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An electron is pulled toward the origin by a central force of magnitude k/r, where k is a constant and r is the electron’s distance from the origin. If the Bohr model is used for this system, the radius of the nth allowed orbit is rₙ and the electron’s kinetic energy is Tₙ. Which relation is correct?

1. Tₙ varies as 1/n², and rₙ varies as n²
2. Tₙ does not depend on n, and rₙ varies as n
3. Tₙ varies as 1/n, and rₙ varies as n
4. Tₙ varies as 1/n, and rₙ varies as n²

Correct answer: Tₙ does not depend on n, and rₙ varies as n

Solution

In the Bohr model, the kinetic energy of the electron in a hydrogen-like atom is constant for all orbits, meaning it does not depend on the principal quantum number n. However, the radius of the allowed orbits increases linearly with n, leading to the conclusion that Tₙ remains constant while rₙ varies with n.

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