An electron of mass m revolves around a stationary neutron of mass M under their mutual gravitational attraction. Assuming Bohr quantization, its minimum possible de Broglie wavelength is h²/(aπGMᵇmᶜ), where a, b and c are positive integers. Find a + b + c.
Physics · Mixed Chapter
JEE Mains

Question

An electron of mass m revolves around a stationary neutron of mass M under their mutual gravitational attraction. Assuming Bohr quantization, its minimum possible de Broglie wavelength is h²/(aπGMᵇmᶜ), where a, b and c are positive integers. Find a + b + c.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — The gravitational attraction provides the required centripetal force: GMm/r² = mv²/r, so v² = GM/r. Step 2 — Bohr quantization imposes mvr = nh/(2π), where n = 1, 2, 3, … . The smallest allowed circular orbit corresponds to n = 1. Step 3 — Substituting v = h/(2πmr) into the force equation gives h²/(4π²m²r²) = GM/r and r₁ = h²/(4π²GMm²). Step 4 — The matter wave fits one circumference at n = 1, hence λₘᵢₙ = 2πr₁ = h²/(2πGMm²). Step 5 — Compare the source expression λₘᵢₙ = h²/(aπGMᵇmᶜ):…