm=3.1 eVE = \frac{hc}{\lambda} = \frac{1240\text{ eV·nm}}{400\text{ nm}} = 3.1\text{ eV}

(The shortcut hc≈1240hc \approx 1240 eV·nm saves serious time.)

KEmax=E−ϕ=3.1−2.0=1.1 eVKE_{max} = E - \phi = 3.1 - 2.0 = 1.1\text{ eV}

Convert to joules: KEmax=1.1×1.6×10−19=1.76×10−19KE_{max} = 1.1 \times 1.6 \times 10^{-19} = 1.76 \times 10^{-19} J.

Using λdB=h/p\lambda_{dB} = h/p and p=2m⋅KEp = \sqrt{2m \cdot KE}:

p=2×9.1×10−31×1.76×10−19≈5.66×10−25 kg⋅m/sp = \sqrt{2 \times 9.1 \times 10^{-31} \times 1.76 \times 10^{-19}} \approx 5.66 \times 10^{-25}\text{ kg·m/s}

λdB=6.63×10−345.66×10−25≈1.17 nm\lambda_{dB} = \frac{6.63 \times 10^{-34}}{5.66 \times 10^{-25}} \approx 1.17\text{ nm}

Final answer: KEmax=1.1 eVKE_{max} = 1.1\text{ eV}, λdB≈1.17 nm\lambda_{dB} \approx 1.17\text{ nm}

Why This Works

The photon model says one photon ejects one electron. The energy budget is simple: photon in = work function (binding energy) + kinetic energy out.

The de Broglie wavelength then treats that same electron as a wave. This is the dual nature in action — same particle, two valid descriptions, and we use whichever is convenient.

Alternative Method

Compute λdB\lambda_{dB} directly using λdB=h/2m⋅KE\lambda_{dB} = h/\sqrt{2m \cdot KE}, plugging KEKE in joules. Avoids computing momentum as a separate step.

Common Mistake

Students convert photon wavelength to frequency, multiply by hh, and then forget to subtract the work function. Or worse — they subtract ϕ\phi in eV from EE in joules. Pick one unit system (eV is faster here) and stay consistent.

The de Broglie wavelength of the photoelectron (1.17 nm) is much larger than the wavelength of light scattered from atoms but much smaller than the original photon (400 nm). This separation is why electron microscopes resolve much finer detail than optical microscopes.

Want to master this topic?

Read the complete guide with more examples and exam tips.

Go to full topic guide →

Try These Next

Calculate de Broglie wavelength of electron accelerated through 100V potential

easy

Calculate de Broglie wavelength of electron accelerated through 100V

medium

Dual Nature of Matter: Application Problems (5)

medium

Dual Nature of Matter: Common Mistakes and Fixes (1)

easy

Dual Nature of Matter: Step-by-Step Worked Examples (4)

easy