Two electrons each are fixed at a distance '2d'. A third charge proton placed at the midpoint is displaced slightly by a distance x (x << d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency: (m = mass of charged particle)

The problem describes a situation where two electrons are fixed at a distance '2d' apart and a proton is placed at the midpoint. The proton is then displaced slightly by a distance 'x' perpendicular to the line joining the two fixed charges. We are asked to find the angular frequency of the simple harmonic motion that the proton will execute.

Here are the steps to solve this problem:

Step 1: Understand the forces acting on the proton. Each electron will exert a force on the proton due to their charges. Since the proton is displaced perpendicular to the line joining the two electrons, the forces will be equal in magnitude but opposite in direction.

Step 2: Calculate the force. The force exerted by each electron on the proton can be calculated using Coulomb's law: F = kq1q2/r^2, where k is Coulomb's constant, q1 and q2 are the charges of the particles, and r is the distance between them. In this case, q1 is the charge of the electron, q2 is the charge of the proton, and r is the distance 'd' plus the displacement 'x'.

Step 3: Find the net force. Since the forces are equal in magnitude but opposite in direction, they will add up to give a net force of 2F.

Step 4: Apply Newton's second law. The net force acting on the proton will cause it to accelerate, according to Newton's second law: F = ma, where m is the mass of the proton and a is its acceleration.

Step 5: Recognize the simple harmonic motion. The acceleration of the proton is proportional to its displacement 'x', but in the opposite direction. This is the characteristic of a simple harmonic motion.

Step 6: Find the angular frequency. The angular frequency 'ω' of a simple harmonic motion is given by the square root of the ratio of the force constant to the mass: ω = sqrt(k/m). In this case, the force constant is the proportionality constant from the force law, and the mass is the mass of the proton.

So, the angular frequency of the simple harmonic motion that the proton will execute is given by the square root of the ratio of the force constant to the mass of the proton.