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 $\lambda$λ. If the work function of this metal is $W$W, the maximum kinetic energy $K_{\text{max}}$Kmax​ of the emitted photoelectrons can be expressed by the following photoelectric equation using Planck's constant $h$h and the speed of light in a vacuum $c$c:

$$K_{\text{max}} = \frac{hc}{\lambda} - W$$Kmax​=λhc​−W

In 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$0) is called the stopping voltage (stopping potential) $V_0$V0​. When an electron with electric charge (absolute value of charge) $e$e moves against an electrostatic force due to a voltage, the magnitude of the work done by the electrostatic force is $eV_0$eV0​.

According to the law of conservation of energy, the maximum kinetic energy of the photoelectron, $K_{\text{max}}$Kmax​, is equal to the magnitude of the work done by this electrostatic force. That is, the following relationship holds:

$$$$

Let $V_0$V0​ be the magnitude of the stopping voltage when a metal is irradiated with monochromatic light of wavelength $\lambda$λ, and let $V_0'$V0′​ be the magnitude of the stopping voltage when the same metal is irradiated with monochromatic light of wavelength $\frac{1}{2}\lambda$21​λ. Find the ratio of the stopping voltages $\frac{V_0'}{V_0}$V0​V0′​​. However, assume that the work function $W$W is $\frac{hc}{3\lambda}$3λhc​.

Constraints

• $h = 6.6 \times 10^{-34} \ \text{J}\cdot\text{s}$h=6.6×10−34 J⋅s
• $c = 3.0 \times 10^8 \ \text{m/s}$c=3.0×108 m/s
• $e = 1.6 \times 10^{-19} \ \text{C}$e=1.6×10−19 C
• $\lambda = 6.0 \times 10^{-7} \ \text{m}$λ=6.0×10−7 m

Input Format

When the calculated ratio $\frac{V_0'}{V_0}$V0​V0′​​ is expressed as an irreducible fraction $\frac{p}{q}$qp​, input the value of the sum of the numerator and denominator $p+q$p+q.