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

<p>In the context of an electron orbiting around a nucleus with a positive charge of $\mathrm{Ze}$, we are dealing with classical physics approximations and the electrostatic force between the electron and the nucleus. In such a setup, the electron's potential energy (U) is due to electrostatic interaction, and it is given by Coulomb's law:</p> <p>$U = -\frac{kZe^2}{r}$</p> <p>Where:</p> <ul> <li>$U$ is the potential energy of the electron,</li> <li>$k$ is Coulomb's constant,</li> <li>$Z$ is the atomic number (number of protons in the nucleus),</li> <li>$e$ is the charge of an electron, and</li> <li>$r$ is the radius of the orbit of the electron around the nucleus.</li> </ul> <p>The negative sign indicates that the potential energy is negative because the electron and nucleus attract each other.</p> <p>The total energy (E) of the electron in orbit is the sum of its kinetic energy (K) and its potential energy (U). Since the electron is in a stable orbit, its kinetic energy can be shown to be exactly half the magnitude of its potential energy but positive:</p> <p>$K = -\frac{1}{2}U$</p> <p>Therefore,</p> <p>$E = K + U = -\frac{1}{2}U + U = \frac{1}{2}U$</p> <p>To find a relation between total energy (E) and potential energy (U), we rearrange the equation as follows:</p> <p>$2E = U$</p> <p>This is to say, the total energy (E) is half the magnitude of potential energy (U) but negative, and the correct relationship between them, when looking for a positive proportionality, yields to $2E = U$. Hence, the correct option is:</p> <p>Option C: $2 \mathrm{E} = \mathrm{U}$</p>