---
title: "An electron of mass \\(m_e\\) and charge magnitude \\(e\\) orbits a stationary proton of charge \\(+e\\) in a circular path. According to the Bohr model of the hydrogen atom, the electron’s orbital angular momentum is quantized such that \\(L = \\dfrac{n h}{2 \\pi}\\), where \\(n\\) is a positive integer and \\(h\\) is Planck’s constant. The electrostatic force between the proton and electron provides the required centripetal force. Which of the following expressions correctly represents the orbital radius \\(r_n\\) of the electron in the \\(n\\text{-th}\\) energy level in terms of \\(n\\), \\(h\\), \\(m_e\\), \\(e\\), and the Coulomb constant \\(k\\)?"
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url: "https://nerd-notes.com/ubq/117375/"
date_modified: "2026-08-04T06:52:00+00:00"
---

# An electron of mass \(m_e\) and charge magnitude \(e\) orbits a stationary proton of charge \(+e\) in a circular path. According to the Bohr model of the hydrogen atom, the electron’s orbital angular momentum is quantized such that \(L = \dfrac{n h}{2 \pi}\), where \(n\) is a positive integer and \(h\) is Planck’s constant. The electrostatic force between the proton and electron provides the required centripetal force. Which of the following expressions correctly represents the orbital radius \(r_n\) of the electron in the \(n\text{-th}\) energy level in terms of \(n\), \(h\), \(m_e\), \(e\), and the Coulomb constant \(k\)?

An electron of mass \(m_e\) and charge magnitude \(e\) orbits a stationary proton of charge \(+e\) in a circular path. According to the Bohr model of the hydrogen atom, the electron's orbital angular momentum is quantized such that \(L = \dfrac{n h}{2 \pi}\), where \(n\) is a positive integer and \(h\) is Planck's constant. The electrostatic force between the proton and electron provides the required centripetal force. Which of the following expressions correctly represents the orbital radius \(r_n\) of the electron in the \(n\text{-th}\) energy level in terms of \(n\), \(h\), \(m_e\), \(e\), and the Coulomb constant \(k\)?

![A schematic diagram showing a central proton labeled +e at the origin. A circular dashed line surrounding the center represents the electron orbit at radius r_n. On the circular path, a small dot representing an electron labeled -e moves counterclockwise with a velocity arrow v_n tangent to the circle. A central arrow labeled F_e points radially inward from the electron toward the proton. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785826320-nQ15D3.jpg)

- **A.** \(\dfrac{n h^2}{4 \pi^2 m_e k e^2}\)
- **B.** \(\dfrac{n^2 h^2}{2 \pi^2 m_e k e^2}\)
- **C.** \(\dfrac{n^2 h^2}{4 \pi^2 m_e k e^2}\)
- **D.** \(\dfrac{4 \pi^2 n^2 h^2}{m_e k e^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117375/*