Bohr Atomic Model, Photoelectric Effect, and Circular Motion of an Electron in a Magnetic Field

Problem Statement

In a hydrogen atom, one photon is emitted when an electron transitions from the energy level $n=4$n=4 to the energy level $n=2$n=2, following conservation of energy. This photon is directed onto a metal surface with work function $W$W, causing electrons to be emitted by the photoelectric effect.

The electron with the maximum kinetic energy among the emitted photoelectrons is selected and sent into a vacuum space with a uniform magnetic field of magnetic flux density $B$B, in a direction perpendicular to the field. This electron undergoes uniform circular motion due to the Lorentz force.

Find the radius $r$r of the circular orbit.

Let $E_0$E0​ be the absolute value of the energy of the hydrogen atom's ground state ($n=1$n=1). The energy level for principal quantum number $n$n is given by $-\frac{E_0}{n^2}$−n2E0​​. Let the electron mass be $m$m, the elementary charge be $e$e. The motion may be treated non-relativistically.

Constraints

• $E_0 = 2.16 \times 10^{-18} \text{ J}$E0​=2.16×10−18 J
• $W = 2.05 \times 10^{-19} \text{ J}$W=2.05×10−19 J
• $m = 9.00 \times 10^{-31} \text{ kg}$m=9.00×10−31 kg
• $e = 1.60 \times 10^{-19} \text{ C}$e=1.60×10−19 C
• $B = 1.40 \times 10^{-4} \text{ T}$B=1.40×10−4 T

Input Format

The orbital radius $r$r of the electron is expressed as $A \times 10^{-2} \text{ m}$A×10−2 m. Express $A$A as an irreducible fraction $\frac{p}{q}$qp​ and find the value of $p+q$p+q.