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Assuming mass m of the charged particle, find the time period of oscillation.
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A. 2πq1q4π0a3mB. 2π4π0a3mq1qCπ2πq1q4π0a3mD. none of these

Answer
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Hint: The acceleration of a charge is equal to the product of square of angular frequency and the displacement. The force exerted on the charge is equal to the product of charge and the net electric field. By equating it with Newton’s second law, we can get acceleration and from acceleration we can get angular frequency and finally the time period.

Formula used:
Newton’s second law of motion:
F=ma
Force on a charge in electric field:
F=qE
Acceleration of an oscillating charge is given as
a=ω2x
Time period is related to angular frequency by following relation:
T=2πω
Here all the symbols have their usual meaning.

Complete answer:
In the given diagram, we have two charges of the same magnitude at (-a,0,0) and (a,0,0). At a point along the line passing through the centre of the line joining the two charges, the various components of the electric field are shown. Since the charges have the same magnitude their electric fields also have the same values.
|E1|=|E2|=E(let)
Now the horizontal components of the electric field cancel each other while the vertical components get added up. Hence, the net electric field is given as
E=2Ecosθ=q4π0(x2+a2)xx2+a2
The force due to this electric field on a charged particle q1 of mass m is given as
F=q1E=q1qx4π0(x2+a2)32
If x << a then, F=q1qx4π0a3
Now using Newton’s second law, we get
mA=q1qx4π0a3A=q1qx4π0ma3
We denote acceleration by A.
As we know that acceleration of a charged particle undergoing oscillations is given as follows:
ω=Ax=q1q4π0ma3
The time period is given as
T=2πω=2π4π0ma3q1q

Hence, the correct answer is option B.

Note:
In this question the charge that is undergoing oscillation is a positive test charge. But in case, we had a negative charge then the directions of resultant forces would have been opposite to that for the positive charge. But the final answer would still remain the same in magnitude.