The Photoelectric Effect Equation- A Complete Guide

What the Photoelectric Effect Actually Is

Light hits metal. Electrons fly out. That's the photoelectric effect in one sentence. Einstein figured out the math behind it in 1905, and it won him the Nobel Prize—not for relativity, weirdly enough.

The phenomenon broke classical physics. Classical theory said light intensity should determine electron ejection. It doesn't. Frequency does. That's the whole problem.

The Photoelectric Effect Equation

Here's the formula:

KE_max = hf - φ

Or the version you'll see more often:

E_photon = KE + φ

Break it down:

• KE_max — maximum kinetic energy of ejected electrons (in Joules)
• h — Planck's constant (6.626 × 10⁻³⁴ J·s)
• f — frequency of incident light (in Hz)
• φ — work function of the metal (in Joules)

Sometimes you'll see it written with wavelength instead:

KE_max = (hc/λ) - φ

Same equation, just rearranged. c is the speed of light (3 × 10⁸ m/s), λ is wavelength.

What Each Piece Means

The Work Function (φ)

This is the minimum energy needed to kick an electron out of the metal surface. Different metals have different work functions. That's why some metals eject electrons easily and others don't.

Lower work function = easier electron ejection.

Threshold Frequency

This is the minimum frequency that causes electron ejection. Below this? Nothing happens, no matter how bright the light.

Calculate it:

f₀ = φ/h

If light frequency is below f₀, the photon doesn't have enough energy to overcome the work function. Electrons stay put.

Kinetic Energy

The kinetic energy of ejected electrons varies. Some electrons need less energy to escape, so they come out slower. The ones with maximum KE are the ones that needed the least extra energy to break free.

How to Solve Photoelectric Effect Problems

Step 1: Identify what's given

Look for: frequency (f) or wavelength (λ), work function (φ), or the metal type.

Step 2: Convert everything to consistent units

Work function might be given in eV (electron volts). Convert to Joules:

1 eV = 1.602 × 10⁻¹⁹ J

Step 3: Plug into the equation

KE_max = hf - φ

Solve for whatever's missing.

Step 4: Find threshold frequency or wavelength if needed

f₀ = φ/h

Then λ₀ = c/f₀

Example Problem

Light with frequency 8.2 × 10¹⁴ Hz hits sodium. The work function of sodium is 2.3 eV. What's the maximum kinetic energy of ejected electrons?

Step 1: Convert work function to Joules

φ = 2.3 × 1.602 × 10⁻¹⁹ = 3.68 × 10⁻¹⁹ J

Step 2: Calculate photon energy

E = hf = (6.626 × 10⁻³⁴)(8.2 × 10¹⁴) = 5.43 × 10⁻¹⁹ J

Step 3: Find KE

KE = E - φ = 5.43 × 10⁻¹⁹ - 3.68 × 10⁻¹⁹ = 1.75 × 10⁻¹⁹ J

Convert to eV: 1.75 × 10⁻¹⁹ / 1.602 × 10⁻¹⁹ = 1.09 eV

Why This Equation Matters

It proved light acts as particles (photons), not just waves. That was the death knell for classical wave theory of light.

It also shows the quantum nature of energy transfer. Photons don't add up over time. One photon interacts with one electron. That's why dim light (low intensity) with high frequency still ejects electrons, but bright light (high intensity) with low frequency doesn't.

Intensity = number of photons, not energy per photon. Frequency = energy per photon.

Common Mistakes

• Using intensity instead of frequency — wrong. Intensity controls electron count, not ejection.
• Forgetting to convert units — eV vs Joules trips up everyone.
• Confusing threshold frequency with actual frequency — threshold is the minimum, not the given value.
• Using the wrong Planck's constant — 6.626 × 10⁻³⁴ J·s, not something else.

Quick Reference

That's the photoelectric effect equation. Memorize the form, understand what each variable means, and convert your units before calculating.