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Question

Two long parallel threads carry charge per unit length. The threads are separated by a distance l. The maximum field strength in the symmetry plane between them is given as λxπε0l. Find x:

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Solution

The electric field at a distance r due to long thread is E1=λ2πϵ0r
The field is symmetry due to both threads at middle of them. so r=z2+(l/2)2
The net field at middle of them is E=2E1sinθ
or E=2(λ2πϵ0r)sinθ=λπϵ0rzr
or E=λπϵ0zz2+l24
For max E, dEdz=0
[1z2+l242z2(z2+l24)2]=0
or 2z2=z2+l24
or z2=l24z=l2
thus, Emax=λπϵ0(l/2)(l/2)2+l24=λπϵ0l
so x=1

232493_142203_ans_3ac9387f959d4806887119b159f0b760.png

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