What Is Stopping Voltage? (Physics Formula)
Stopping voltage is the smallest retarding potential that keeps even the fastest photoelectrons from reaching a collector. Its positive magnitude is (V_s=K_{\max}/e=(hf-\phi)/e), when light causes electrons to escape. Here, (h) is Planck’s constant, (f) is light frequency, (\phi) is the material’s work function, and (e) is the electron-charge magnitude.
Understanding this formula is a useful investment of study time: it connects a voltage reading to the energy carried by light and helps you check whether a calculation or experiment makes sense. The key is to keep two ideas separate: the light must have enough energy to release electrons, and the stopping voltage measures the greatest kinetic energy those electrons have after escape.
Stopping Voltage: Definition, Photoelectric-Effect Context & Formula
Stopping voltage is the retarding voltage needed to prevent the most energetic emitted electrons from reaching a collector. It is measured in volts and is usually written as a positive magnitude, (V_s). It does not stop the light; it opposes the motion of electrons released in the photoelectric effect.
In a basic photoelectric experiment, light shines on a metal surface called the emitter. If the light has enough energy, some electrons leave that surface and travel toward a collector. A retarding electric potential opposes their motion. As the retarding potential grows, fewer electrons reach the collector. The stopping voltage is the value at which even the fastest emitted electrons cannot reach it.
The relationship is:
[ eV_s=K_{\max} ]
So:
[ V_s=\frac{K_{\max}}{e} ]
Here, (K_{\max}) is the maximum kinetic energy of an emitted electron, and (e) is the magnitude of the electron’s charge. Since a potential difference transfers energy (eV_s) to a charge of magnitude (e), the equation links voltage directly to electron energy.
Einstein’s photoelectric equation gives that maximum kinetic energy:
[ K_{\max}=hf-\phi ]
Combining the equations gives the stopping-voltage formula:
[ V_s=\frac{hf-\phi}{e} =\frac{h}{e}f-\frac{\phi}{e} ]
This formula applies when electrons are emitted. It also makes the roles clear: at a fixed material, higher light frequency means a higher stopping voltage. The work function sets how much energy the light must first supply to free an electron.
Derivation, Variables, Units & Threshold Conditions
The stopping-voltage equation follows from energy conservation. A photon supplies energy (hf); the material uses (\phi) to release an electron; the remaining energy becomes that electron’s kinetic energy. A retarding potential removes this kinetic energy, so the stopping condition is (eV_s=K_{\max}).
The symbols and units are:
| Symbol | Meaning | SI unit |
|---|---|---|
| (V_s) | Positive magnitude of stopping voltage | volt (V) |
| (K_{\max}) | Maximum photoelectron kinetic energy | joule (J) |
| (e) | Magnitude of electron charge | coulomb (C) |
| (h) | Planck’s constant | joule-second (J·s) |
| (f) | Frequency of incident light | hertz (Hz) |
| (\phi) | Work function of the material | joule (J) or electronvolt (eV) |
A work function is the minimum energy needed to free an electron from a material’s surface. The threshold frequency is the lowest frequency that can supply this energy:
[ f_0=\frac{\phi}{h} ]
Photoemission requires (f\geq f_0) in the ideal single-photon model. At exactly the threshold, the emitted electron has no kinetic energy left over, so the ideal stopping voltage is zero. Below threshold, the photon energy is too small to release electrons. Raising intensity does not make up for that missing energy in this model.
If a problem gives wavelength instead of frequency, use:
[ f=\frac{c}{\lambda} ]
where (c) is the speed of light and (\lambda) is wavelength. Shorter wavelength means higher frequency and greater photon energy. Convert wavelength to metres before using the SI form of the equation.
Units matter. If (K_{\max}) is in joules, divide by (e) in coulombs to obtain volts. If energy is in electronvolts, the numerical value of the stopping voltage in volts is the same. For example, an electron with (0.8\ \text{eV}) of maximum kinetic energy has a stopping-voltage magnitude of (0.8\ \text{V}).
Worked Calculations, Graphs & Common Conceptual Errors
A reliable calculation follows a short sequence: identify the material and light, check for emission, calculate the maximum kinetic energy, then convert it to stopping voltage. I use this order because it catches a common mistake: calculating a negative kinetic energy when the photon does not have enough energy to release electrons.
Example: 400 nm light on a surface with a 2.30 eV work function
- Convert wavelength to photon energy. A handy equivalent is (hc\approx1240\ \text{eV·nm}):
[ E_{\text{photon}}=\frac{1240}{400}=3.10\ \text{eV} ]
-
Compare photon energy with the work function. Since (3.10>2.30), emission is possible.
-
Find the maximum kinetic energy:
[ K_{\max}=3.10-2.30=0.80\ \text{eV} ]
- Convert to stopping voltage:
[ V_s=0.80\ \text{V} ]
This result is the positive magnitude of the retarding potential.
