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Three infinitely long thin wires each ca...

Three infinitely long thin wires each carrying current `I` in the same direction are in `x-y` plane of a gravity free space. The central wire is along the `y`-axis while the other two are along `x=+-d`. If the central wire is slightly displaced along `z`-axis and released, show that it will execute `S.H.M`. The linear mass density of the wire is `lambda`. If the time period of this small oscillation is `(kpid)/(I)sqrt((pilambda)/(mu_(0)))`, then find the value of `k`.

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To solve the problem, we need to analyze the forces acting on the central wire when it is displaced along the z-axis and show that it executes simple harmonic motion (SHM). We will also derive the expression for the time period of the oscillation and find the value of \( k \). ### Step-by-step Solution: 1. **Understanding the Setup**: - We have three infinitely long wires carrying current \( I \) in the same direction. - The central wire is along the y-axis, while the other two wires are at \( x = +d \) and \( x = -d \). - The central wire is displaced slightly along the z-axis. ...
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