# Physics307L F08:People/Mondragon/Notebook/071017/dataprocessing

## Theory

The ${\displaystyle {\tfrac {h}{e}}}$ apparatus is basically a capacitor. One plate is a photocathode. Light shining on the photocathode gives electrons the energy necessary to escape the surface of the cathode and to possibly cross the gap to the anode. As charge builds up on the anode, the voltage across the capacitor will increase, as will the energy necessary for electrons to cross the gap.

Just for future reference, I doubt I'll be using it though ${\displaystyle q=C*V}$, where ${\displaystyle q}$ is the absolute value of charge on one plate of the capacitor, ${\displaystyle V}$ is the absolute value of the voltage across the capacitor, and ${\displaystyle C}$ is the capacitance of the capacitor.

The energy needed by an electron to escape the photocathode's surface is ${\displaystyle W_{0}}$. ${\displaystyle W_{0}}$ is a property of the photocathode. After light imparts its energy onto en electron, part of the energy will be used to escape the photocathode (if ${\displaystyle E_{light}\geq W_{0}}$), and the rest manifests itself as the kinetic energy of the now free electron, ${\displaystyle KE_{max}}$. ${\displaystyle E_{light}=W_{0}+KE_{max}}$

Because of the voltage across the capacitor, ${\displaystyle KE_{max}}$ must be more than a certain potential energy ${\displaystyle PE_{gap}}$ created by the voltage difference. The potential energy that needs to be overcome is proportional to the voltage difference. When ${\displaystyle PE_{gap}\geq KE_{max}}$, electrons can no longer reach the anode so the voltage no longer rises. This final voltage is called ${\displaystyle V_{stop}}$. Because of the way capacitors charge, ${\displaystyle V_{stop}}$ is never actually reached in a finite amount of time. ${\displaystyle V_{stop}}$ is proportional to ${\displaystyle KE_{max}}$.

Electrons can only make it to the anode if ${\displaystyle KE_{max}. ${\displaystyle KE_{max}}$ should, in this experiment, stay the same thoughout a test, but ${\displaystyle PE_{gap}}$ changes as ${\displaystyle V}$ changes. When ${\displaystyle V_{stop}}$ is reached, it is because ${\displaystyle KE_{max}=PE_{gap}=e\,V_{stop}}$

{\displaystyle {\begin{aligned}E_{light}&=W_{0}+KE_{max}\\E_{light}&=W_{0}+e\,V_{stop}\\{\frac {E_{light}}{e}}&={\frac {W_{0}}{e}}+V_{stop}\\\end{aligned}}}

According to the particle theory of light, the energy imparted by light is proportional to the frequency of the light, the constant of proportionality being Plank's constant. According to the wave theory of light, the energy imparted by light is proportional to the square root of the intensity of the light. The data shows that the stopping voltage doesn't change when the light intensity changes, but it does when the light frequency changes. So

{\displaystyle {\begin{aligned}E_{light}&=W_{0}+KE_{max}\\E_{light}&=W_{0}+e\,V_{stop}\\{\frac {E_{light}}{e}}&={\frac {W_{0}}{e}}+V_{stop}&={\frac {h\nu }{e}}\\V_{stop}&={\frac {h\nu }{e}}-{\frac {W_{0}}{e}}\\\end{aligned}}}

ν is the frequency of the light falling on the photocathode and ${\displaystyle h}$ is Plank's constant.