Einstein's Photoelectric Equation — Revision Notes

⚡ 30-Second Revision

Einstein's Equation: — $K_{max} = h\nu - \phi$
Photon Energy: — $E = h\nu = hc/\lambda$
Work Function: — $\phi = h\nu_0 = hc/\lambda_0$
Stopping Potential: — $K_{max} = eV_0$
Constants: — $h = 6.63 \times 10^{-34},\text{J}\cdot\text{s}$, $c = 3 \times 10^8\,\text{m/s}$, $e = 1.6 \times 10^{-19},\text{C}$
Useful Conversion: — $1\,\text{eV} = 1.6 \times 10^{-19},\text{J}$
Shortcut: — $hc \approx 1240\,\text{eV}\cdot\text{nm}$

2-Minute Revision

Einstein's Photoelectric Equation, $K_{max} = h\nu - \phi$, is central to understanding the photoelectric effect. It states that the maximum kinetic energy of an emitted electron ($K_{max}$) equals the energy of the incident photon ($h\nu$) minus the work function ($\phi$) of the metal.

The photon energy $E = h\nu$ is directly proportional to frequency ($\nu$) and inversely proportional to wavelength ($\lambda$, as $E = hc/\lambda$). The work function $\phi$ is the minimum energy required for an electron to escape, defining the threshold frequency ($\nu_0 = phi/h$) and threshold wavelength ($\lambda_0 = hc/\phi$).

No emission occurs below $\nu_0$ or above $\lambda_0$. The stopping potential ($V_0$) is the retarding voltage needed to stop the most energetic electrons, where $K_{max} = eV_0$. Crucially, light intensity affects the number of emitted electrons (photoelectric current), while light frequency affects their maximum kinetic energy.

Emission is instantaneous, contradicting classical wave theory.

5-Minute Revision

The photoelectric effect, where electrons are ejected from a metal by light, is explained by Einstein's Photoelectric Equation: $K_{max} = h\nu - \phi$. This equation is based on the quantum nature of light, where light consists of discrete energy packets called photons.

Each photon carries energy $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the light's frequency. When a photon strikes an electron, it transfers all its energy. A part of this energy, the work function ($\phi$), is used to liberate the electron from the metal surface.

The remaining energy becomes the electron's maximum kinetic energy ($K_{max}$). Electrons deeper in the metal or those undergoing collisions will have less than $K_{max}$.

Key implications:

1
Threshold Frequency ($\nu_0$): — Emission only occurs if $h\nu \ge \phi$. The minimum frequency for emission is $\nu_0 = phi/h$. Below $\nu_0$, no electrons are emitted, regardless of light intensity.
2
Threshold Wavelength ($\lambda_0$): — Correspondingly, the maximum wavelength for emission is $\lambda_0 = hc/\phi$. If $\lambda > \lambda_0$, no emission.
3
Instantaneous Emission: — The photon-electron interaction is immediate, explaining why electrons are ejected almost instantly (within $10^{-9}$ s) if $\nu \ge \nu_0$.
4
Intensity vs. Frequency: — Light intensity (number of photons) determines the photoelectric current (number of emitted electrons), while light frequency (photon energy) determines the maximum kinetic energy of the emitted electrons.

Experimentally, $K_{max}$ can be measured using the stopping potential ($V_0$), where $K_{max} = eV_0$. This leads to the relation $eV_0 = h\nu - \phi$. A plot of $V_0$ vs. $\nu$ yields a straight line with slope $h/e$ and x-intercept $\nu_0$. Remember to use consistent units in calculations; $hc \approx 1240\,\text{eV}\cdot\text{nm}$ is a handy constant for wavelength-energy conversions.

Prelims Revision Notes

1
Photoelectric Effect: — Emission of electrons from a metal surface when light falls on it.
2
Einstein's Photoelectric Equation: — $K_{max} = h\nu - \phi$

* $K_{max}$: Maximum kinetic energy of emitted photoelectron. * $h$: Planck's constant ($6.626 \times 10^{-34},\text{J}\cdot\text{s}$ or $4.136 \times 10^{-15},\text{eV}\cdot\text{s}$). * $\nu$: Frequency of incident light. * $\phi$: Work function of the metal (minimum energy to escape).

1
Photon Energy: — $E = h\nu = hc/\lambda$. (Use $hc \approx 1240\,\text{eV}\cdot\text{nm}$ for quick calculations).
2
Work Function ($\phi$): — Characteristic property of the metal. Determines how tightly electrons are bound.
3
Threshold Frequency ($\nu_0$): — Minimum frequency for emission. $\phi = h\nu_0$. If $\nu < \nu_0$, no emission.
4
Threshold Wavelength ($\lambda_0$): — Maximum wavelength for emission. $\phi = hc/\lambda_0$. If $\lambda > \lambda_0$, no emission.
5
Stopping Potential ($V_0$): — Minimum retarding potential to stop $K_{max}$ electrons. $K_{max} = eV_0$.
6
Key Observations Explained by Einstein's Equation:

* Threshold Frequency: Explained by $\phi$. * Instantaneous Emission: Photon-electron interaction is one-to-one and immediate. * **$K_{max}$ depends on $\nu$ (not intensity):** Each photon's energy $h\nu$ determines $K_{max}$. * Photoelectric Current depends on Intensity: More photons (higher intensity) means more electrons, thus higher current.

1
Graphical Representations:

* **$K_{max}$ vs. $\nu$:** Straight line with slope $h$, x-intercept $\nu_0$, y-intercept $-\phi$. * **$V_0$ vs. $\nu$:** Straight line with slope $h/e$, x-intercept $\nu_0$, y-intercept $-phi/e$. * Photoelectric Current vs. Intensity: Straight line through origin (for $\nu > \nu_0$). * Photoelectric Current vs. Potential: Saturates at positive potential, becomes zero at $V_0$ (negative potential).

1
Unit Conversions: — Be proficient in converting between Joules and electron volts ($1\,\text{eV} = 1.6 \times 10^{-19},\text{J}$).

Vyyuha Quick Recall

Einstein's Photoelectric Equation: Kids Have Nice Photos.

K ($K_{max}$) = H ($h$) N ($\nu$) - P ($\phi$)

This helps remember the main variables and their relationship in the equation.