Chapter 7, Problem 7.122SP

Interpretation Introduction

Interpretation:

The velocity (in $\text{m/s}$) of an ejected electron when the light with a wavelength of $\text{313 nm}$ is employed should be calculated using the given equation.

Concept Introduction:

The speed, wavelength and frequency of a wave are interrelated by $\text{c }=\text{ λν}$ where $\text{λ and ν}$ are mentioned in meters ($\text{m}$) and reciprocal seconds (${\text{s}}^{-1}$).

The photoelectric effect was not explained by the wave theory of light which is associated with the energy of light to its intensity.  Einstein prepared a best guess.  He suggested that a beam of light is a stream of particles.  These “particles” are known as photons.  Einstein worked out that each photon should possess some energy $\text{E}$,

${\text{E}}_{\text{photon}}=\text{ hν}$

Where, $\text{h}$ is called Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$) and $\text{ν}$ is the frequency of the light.  Electrons are arranged in a metal by their attractive forces.  Light of a high frequency (which corresponds to a high energy) requires removing the electrons from the metal which break them free.  If the frequency of the photons is $\text{hν}$ which exactly equals the energy which binds the electrons in the metal, the light have sufficient energy to move the electrons loose.  If a light of higher frequency is used, the electrons are removed form the metal and also some kinetic energy is acquired.  This can be summarized by the equation:

$\text{hν }=\text{ KE + W}$

Where, $\text{KE}$ is the kinetic energy of the ejected electron and $\text{W}$ is the binding energy of the electron in the metal.  Rearrange the above equation as

$\text{KE }=\text{ hν }-\text{ W}$

From this equation, when the photon is more energetic (i.e., the higher its frequency), the kinetic energy of the ejected electron is larger.  If the frequency of light is below the threshold frequency, the photon moves away from the surface and no electrons will be ejected.  If the frequency is equal to the threshold frequency, it removes the most loosely attached electron.  If the frequency is above the threshold frequency, it will not only remove the electron, but also require certain kinetic energy to the ejected electron.

To calculate: Calculate the velocity (in $\text{m/s}$) of an ejected electron when the light with a wavelength of $\text{313 nm}$