CBSE Class 12 · 2014 · Delhi · Modern PhysicsShort Answer

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Using the Rutherford model of the atom, derive the expression for the total energy of the electron in a hydrogen atom. What is the significance of the total negative energy possessed by the electron?

Answer: $E = - \frac{k e ^{2}}{2 r}$ ; negative energy $\Rightarrow$ electron is bound.

Solution

In the Rutherford model the Coulomb attraction supplies the centripetal force for the electron (charge

- e

) orbiting a nucleus of charge

+ e

\frac{k e ^{2}}{r ^{2}} = \frac{m v ^{2}}{r} \Rightarrow m v^{2} = \frac{k e ^{2}}{r} .

Kinetic energy:

K = \frac{1}{2} m v^{2} = \frac{k e ^{2}}{2 r}

.
Potential energy:

U = - \frac{k e ^{2}}{r}

.
Total energy:

E = K + U = \frac{k e ^{2}}{2 r} - \frac{k e ^{2}}{r} = - \frac{k e ^{2}}{2 r} .

The negative total energy means the electron is bound to the nucleus; energy must be supplied to remove it to infinity.

E = - \frac{k e ^{2}}{2 r} (< 0 \Rightarrow bound)

CBSE Class 12 Board · 2014 · Delhi · Subjective

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CBSE Class 12 · 2014 · Delhi · Modern PhysicsShort Answer

Solution

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