Bohr Model – Stability Condition for a Hydrogen Electron Orbit Let v be the electron speed, m its mass, e its charge, ε0 the permittivity of free space, and r the orbital radius in a hydrogen atom. For a stable circular orbit, which force-balance equation must hold?

Introduction / Context:
In the Bohr model of the hydrogen atom, an electron executes uniform circular motion around the proton. Stability of the orbit requires that the inward Coulomb attraction provide the necessary centripetal force. Writing and simplifying this force balance yields the fundamental relation connecting speed and radius for hydrogenic orbits.

Given Data / Assumptions:

• Single electron, single proton (hydrogen), circular orbit.
• Electrostatic attraction provides centripetal force.
• Non-relativistic speeds; classical centripetal force expression valid.

Concept / Approach:

Coulomb force magnitude between charges +e and −e separated by r is F_C = e^2 / (4 * π * ε0 * r^2). For circular motion at speed v and radius r, centripetal force is F_cen = m * v^2 / r. Equating F_C = F_cen gives the stability requirement. This relation, combined with Bohr’s angular momentum quantization m * v * r = n * h / (2π), yields the allowed radii and energy levels.

Step-by-Step Solution:

Verification / Alternative check:

Solving together with L = n * h / (2π) reproduces the Bohr radius a0 and hydrogen spectrum, verifying the consistency of the force balance.

Why Other Options Are Wrong:

• Option B mixes dimensions incorrectly.
• Option C places r^2 in the denominator on the RHS, breaking force units.
• Option D omits one factor of e, altering the force.
• Option E uses angular momentum-like form, not a force equality.

Common Pitfalls:

Dropping π or ε0; confusing the 1/r vs 1/r^2 dependences; forgetting to keep units consistent (Newtons for both sides).

Final Answer:

m * v^2 / r = e^2 / (4 * π * ε0 * r^2)