A particle of mass m moves in a circular orbit under a constant-magnitude force F toward the centre. Applying Bohr quantization mvr = nh/(2π), determine the dependence of orbital radius r and speed v on principal quantum number n.
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Question

A particle of mass m moves in a circular orbit under a constant-magnitude force F toward the centre. Applying Bohr quantization mvr = nh/(2π), determine the dependence of orbital radius r and speed v on principal quantum number n.

Options

G1 Physics · Accepted Answer

Step 1: Circular centripetal equation for a constant inward force is mv²/r = F. Step 2: Bohr quantization: mvr = nh/(2π), so v = nh/(2πmr). Step 3: Substitute into centripetal equation: m[n²h²/(4π²m²r²)]/r = F. Step 4: Rearrange: r³ = n²h²/(4π²mF), so r ∝ n²ᐟ³. Step 5: From v ∝ n/r, v ∝ n/n²ᐟ³ = n¹ᐟ³. Final answer: r ∝ n²ᐟ³ and v ∝ n¹ᐟ³, Option C.