Question

Consider an electron with mass \(m\) and charge \(-e\) located a distance \(r\) from a fixed proton with mass \(M\) and charge \(+e .\) The electron is released from rest. Which one of the following expressions for the magnitude of the initial acceleration of the electron is correct? a) \(a=\frac{2 k e^{2}}{m M r}\) c) \(a=\frac{1}{2} m e^{2} k^{2}\) e) \(a=\frac{k e^{2}}{m r^{2}}\) b) \(a=\sqrt{\frac{2 e^{2}}{m k r}}\) d) \(a=\frac{2 k e^{2}}{m r}\)

Short answer

Answer: The magnitude of the initial acceleration of the electron is given by the expression \(a = \frac{k e^2}{m r^2}\), which shows that it is directly proportional to the square of the charge and inversely proportional to the mass of the electron and the square of the distance between the electron and the proton.

Step-by-step solution

Understand the given data

We are given the following information: - Electron has mass m and charge -e - Proton has mass M and charge +e - The distance between them is r

Calculate the force between the electron and the proton

We can calculate the force between the two charges using Coulomb's law: Force, \(F = k\frac{|q_1||q_2|}{r^2}\) Where - F is the force; - k is the electrostatic constant (\(k \approx 8.99 \times 10^9 N m^2/C^2\)); - \(q_1\) and \(q_2\) are the charges of the electron and proton respectively; - r is the distance between the charges. In our case, \(q_1 = -e\) and \(q_2 = +e\). So, \(F = k\frac{-e \times e}{r^2} = k\frac{e^2}{r^2}\)

Calculate the acceleration of the electron

Now, we apply Newton's second law of motion to calculate the acceleration of the electron: Force, \(F = m \times a\) Where - F is the force acting on the electron; - m is the mass of the electron; - a is the acceleration. Recall that we found the force in step 2: \(F = k\frac{e^2}{r^2}\). We can now substitute this into Newton's second law equation to solve for acceleration: \(k\frac{e^2}{r^2} = m \times a\) Solving for a, we get: \(a = \frac{k e^2}{m r^2}\) The correct expression for the magnitude of the initial acceleration of the electron is therefore (e) \(a = \frac{k e^2}{m r^2}\).