Modern Physics Concept Page - 4

Formula

Introduction to maximum kinetic energy

What happens when a light of is incident on a metal? Let us say that the frequency of the light is more than threshold frequency. Hence we know that electrons may emit from the metal surface. But we haven't covered the further story. Will the electron come out for sure? No. There is no surity of ejection of the electron. Why?
Let us understand this. When a light is incident on the metal, the electrons may or may not acquire the energy of the light. What if an electron acquires all the energy of the light. This electron now has capability to come out. But electrons oscillate continously.  So this electron may collide with other electrons. Some energy will be lost in this process. Hence the electrons which are on the surface of the metal and headed out of the metal surface will come out.  

Example

Example

Example: A radiation of wave length 2500 A0$A^0$ is incident on a metal plate whose work function is 3.5 eV. Then the energy and momentum of the photo electrons emitted by the surface is (h=6.63×10−34$h=6.63\times {10}^{-34}$Js and c=3×108$c=3\times {10}^8$ m/s)Given:λ=2500×10−10m$\lambda =2500\times {10}^{-10}m$E=6.63×3×10−262500×10−10=7.95×10−19J$E={\displaystyle \frac{6.63\times 3\times {10}^{-26}}{2500\times {10}^{-10}}}=7.95\times {10}^{-19}J$p=Ec=7.95×10−193×108=2.65×10−27kg.m.s−1$p=\frac{E}{c}=\frac{7.95\times {10}^{-19}}{3\times {10}^8}=2.65\times {10}^{-27}kg.m.s^{-1}$

Formula

Energy and momentum of photoelectrons

Energy:
E=hν=hcλ$E=h\nu ={\displaystyle \frac{hc}{\lambda }}$
Momentum:
p=hλ=Ec$p={\displaystyle \frac{h}{\lambda }}={\displaystyle \frac{E}{c}}$

Definition

Photocell

A photocell is a technological application of the photoelectric effect.A photocell consists of a semi-cylindrical photo-sensitive metal plate C (emitter) and a wire loop A(collector) supported in an evacuated glass or quartz bulb. It is connected to the external circuit having a high-tension battery B and microammeter (μ$\mu$A). When light of suitable wavelength falls on the emitter C, photoelectrons are emitted. These photoelectrons are drawn to the collector A.A photocell converts a change in intensity of illumination into a change in photocurrent.

Definition

Saturation current

If the potential of the anode is increased gradually, a situation arrives when the effect of the space charge becomes negligible and an electron that is emitted from the cathode is able to reach the anode. The current then becomes constant and is known as the saturation current. Further increase in the anode potential does not change the magnitude of the photocurrent.

Definition

Stopping potential

The smallest magnitude of the anode potential which just stops the photocurrent is called the stopping potential.
This potential should stop even the ost energitic photoelectron. Hence 
Tha value of maximum kinetic energy should be equal to eV0$e{V}_0$
We know Kmax=hν−ϕ=hcλ−ϕ$K_{m}ax=h\nu -\varphi ={\displaystyle \frac{hc}{\lambda }}-\varphi$
V0=hcλe−ϕe$V_0={\displaystyle \frac{hc}{\lambda e}}-{\displaystyle \frac{\varphi }{e}}$

Definition

Factors affecting stopping potential

The stopping potential V0$V_0$ depends on the wavelength of light and the work function of the metal. It doesn't depend on the intensity of light.

Definition

frequency vs potential

The maximum kinetic energy of the photoelectrons varies linearly with the frequency of incident radiation, but is independent of its intensity.
For a frequency of ν$\nu$ of incident radiation, lower than the threshold frequency ν0${\nu }_0$ no photoelectric emission is possible even if the intensity is large.

Definition

Einstein's photoelectric equation

Einstein's photoelectric equation :
Kmax=hcλ−φ=hν−φ$K_{max}={\displaystyle \frac{hc}{\lambda }}-\phi =h\nu -\phi$
The kinetic energy of the photoelectron coming out may be anything between zero and (E−φ)$(E-\phi )$ where E=hcλ$E={\displaystyle \frac{hc}{\lambda }}$ is the energy supplied to the individual electrons.
Kmax=E−φ$K_{max}=E-\phi$

Diagram

Plot of stopping potential vs frequency of incident light

We know the relation of stopping potential V0$V_0$ with frequency ν$\nu$ is 
V0=heν−We$V_0={\displaystyle \frac{h}{e}}\nu -{\displaystyle \frac{W}{e}}$
where W$W$ is the work function of the metal.
If we want to plot the stopping potential V0$V_0$ vs frequency ν$\nu$, it will be a straight line with slope he${\displaystyle \frac{h}{e}}$ and negative y intercept We

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