Question

Consider an electron with mass \(m\) and charge \(-e\) moving in a circular orbit with radius \(r\) around a fixed proton with mass \(M\) and charge te. The electron is held in orbit by the electrostatic force between itself and the proton. Which one of the following expressions for the speed of the electron is correct? a) \(v=\sqrt{\frac{k e^{2}}{m r}}\) c) \(v=\sqrt{\frac{2 k e^{2}}{m r^{2}}}\) e) \(v=\sqrt{\frac{k e^{2}}{2 M r}}\) b) \(v=\sqrt{\frac{G M}{r}}\) d) \(v=\sqrt{\frac{m e^{2}}{k r}}\)

Short answer

Answer: \(v = \sqrt{\frac{ke^2}{mr}}\)

Step-by-step solution

Write down the expression for the centripetal force

The centripetal force required for any object (of mass m) to move in a circular orbit of radius r at a speed v is given by: \(F_{centripetal} = \frac{m v^2}{r}\).

Write down the expression for the electrostatic force

The electrostatic force between two charged particles with charges \(q_1\) and \(q_2\) separated by a distance r is given by Coulomb's law: \(F_{electrostatic} = \frac{k q_1 q_2}{r^2}\), where k is the electrostatic constant. In our case, the charges are those of the electron (-e) and the proton (te), so we will have: \(F_{electrostatic} = \frac{k \times (-e) \times te}{r^2}\). We will use this expression for the electrostatic force acting between the electron and proton.

Equate the centripetal force and electrostatic force

Since the electrostatic force is providing the centripetal force for the electron's circular motion, we can equate the two forces: \(\frac{m v^2}{r} = \frac{k \times (-e) \times te}{r^2}\).

Solve for the speed of the electron

Let's now solve this equation for the velocity v: \(m v^2 = k \times (-e) \times te \times r\). Divide both sides of the equation by m: \(v^2 = \frac{k(-e)(te)r}{m}\). Now, take the square root of both sides to obtain the expression for v: \(v = \sqrt{\frac{k(-e)(te)r}{m}}\). Since the question gives the proton's charge te = +e, the expression for v simplifies to: \(v = \sqrt{\frac{ke^2}{mr}}\).

Compare the obtained expression with the given options

We have found the expression for the electron's speed to be: \(v = \sqrt{\frac{ke^2}{mr}}\). Comparing it with the given options, we see that it matches with option (a). Therefore, the correct expression for the electron's speed is: \(v = \sqrt{\frac{ke^2}{mr}}\).