The following question was included in the AIEEE 2008 question paper:

Suppose an electron is attracted towards the origin by a force k/r where ‘k’ is a constant and ‘r’ is the distance of the electron from the origin. By applying Bohr’s model to this system, the radius of the n^{th} orbital of the system is found to be ‘r_{n}’ and the kinetic energy of the electron to be ‘T_{n}’. Then which of the following is true?

(1) T_{n} α** **1/n^{2}, r_{n} α** **n^{2}

(2) T_{n} independent of n, r_{n} α** **n

(3) T_{n} α** **1/n, r_{n} α** **n

(4) T_{n} α** **1/n, r_{n} α** **n^{2}

The force k/r supplies the centripetal force for the circular motion of the electron so that we have

k/r = mv^{2}/r where ‘m’ is the mass and ‘v’ is the speed of the electron.

Therefore, mv^{2} = k which is _{n} of the electron is ½ mv^{2} which is therefore

Also, v = √(k/m).

The angular momentum of the electron in the n^{th} orbit of radius r_{n} is mvr_{n} and in the Bohr model mvr_{n} = nh/2π where ‘h’ is Planck’s constant. Substituting for ‘v’ we have

m×√(k/m) ×r_{n} = nh/2π

This gives r_{n} α n. So the correct option is (2).

In the above question the attractive force on the electron was imagined to be inversely proportional to the distance just for the sake of testing your problem solving skill. In a real hydrogen atom the force is certainly inversely proportional to the *square** *of

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