38-2 The Photoelectric Effect

38-2 The Photoelectric Effect#

Prompts

  • Sketch a photoelectric setup: incident light, metal target, ejected electrons, collector. What is the stopping potential \(V_{\text{stop}}\) and how does it relate to \(K_{\max}\)?

  • Classical puzzle: Why does \(K_{\max}\) not depend on light intensity? Why can light below a cutoff frequency \(f_0\) never eject electrons, no matter how bright?

  • Write Einstein’s photoelectric equation \(hf = K_{\max} + \Phi\). What is the work function \(\Phi\)? What is \(f_0\) in terms of \(\Phi\)?

  • A plot of \(V_{\text{stop}}\) vs \(f\) is a straight line. What is its slope? How can this be used to measure \(h\)?

  • Light of wavelength 400 nm strikes a metal with work function 2.0 eV. What is \(K_{\max}\) for the ejected electrons?

Lecture Notes#

Overview#

  • The photoelectric effect: light of short enough wavelength incident on a metal surface ejects electrons (photoelectrons).

  • Einstein’s photon model explains two results that baffled classical wave theory: (1) \(K_{\max}\) is independent of light intensity; (2) there is a cutoff frequency \(f_0\) below which no electrons are ejected, regardless of intensity.

  • Einstein’s equation \(hf = K_{\max} + \Phi\) expresses energy conservation. The work function \(\Phi\) is the minimum energy needed for an electron to escape the metal.

  • A plot of stopping potential \(V_{\text{stop}}\) vs frequency \(f\) is linear with slope \(h/e\), providing a direct measurement of Planck’s constant.

The experiment#

Light of frequency \(f\) shines on a clean metal target. Electrons are ejected and collected by a collector cup. A potential difference \(V\) between target and collector can retard the electrons. The stopping potential \(V_{\text{stop}}\) is the value of \(V\) at which the photocurrent drops to zero—the most energetic electrons are turned back just before reaching the collector.

The maximum kinetic energy of the ejected electrons is

(352)#\[ K_{\max} = eV_{\text{stop}} \]

Two puzzles for classical physics#

Puzzle 1: For a given frequency, \(K_{\max}\) does not depend on the intensity of the light. Dazzling light and dim light give the same maximum kinetic energy. Classically, a stronger oscillating field should give a bigger “kick” to the electron.

Puzzle 2: Below a certain cutoff frequency \(f_0\) (or above a cutoff wavelength \(\lambda_0 = c/f_0\)), no electrons are ejected—no matter how intense the light. Classically, one would expect that sufficiently bright light of any frequency could supply enough energy to eject electrons.

Einstein’s photon explanation#

Photon model: Each photon has energy \(hf\). One photon is absorbed by one electron. Increasing intensity increases the number of photons, not the energy per photon. So \(K_{\max}\) (from one photon–one electron) is unchanged.

Electrons in the metal are bound; they need a minimum energy \(\Phi\) (the work function) to escape. If \(hf < \Phi\), a single photon cannot supply enough energy—no electron escapes, regardless of how many photons arrive. If \(hf \geq \Phi\), electrons can escape; any energy beyond \(\Phi\) becomes kinetic energy.

The photoelectric equation#

Energy conservation for one photon absorbed by one electron:

(353)#\[ hf = K_{\max} + \Phi \]

or

(354)#\[ K_{\max} = hf - \Phi \]

  • Work function \(\Phi\): minimum energy to escape; property of the metal.

  • Cutoff frequency \(f_0 = \Phi/h\): for \(f < f_0\), \(hf < \Phi\), so no ejection.

  • Cutoff wavelength \(\lambda_0 = hc/\Phi\): for \(\lambda > \lambda_0\), no ejection.

Using \(K_{\max} = eV_{\text{stop}}\):

(355)#\[ V_{\text{stop}} = \frac{h}{e}\,f - \frac{\Phi}{e} \]

A plot of \(V_{\text{stop}}\) vs \(f\) is a straight line with slope \(h/e\) and vertical intercept \(-\Phi/e\). Millikan’s 1916 experiment confirmed this and measured \(h\).

Example

Light \(\lambda = 400\) nm, work function \(\Phi = 2.0\) eV. Photon energy \(hf = hc/\lambda = 1240\,\text{eV}\cdot\text{nm}/400\,\text{nm} = 3.1\) eV. So \(K_{\max} = hf - \Phi = 3.1 - 2.0 = 1.1\) eV. (Use \(hc \approx 1240\) eV·nm.)

Summary#

  • Photoelectric effect: light ejects electrons from metals; \(K_{\max} = eV_{\text{stop}}\).

  • Einstein’s equation \(hf = K_{\max} + \Phi\); work function \(\Phi\); cutoff \(f_0 = \Phi/h\).

  • Photon model explains: \(K_{\max}\) independent of intensity; cutoff frequency exists.

  • \(V_{\text{stop}}\) vs \(f\): linear, slope \(h/e\); measure \(h\) from photoelectric data.