29.2 The Photoelectric Effect

The photoelectric effect reveals light's particle-like nature. When light hits a metal surface, it can knock electrons loose, but only if the light has a high enough frequency. The energy that matters here depends on frequency, not brightness. This was one of the key observations that broke classical physics and launched quantum theory.

Understanding the photoelectric effect is crucial for grasping quantum mechanics. It shows that light can behave as a stream of particles (photons), each carrying a discrete packet of energy. Einstein's explanation of this effect earned him the Nobel Prize, and it remains one of the clearest demonstrations of wave-particle duality.

The Photoelectric Effect

Photoelectric effect and light particles

Light behaves as discrete particles called photons. Each photon carries a specific amount of energy determined by its frequency. When photons strike a metal surface (like zinc or gold), they can transfer their energy to electrons, ejecting them as photoelectrons.

Two observations here are critical, and they're what classical wave theory could not explain:

• Increasing light intensity increases the number of electrons ejected, but does not increase their kinetic energy. More intensity just means more photons hitting the surface, so more electrons get kicked out. But each individual photon still carries the same energy, so each ejected electron leaves with the same maximum speed.
• A minimum frequency (the threshold frequency) is required to eject any electrons at all. This threshold is specific to each metal and relates to its work function, which is the minimum energy needed to free an electron from the surface. Below the threshold frequency, no electrons are ejected no matter how bright the light is. For example, infrared light won't eject electrons from zinc regardless of intensity, but ultraviolet light will.

This is the key insight: each photon acts individually. A single photon either has enough energy to free an electron, or it doesn't. You can't "add up" the energy of many weak photons.

Energy calculations for photons

The energy of a photon $E$ is directly proportional to its frequency $f$:

$E = hf$

where $h$ is Planck's constant, $h = 6.626 \times 10^{-34}$ J$\cdot$s.

Since frequency and wavelength are related by $c = f\lambda$, you can also express photon energy in terms of wavelength $\lambda$:

$E = \frac{hc}{\lambda}$

where $c = 3.0 \times 10^8$ m/s is the speed of light. Higher frequency (shorter wavelength) means more energy per photon.

Calculating energy from frequency:

1. Multiply the frequency by Planck's constant.
2. Example: A photon with $f = 6.0 \times 10^{14}$ Hz has energy $E = (6.626 \times 10^{-34})(6.0 \times 10^{14}) = 3.98 \times 10^{-19}$ J.

Calculating energy from wavelength:

1. Divide $hc$ by the wavelength (make sure the wavelength is in meters).
2. Example: A photon with $\lambda = 500$ nm $= 500 \times 10^{-9}$ m has energy $E = \frac{(6.626 \times 10^{-34})(3.0 \times 10^8)}{500 \times 10^{-9}} = 3.98 \times 10^{-19}$ J.

Notice both examples give the same energy. That's because a 500 nm photon has a frequency of $6.0 \times 10^{14}$ Hz. The two formulas are just different ways to get the same answer.

Light properties in the photoelectric effect

The photoelectric equation ties everything together:

$KE_{max} = hf - \phi$

where $KE_{max}$ is the maximum kinetic energy of ejected electrons and $\phi$ (phi) is the metal's work function.

Here's how to think about it: the photon delivers energy $hf$. Some of that energy goes toward breaking the electron free (that's $\phi$). Whatever energy is left over becomes the electron's kinetic energy.

Effect of changing frequency:

• If $hf > \phi$, electrons are ejected. Increasing frequency further increases $KE_{max}$ linearly.
• If $hf < \phi$, no electrons are ejected at all. For instance, red light on zinc doesn't have enough energy per photon to overcome zinc's work function.

Effect of changing intensity:

• Higher intensity means more photons per second hitting the surface, so more electrons are ejected (greater current).
• But intensity does not change the energy of each individual photon, so $KE_{max}$ stays the same.

Experimental setup and measurements

The photoelectric effect is typically studied using a vacuum tube with two electrodes:

• The cathode is the metal surface that light shines on. This is where electrons get ejected.
• The anode collects the ejected electrons, producing a measurable electric current.

To measure $KE_{max}$, you apply a reverse voltage (called the stopping potential, $V_0$) that opposes the electrons' motion. You gradually increase this voltage until the current drops to zero. At that point:

$KE_{max} = eV_0$

where $e = 1.6 \times 10^{-19}$ C is the electron charge. The stopping potential gives you a direct measurement of the maximum kinetic energy without needing to measure electron speeds.

Different types of electromagnetic radiation (visible light, ultraviolet, X-rays) have different frequencies and therefore different photon energies. Higher-frequency radiation is more effective at ejecting electrons and gives them more kinetic energy.