If \(\varepsilon_{0}\) be the permittivity of vacuum and \(r\) be the radius of orbit of \(\mathrm{H}\) -atom in which electron is revolving then velocity of electron is given by : (a) \(v=\frac{e}{\sqrt{4 \pi \varepsilon_{0} r m}}\) (b) \(v=e \times \sqrt{4 \pi \varepsilon_{0} r m}\) (c) \(v=\frac{4 \pi \varepsilon_{0} r m}{e}\) (d) \(v=\frac{4 \pi \varepsilon_{0} r m}{e^{2}}\)

Understanding Coulomb's Law is essential for comprehending the forces at play within an atom. Imagine the atom as a miniature solar system, with the electron orbiting the nucleus much like a planet orbits the sun. Coulomb's Law mathematically describes the electrostatic force of attraction or repulsion between two charged particles. In our solar system analogy, this law explains the gravitational force keeping the planets in orbit, but in the atomic realm, it describes the attraction between the positively charged nucleus and the negatively charged electrons.

The law is expressed as \( F = \frac{k_e \cdot q_1 \cdot q_2}{r^2} \), where \(F\) is the force between the charges, \(k_e\) is Coulomb's constant, \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between the centers of the two charges. Coulomb's Law tells us that the force is proportional to the product of the two charges and inversely proportional to the square of the distance between them. In the context of a hydrogen atom, it's this law that dictates the electrostatic force experienced by an electron as it moves around the nucleus.

Centripetal force is not a force in itself but refers to the 'center-seeking' force that keeps an object moving in a circular path. This force is always directed towards the center of the orbit. Imagine a ball on a string being swung in a circle; the tension in the string is providing the centripetal force that keeps the ball in its circular path. For an electron in a hydrogen atom, this force results from the electrostatic attraction between the electron and the nucleus, acting towards the atom's center.

Mathematically, the centripetal force is given by \( F = \frac{mv^2}{r} \), where \(m\) is the mass of the electron, \(v\) is the velocity of the electron in its orbit, and \(r\) is the radius of the orbit. This is critical for determining the velocity of an electron in a hydrogen atom because we can set this expression equal to the electrostatic force provided by Coulomb's Law to solve for the electron's velocity.

Electrostatic force is the force exerted by static electric charges. In atoms, this is the force that holds the electrons in orbit around the nucleus. This force is also described by Coulomb's Law, which we previously discussed, and is the foundation for understanding the behavior of an electron within an atom.

The specific electrostatic force between an electron and the proton in the hydrogen nucleus can be considered the 'glue' that binds the electron to the atom. Without this force, the electron would not be bound to the nucleus and would fly off into space. This force is always directed radially inward towards the center of the atom's nucleus. It's critical for predicting the behavior of an electron, as it influences how close the electron can be to the nucleus and the energy levels it can occupy.