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Question

A charge (q, m) is projected perpendicular away from an infinitely long wire carrying a current i with a speed v0. Find the maximum separation between the charge and the wire in the subsequent motion if the initial separation is x0.

A
3x0
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B
x0eπmv0μ0iq
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C
x0e2πmv0μ0iq
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D
2x0
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Solution

The correct option is C x0e2πmv0μ0iq
The situation from the question is as follows:
Let the velocity and acceleration of the charge be v and a respectively and the magnetic field be B.
The force F on the charge = qv×B = q(vxi+vyj)×μ0i2πx(k) = m (axi+ayj)
So,
ax=qvyμ0i2πxm

ay=qvxμ0i2πxm=dvydt.dxdx = vxdvydx

qμ0i2πxm=dvydx

Now as the magentic field is always normal to the velocity of the charge, the speed of the charge is always v0.
At the instant of maximum seperation between the charge and wire, vx=0 such that vy=v0 directed in the positive y direction and x=x1; and initially vx=v0 directed towrads the wire, vy=0 and x=x0.
So,
x1x0qμ0i2πxm.dx=v00dvy

Solving the above integral equation we get:
x1=x0e2πmv0μ0iq

Hence, option (c) is correct.


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