Problem 146 - A Complete Resource Book in Physics for JEE Main

Question

Two identical particles each having charge \(q\) are very far apart. They are given velocity \(v_{0}\) parallel to each other such that initial perpendicular separation between them is \(d\). If the subsequent minimum separation between them is \(2 d\), find the initial velocity \(v_{0}\) and the loss in their total kinetic energies.

Step-by-Step Solution

Answer

The initial velocity \(v_0\) of the identical particles is given by: \[v_0 = \sqrt{\frac{kq^2}{md}}\] And the loss in their total kinetic energies is: \[Loss_{KE} = k \frac{q^2}{2d}\]

Step 1: Determine the initial electric potential energy

The electric potential energy U between two charged particles is given by the formula: \[U = k \frac{q_1 q_2}{r}\] Where k is the electrostatic constant, q1 and q2 are the charges of the particles, and r is the distance between them. Since the particles are very far apart at the beginning, U_initial is approximately equal to 0.

Step 2: Determine the initial kinetic energy

The initial kinetic energy KE of each particle is given by the formula: \[KE = \frac{1}{2}m v^2\] where m is the mass of the particle and v is the velocity. Since both particles have the same mass and initial velocity, their initial combined kinetic energies will be: \[KE_{total\_initial} = 2\times \frac{1}{2} m v_0^2= m v_0^2\]

Step 3: Determine the final electric potential energy

At the point of minimum separation, the distance between the particles is 2d. We can use the same formula for electric potential energy, and plug in the given values: \[U_{final} = k \frac{q^2}{2d}\]

Step 4: Determine the final kinetic energy

At the point of minimum separation, the particles would have maximum potential energy but their combined kinetic energies would be less than the initial kinetic energies. Therefore, we have: \[KE_{total\_final} = KE_{total\_initial} - U_{final}\] \[KE_{total\_final} = m v_0^2 - k \frac{q^2}{2d}\]

Step 5: Determine the difference between initial and final kinetic energies

To find the loss in total kinetic energies, we can subtract the final kinetic energy from the initial kinetic energy: \[Loss_{KE} = KE_{total\_initial} - KE_{total\_final}\] \[Loss_{KE} = m v_0^2 - (m v_0^2 - k \frac{q^2}{2d})\] \[Loss_{KE} = k \frac{q^2}{2d}\] Now that we have found the loss in total kinetic energies, we can use this value to find the initial velocity v0. Since the final kinetic energy is the only other unknown in the equation, we can solve for v0 using the equation \[m v_0^2 - k \frac{q^2}{2d} = Loss_{KE}\] \[v_0^2 = \frac{kq^2}{md}\] \[v_0 = \sqrt{\frac{kq^2}{md}}\] Now, we have found both the initial velocity v0 and the loss in total kinetic energies.