Determining the Planck Constant

Planck Constant ($h$): A fundamental constant in quantum mechanics, approximately $6.63 \times 10^{-34} \text{ J} \cdot \text{s}$, which defines the scale of energy quantization.

Photon Energy: The energy of a single quantum of light is directly proportional to its frequency ($f$) and inversely proportional to its wavelength ($\lambda$), expressed as $E = hf = \frac{hc}{\lambda}$.

Threshold Voltage ($V_{th}$): The minimum potential difference required across an LED for it to begin emitting light, representing the point where electrons have sufficient energy to cross the semiconductor bandgap.

Energy Conversion: The experiment relies on the principle that the electrical potential energy gained by an electron ($E = eV$) is converted entirely into the energy of a single emitted photon ($E = \frac{hc}{\lambda}$).

Quantization: By assuming a one-to-one relationship where one electron crossing the junction produces one photon, we can equate the two energy expressions: $eV = \frac{hc}{\lambda}$.

Linear Relationship: Rearranging the energy equation into the form $V = (\frac{hc}{e}) \cdot \frac{1}{\lambda}$ shows that the threshold voltage is directly proportional to the reciprocal of the wavelength.

Wavelength vs. Frequency: While both can be used for the x-axis, plotting against $1/\lambda$ is more common in LED experiments because LED specifications are typically provided in nanometers.

Gradient Components: The slope of the $V$ vs. $1/\lambda$ graph is $\frac{hc}{e}$, meaning the Planck constant is derived by multiplying the gradient by $\frac{e}{c}$.

Unit Consistency: Always ensure wavelengths are converted from nanometers ($10^{-9} \text{ m}$) to meters before calculating the reciprocal for the x-axis.

Slope Calculation: When determining the gradient from a graph, use a large triangle covering at least half of the line of best fit to minimize percentage uncertainty.

Percentage Error: Be prepared to compare your experimental value of $h$ with the accepted value ($6.63 \times 10^{-34} \text{ J} \cdot \text{s}$) to evaluate the accuracy of the experiment.