Q.In a photoelectric experiment carried out with two different metals A and B, the stopping potential $V_{stop}$ (in volts) is measured as a function of the frequency $\nu$ (in Hz) of the incident light. For each metal the measured points lie on a straight line, and the two straight lines are parallel to each other (they have the same slope). The line for metal A lies to the left of the line for metal B. Extrapolated down to zero stopping potential, the line for A meets the frequency axis at a threshold frequency of about $5\times10^{14}$ Hz, while the line for B meets it at about $10\times10^{14}$ Hz; each line then rises with the same slope (for instance, line A passes close to the point $\nu=10\times10^{14}$ Hz, $V_{stop}=2$ V, and line B passes close to the point $\nu=15\times10^{14}$ Hz, $V_{stop}=2$ V).

Photoelectric Effect: Why the Key Formulas Hold

The photoelectric effect is a cornerstone of quantum physics. It showed that light behaves as particles (photons) , not just waves. Let's build the reasoning step-by-step.

1. The Core Idea: Energy Conservation

When a photon hits a metal surface, it transfers all its energy to a single electron inside the metal.

• The photon's energy is $E = hf$, where $h$ is Planck's constant and $f$ is the frequency of light.
• The electron needs a minimum energy to escape the metal — this is called the work function, $\phi$.

Why only one electron?

Einstein proposed that light is quantized into discrete packets (photons). A single photon cannot split its energy among multiple electrons — it interacts with one electron at a time.

2. The Photoelectric Equation

If the photon's energy is greater than the work function, the excess energy becomes the electron's kinetic energy after escape:

Where:

• $hf$ = energy of incident photon
• $\phi$ = work function (minimum energy to remove electron)
• $K_{\text{max}}$ = maximum kinetic energy of ejected electron

Why "maximum" kinetic energy?

• Electrons inside the metal have different binding energies.
• Some electrons are near the surface (loosely bound) → get maximum $K$.
• Others are deeper → lose energy in collisions before escaping → lower $K$.

3. The Stopping Potential Connection

We measure $K_{\text{max}}$ using a stopping potential $V_s$:

Where $e$ is the electron charge. This is because:

• An electric field opposing the electron's motion does work $eV_s$ to stop it.
• At the stopping potential, the electron's kinetic energy is exactly balanced by the electric potential energy.

Combining:

This is the Einstein photoelectric equation in its most testable form.

4. Why the Threshold Frequency Exists

From the equation:

If $f$ is too low, $hf < \phi$. Then:

• The photon cannot supply enough energy to overcome the work function.
• No electron is ejected, regardless of light intensity.

The threshold frequency $f_0$ is when $K_{\text{max}} = 0$:

Why intensity doesn't matter for ejection?

• Intensity = number of photons per second.
• Each photon still has energy $hf$. If $hf < \phi$, even a billion photons won't eject an electron — each photon is individually too weak.

5. Why Kinetic Energy Depends on Frequency, Not Intensity

From $K_{\text{max}} = hf - \phi$:

• Frequency $f$ directly determines $K_{\text{max}}$.
• Intensity only affects the number of electrons ejected (more photons → more electrons), not their individual energy.

This was the key experimental contradiction with classical wave theory:

• Classical: Higher intensity = bigger wave amplitude = more energy to electrons.
• Reality: Higher frequency = more energy per electron; intensity only changes current.

6. Summary of Key Relationships …