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+1 vote
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in Physics by (75.0k points)

Two long parallel wires of negligible resistance are connected at one end to a resistance R and at the other end to a dc voltage source. The distance between the axes of the wires is η = 20 times greater than the cross-sectional radius of each wire. At what value of resistance R does the resultant force of interaction between the wires turn into zero?

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There are excess surface charges on each wire (irrespective of whether the current is flowing through them or not). Hence in addition to the magnetic force vector Fm we must take into account the electric force vector Fe Suppose that an excess charge λ corresponds to a unit length of the wire, then electric force exerted per unit length of the wire by other wire can be found with the help of Gauss’s theorem. 

where l is the distance between the axes of the wires. The magnetic force acting per unit length of the wire can be found with the help of the theorem on circulation of vector 

     .......(2)  

where i is the current in the wire.

Now, from the relation,

where λ = C φ, where C is the capacitance of the wires per unit lengths and is given in problem 3.108 and φ = iR

Dividing (2) by (1) and then substituting the value of i/λ from (3), we get 

The resultant force of interaction vanishes when this ratio equals unity. This is possible when R = R0 , where 

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