Compare different materials and wavelengths
| Light and surface | Photon energy | Work function | Result |
|---|---|---|---|
| 400 nm, (\phi=2.30\ \text{eV}) | 3.10 eV | 2.30 eV | Emission; (V_s\approx0.80\ \text{V}) |
| 500 nm, (\phi=2.30\ \text{eV}) | 2.48 eV | 2.30 eV | Emission; (V_s\approx0.18\ \text{V}) |
| 400 nm, (\phi=4.50\ \text{eV}) | 3.10 eV | 4.50 eV | No emission in the ideal model |
The table shows why wavelength alone does not determine stopping voltage. The material’s work function matters too. The same light can release electrons from one surface but not another.
What the stopping-voltage graph shows
Rearrange the formula as:
[ V_s=\frac{h}{e}f-\frac{\phi}{e} ]
A graph of stopping voltage against frequency is a straight line in the ideal model. Its slope is:
[ \frac{h}{e}\approx4.136\times10^{-15}\ \text{V·s} ]
The line’s frequency-axis intercept is the threshold frequency (f_0), where (V_s=0). If you measure the slope, you can estimate Planck’s constant using (h=e\times\text{slope}). This is one reason the photoelectric effect is useful in physics experiments.
Errors to avoid
- Do not treat intensity as a way to raise stopping voltage. At a fixed frequency above threshold, greater intensity mainly increases the number of emitted electrons per unit time. Stopping voltage tracks the maximum electron energy, which depends on frequency and work function.
- Do not confuse the sign of a circuit setting with (V_s). A diagram may show the collector at a negative voltage relative to the emitter. The formula here uses the positive magnitude of the retarding potential.
- Do not use the equation to claim emission when (hf<\phi). In that case, the ideal model predicts no photoelectrons and no measured photoelectric stopping voltage.
- Do not mix joules and electronvolts without conversion. Either unit system works, but keep the energy terms consistent.
Experimental Applications, Measurement Limits & Next Physics Topics
A photoelectric experiment tests how light frequency relates to electron energy. The measured stopping voltage can be used to estimate the maximum kinetic energy, identify a threshold frequency, or estimate Planck’s constant from a graph. These are direct uses of the formula, not separate rules.
A typical measurement records the photocurrent, the current caused by photoelectrons reaching the collector, as the retarding voltage changes. The current falls as the retarding potential increases. The stopping voltage is found where the photocurrent reaches zero, or is estimated from the trend near zero. The exact method depends on the apparatus and the quality of the readings.
Two useful experimental scenarios
- Changing frequency: Keep the material and other conditions the same, then measure stopping voltage for several light frequencies above threshold. A rising, nearly straight-line trend supports the predicted relation. The slope can be used to estimate (h/e).
- Changing intensity: Keep frequency above threshold and change intensity. The current may change because the emission rate changes, while the stopping-voltage magnitude should remain about the same in the ideal model.
Real measurements may not match the simple line perfectly. Surface condition can affect the work function, while contact potentials between materials can shift measured voltages. Weak signals, instrument limits, and uncertainty in locating the zero-current point can also affect results. These effects do not change the basic model, but they matter when interpreting experimental data.
For a careful analysis, record the light frequency, material, applied-voltage convention, and uncertainty in each reading. Check that the fitted graph uses frequency on the horizontal axis and stopping-voltage magnitude on the vertical axis. Then compare its slope and frequency intercept with the theoretical values. A discrepancy is a prompt to examine measurement conditions, not an automatic reason to discard the model.
Frequently asked questions
What is stopping voltage in simple terms?
It is the retarding voltage that prevents even the fastest photoelectrons from reaching the collector.
What is the stopping-voltage formula?
Use (V_s=K_{\max}/e=(hf-\phi)/e), for light that causes photoemission. (V_s) is the positive magnitude.
Why is stopping voltage measured in volts?
Voltage is energy transferred per unit charge. A retarding potential removes the photoelectrons’ kinetic energy.
Does brighter light increase stopping voltage?
Not in the ideal model if frequency and material stay fixed. Brighter light mainly increases the rate of electron emission.
What happens below the threshold frequency?
The photon energy is below the work function, so the ideal single-photon model predicts no photoemission, even if intensity increases.
What is the threshold frequency formula?
It is (f_0=\phi/h), where (\phi) is the material’s work function and (h) is Planck’s constant.
Can stopping voltage be negative?
The retarding voltage in a circuit may be shown with a negative sign, depending on how voltage is defined. In the formula here, (V_s) means its positive magnitude.
How does wavelength affect stopping voltage?
For a fixed material, shorter wavelength means higher frequency and more photon energy. Above threshold, this generally increases stopping voltage.
What does the slope of a stopping-voltage graph mean?
For a graph of (V_s) against (f), the ideal slope is (h/e), about (4.136\times10^{-15}\ \text{V·s}).
Why might a measured graph differ from the ideal line?
Surface conditions, contact potentials, measurement uncertainty, and instrument limits can shift or spread the data.
(This article was written by one of our staff writers, Nathan Sterling. Visit our Meet the Team page.)
