Problem 9 - A Complete Resource Book in Physics for JEE Main

Question

A charged particle of mass \(m\) and charge \(q\) is released from rest in an electric field of constant magnitude \(E\). The kinetic energy of the particle after time \(t\) will be (A) \(\frac{2 E^{2} t^{2}}{m q}\) (B) \(\frac{E q^{2} m}{2 t^{2}}\) (C) \(\frac{E^{2} q^{2} t^{2}}{2 m}\) (D) \(\frac{E q m}{2 t}\)

Step-by-Step Solution

Answer

The kinetic energy of the charged particle after time $t$ is given by \(\frac{E^{2} q^{2} t^{2}}{2 m}\).

Step 1: Determine the electric force on the particle

In the presence of an electric field 'E', a charged particle with charge 'q' will experience an electric force 'F' given by the equation: \(F = qE\)

Step 2: Use Newton's second law to find the acceleration

Newton's second law states that the acceleration 'a' of an object is directly proportional to the net force acting on it and inversely proportional to its mass 'm'. Using this law, we can determine the acceleration of the particle: \(a = \frac{F}{m}\) Substitute the electric force from Step 1: \(a = \frac{qE}{m}\)

Step 3: Use equation of motion to find velocity after time 't'

As the particle starts from rest, the equation of motion for its velocity 'v' after time 't' can be given by: \(v = at\) Substitute the acceleration from Step 2: \(v = \frac{qE}{m}t\)

Step 4: Calculate the kinetic energy

The kinetic energy 'K' of the particle can be given by the equation: \(K = \frac{1}{2}mv^{2}\) Substitute the velocity from Step 3: \(K = \frac{1}{2}m\left(\frac{qE}{m}t\right)^{2}\)

Step 5: Simplify the expression for kinetic energy

Now, simplify the expression for kinetic energy: \(K = \frac{1}{2}m\left(\frac{q^{2}E^{2}t^{2}}{m^{2}}\right)\) \(K = \frac{q^{2}E^{2}t^{2}}{2m}\) Comparing the simplified expression with the given options, we can see that the correct answer is: (C) \(\frac{E^{2} q^{2} t^{2}}{2 m}\)