Einstein's Photoelectric Equation — Revision Notes

⚡ 30-Second Revision

Einstein's Equation: — $K_{max} = h

u - phi$

Photon Energy: — $E = h

u = hc/lambda$

Work Function: — $phi = h

u_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, $K_{max} = h u - phi$, 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 ($h u$) minus the work function ($phi$) of the metal.

The photon energy $E = h u$ is directly proportional to frequency ($u$) 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 ($u_0 = phi/h$) and threshold wavelength ($lambda_0 = hc/phi$).

No emission occurs below $u_0$ 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: $K_{max} = h u - phi$. This equation is based on the quantum nature of light, where light consists of discrete energy packets called photons.

Each photon carries energy $E = h u$, where $h$ is Planck's constant and $u$ 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 ($

u_0$):** Emission only occurs if$h u ge phi$. The minimum frequency for emission is$ u_0 = phi/h$. Below$ u_0$, no electrons are emitted, regardless of light intensity.

1
Threshold Wavelength ($lambda_0$): — Correspondingly, the maximum wavelength for emission is $lambda_0 = hc/phi$. If $lambda > lambda_0$, no emission.
2
Instantaneous Emission: — The photon-electron interaction is immediate, explaining why electrons are ejected almost instantly (within $10^{-9}$ s) if $

u ge u_0$.

1
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 $eV_0 = h u - phi$. A plot of $V_0$ vs. $u$ yields a straight line with slope $h/e$ and x-intercept $u_0$. 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

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

1
Photon Energy: — $E = h

u = hc/lambda$. (Use$hc approx 1240, ext{eV}cdot ext{nm}$ for quick calculations).

1
Work Function ($phi$): — Characteristic property of the metal. Determines how tightly electrons are bound.
2
**Threshold Frequency ($

u_0$):** Minimum frequency for emission.$phi = h u_0$. If$ u < u_0$, no emission.

1
Threshold Wavelength ($lambda_0$): — Maximum wavelength for emission. $phi = hc/lambda_0$. If $lambda > lambda_0$, no emission.
2
Stopping Potential ($V_0$): — Minimum retarding potential to stop $K_{max}$ electrons. $K_{max} = eV_0$.
3
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 $u$ (not intensity):** Each photon's energy $h u$ determines $K_{max}$. * Photoelectric Current depends on Intensity: More photons (higher intensity) means more electrons, thus higher current.

1
Graphical Representations:

* **$K_{max}$ vs. $u$:** Straight line with slope $h$, x-intercept $u_0$, y-intercept $-phi$. * **$V_0$ vs. $u$:** Straight line with slope $h/e$, x-intercept $u_0$, y-intercept $-phi/e$. * Photoelectric Current vs. Intensity: Straight line through origin (for $u > u_0$). * 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 ($u$) - P ($phi$)

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