CHAPTER 21         

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Two long parallel wires of copper of radius R are in contact along their full length.  The wires carry equal currents i in opposite directions (like Fig. 21.53 with the bottom current toward negative x).  The currents are uniformly distributed over the volumes of the wires.  Find the magnetic field in the midplane (x-y plane) of the wires as a function of the distance y from the point of contact.  Where is the magnetic field maximum and what is the value of the maximum magnetic field?

We begin by finding the magnetic field of a single long, straight, current carrying wire as shown below.



Using Ampere's law with the path as shown above, we have

The diagram below depicts the given situation for y greater than or equal to zero.



In the sketch above, the size of the magnetic field of the top wire, with a current out of the page, is given by



Referring to the diagram above again, we compute the magnetic field for y greater than or equal to zero.



The case of y less than or equal to zero is similar.  (You may find it helpful to construct a sketch of your own.)



Noting that the magnetic field in the problem at hand has the same form for all values of y (with z=0), we write the expression for the magnetic field.



A graph of the magnetic field strength versus y/R clearly shows that the maximum magnetic field strength is at y=0.  The value of the magnetic field at y=0 is as given above.