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If four de Broglie wavelengths fit into the circumference of the orbit of radius r, then λ= ______.

Asked in GSEB Board July 2022 · Postulates, radius, speed and angular momentum

(1) 4π r
(2) (π r)/4
(3) 2π r
(4) (π r)/2

Answer: (4) (π r)/2

Step-by-step solution

Idea: an allowed Bohr orbit holds a whole number n of de Broglie wavelengths, nλ=2π r.

Here n=4: 4λ=2π r.

λ=(2π r)/4=(π r)/2.

Why the other options are wrong

(1) No single slip from 4λ=2π r gives 4π r. Sanity check: 4π r is twice the whole circumference, so not even one such wavelength would fit round the orbit, let alone four.
(2) (π r)/4 takes the circumference as π r, dropping the 2 from 2π r.
(3) 2π r is the whole circumference: it fits one wavelength into the orbit (n=1), not four.

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