Photoelectric effect and stopping voltage
Problem Statement
Photoelectric effect and stopping voltage
Problem Statement
The phenomenon in which electrons are emitted from a metal surface when light is shone on it is called the photoelectric effect. These emitted electrons are called photoelectrons.
In a vacuum, a metal plate (cathode) is shone with monochromatic light of wavelength λ. If the work function of this metal is W, the maximum kinetic energy Kmax of the emitted photoelectrons can be expressed by the following photoelectric equation using Planck's constant h and the speed of light in a vacuum c:
Kmax=λhc−WIn this case, by applying an appropriate positive or negative voltage (potential difference) to the cathode, it is possible to prevent photoelectrons from reaching the opposing electrode (anode). The magnitude of the voltage at which photoelectrons can no longer reach the anode (the photocurrent becomes 0) is called the stopping voltage (stopping potential) V0. When an electron with electric charge (absolute value of charge) e moves against an electrostatic force due to a voltage, the magnitude of the work done by the electrostatic force is eV0.
According to the law of conservation of energy, the maximum kinetic energy of the photoelectron, Kmax, is equal to the magnitude of the work done by this electrostatic force. That is, the following relationship holds:
Let V0 be the magnitude of the stopping voltage when a metal is irradiated with monochromatic light of wavelength λ, and let V0′ be the magnitude of the stopping voltage when the same metal is irradiated with monochromatic light of wavelength 21λ. Find the ratio of the stopping voltages V0V0′. However, assume that the work function W is 3λhc.
Constraints
- h=6.6×10−34 J⋅s
- c=3.0×108 m/s
- e=1.6×10−19 C
- λ=6.0×10−7 m
Input Format
When the calculated ratio V0V0′ is expressed as an irreducible fraction qp, input the value of the sum of the numerator and denominator p+q.
Solution
Explanation
Application of the Photoelectric Equation
From the relationship between the photoelectric equation and the stopping voltage given in the problem statement, the stopping voltage V0 can be expressed as follows:
eV0=λhc−WSubstitute the work function of the metal, W=3λhc, here:
eV0=λhc−3λhc=3λ2hcTherefore, the stopping voltage V0 at wavelength λ is as follows:
V0=3eλ2hcStopping Voltage When Wavelength Changes
Next, we find the stopping voltage V0′ when the same metal is irradiated with monochromatic light of wavelength 21λ. Substitute 21λ into the wavelength part of the photoelectric equation.
eV0′=21λhc−W eV0′=λ2hc−3λhc=3λ5hcTherefore, the stopping voltage V0′ at wavelength 21λ is as follows:
V0′=3eλ5hcCalculation of the Ratio of Blocking Voltages
Calculate the ratio of the two blocking voltages V0V0′.
V0V0′=3eλ2hc3eλ5hc=25This result is an irreducible fraction 25 and does not depend on the specific values of Planck's constant h, the speed of light c, the electric charge e, and the wavelength λ.
Determination of Input Numbers
The obtained ratio is an irreducible fraction qp=25.
The numerator p=5 and the denominator q=2 give their sum 5+2=7.
The natural number to enter is 7.