A charged particle moves in a circular orbit perpendicular to a uniform magnetic field B. Suppose Bohr quantization mvr = nħ is artificially imposed while magnetic force qvB supplies the centripetal force. If the orbit changes from n = 1 to n = 16, by what integer factor does the orbital radius increase?
Physics · Atomic Structure and X-rays · atomic structure and bohr atomic model
JEE Mains NEET JEE Advanced

Question

A charged particle moves in a circular orbit perpendicular to a uniform magnetic field B. Suppose Bohr quantization mvr = nħ is artificially imposed while magnetic force qvB supplies the centripetal force. If the orbit changes from n = 1 to n = 16, by what integer factor does the orbital radius increase?

Physics · Atomic Structure and X-rays · atomic structure and bohr atomic model

G1 Physics · Accepted Answer

Magnetic force provides centripetal force: qvB = mv² ÷ r. For nonzero v: qBr = mv. Multiply both sides by r: qBr² = mvr. Using Bohr quantization, mvr = nħ. Therefore, rₙ² = nħ ÷ (qB). Hence, rₙ ∝ √n. Thus, r₁₆ ÷ r₁ = √16 ÷ √1 = 4. Hence N = 4.