Q. 323.9( 7 Votes )

# Two particles A and B, each carrying a charge Q, are held fixed with a separation d between them. A particle C having mass m and charge q is kept at the middle point of the line AB.

(a) If it is displaced through a distance x perpendicular to AB, what would be the electric force experienced by it.

(b) Assuming x << d, show that this force is proportional to x.

(c) Under what conditions will the particle C execute simple harmonic motion if it is released after such a small displacement?

Find the time period of the oscillations if these conditions are satisfied.

Answer :

**Given:**Charge of particles A and B : q = Q

Separation between A and B : d

(a)

here ‘x’ is the distance at which particle C of mass m and charge ‘q’ is place on the perpendicular bisector of AB.

**Formula used:**

From the figure we can find sinθ :

The horizontal components of force cancel each other.

Total vertical component of force is: F’= 2Fsinθ

We use Coulomb’s Law:

Where F is the electrostatic force on b due to A, k is a constant .

k = = 9× 10

^{9}Nm

^{2}C

^{-2}and r is the distance between the two charges.

Here r =

Substituting we get,

F’ is the net electric force experience by particle C of charge q.

(b)

When x ≪ d,

Thus, x

^{2}can be neglected.

Substituting we get,

Thus, force is proportional to x.

(c)

The condition for Simple harmonic Motion of a particle is:

Here, m is the mass of the particle C, ω is the angular frequency.

Thus comparing two equations of F’, we get

We know that Where t is the Time Period.

Hence time period when the particle is released after a small displacement under SHM is

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