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Question

Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle 'θ' with the vertical. If mass per unit length of wire is λ, then the value of I is (g = gravitational acceleration)


A
2πgLμ0tanθ
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B
2sinθπλgLμ0cosθ
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C
sinθπλgLμ0cosθ
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D
πλgLμ0tanθ
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Solution

The correct option is B 2sinθπλgLμ0cosθ

Consider a free body diagram wire.
When the wire is in equilibrium position, it will carry current in opposite direction.
FB=μoI2L2πd
Where L is the length of each wire and d is the separation between the wires.

From the diagram, we can get d = 2Lsinθ
Tcosθ=mg=λLg-----(1)
(In the vertical direction)
Tsinθ=FB=μ0I2L4πLsinθ ----(2)
(In horizontal direction)

From (1) and (2):
TsinθTcosθ=μ0I2L4πLsinθλLgI=4πλLgsin2θμ0cosθ=2sinθπλgLμ0cosθ

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