Question

Orbits of a particle moving in a circle are such that the perimeter of the orbit equals an integer number of de-Broglie wavelengths of the particle. For a charged particle moving in a plane perpendicular to a magnetic field, the radius of the n^th orbital will therefore be proportional to:

Options

• n²

• n

• n^1/2

• n^1/4

Answer 1

n^1/2

Explanation:

According to the question,

2πr = nλ = `"nh"/"p" = "nh"/"mv"`

or mvr = `"nh"/(2pi)` or mv = `"nh"/(2pi"r")`

F = qv_B = `"mv"^2/"r"`  or,  q_B = `"mv"/"r" ="nh"/(2pi"r.r")`

or, r₂ = `"nh"/(2pi"qB")`  or,  r = `sqrt("nh"/(2pi"qB"))`

i.e., r ∝ n^1/2