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Question

A long, straight wire lies along the y-axis and carries a current of 9.5 A in the positive y direction. A second long, straight wire, parallel to the first and 9.8  cm to the right, carries a current of 5.40 A in the negative y direction. At what value of the x coordinate does the resultant magnetic field vanish?

deuces · Accepted Answer

Given: 
   d=distance between wires 
   I1=current for the first wire 
   I2=current for the second wire 
   k=2e-7  
  
Need to know: 
   x = r1 = distance from the first wire 
   x-d = r2 = distance from second wire 
   B=kI/r 
  
To Solve:    
   k I1 / r1   =   k I 2/ r2  
. 
  The k's cancel out. Insert definitions for r1 and r2: 
        9.5/ x   =   5.4/ (x - d) 
. 
  cross multiply the equations, and put in the value of d: 
          9.5(x-9.8) = 5.4(x) 
  
  Distribute the current for wire 1: 
         9.5(x) - 93.1 = 5.4(x) 
  
Solve for x. 
        93.1=4.1x
        x = 22.7 cm = .227 m

kirchwey · Answer

Magnetic field B from a single wire = mu0*I/(2pi*d) (ref.)
At the null-field point between two current-carrying wires, I1/d1 = -I2/d2 or d1/d2 = -I1/I2 = 9.5/5.4 = 1.7592593
We also know that d1+d2 = 0.098. Given d1/d2 and d1+d2 we have
d1+d2 = d1(1+d2/d1) ==> 
d1 = (d1+d2)/(1+d2/d1) = (d1+d2)/(1+1/(d1/d2))
d1 (x coordinate of null field) = 6.2483221E-2 m

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