Question

An electron rotates in a circle around a nucleus having positive charge Ze. Correct relation between total energy (E) of electron to its potential energy (U) is:

• A. E = 2U
• B. 2E = 3U
• C. E = U
• D. 2E = U

Answer 1

The Correct Option is D

The electrostatic force between the electron and nucleus is described by:

\[ F = \frac{K(Ze)(e)}{r^2} = \frac{mv^2}{r} \]

The electron's kinetic energy (KE) is calculated as:

\[ \text{KE} = \frac{1}{2}mv^2 = \frac{1}{2} \frac{K(Ze)(e)}{r} \]

Potential energy (PE) is defined as:

\[ \text{PE} = -\frac{K(Ze)(e)}{r} \]

Total energy (TE) is the sum of KE and PE:

\[ \text{TE} = \text{KE} + \text{PE} = \frac{K(Ze)(e)}{2r} + \left( -\frac{K(Ze)(e)}{r} \right) = -\frac{K(Ze)(e)}{2r} \]

Consequently, the relationship between total energy and potential energy is:

\[ 2 \times \text{TE} = \text{PE} \]

This simplifies to \( 2E = U \), aligning with Option (4).