C\" N-m 99 x10 C oful equations 6. (15 pts) Magnetic quadrupole. Three wires are
ID: 1770665 • Letter: C
Question
C" N-m 99 x10 C oful equations 6. (15 pts) Magnetic quadrupole. Three wires are arranged along the y-axis as in the figure. The top and bottom wires are each a distance d from the origin and each carries a current I flowing out of the page. The middle wire, at the origin, carries a current 21 flowing into the page. (a) Write an expression for the magnetic field along the x-axis in terms of x and appropriate constants. In a picture, CLEARLY show the directions of the fields generated by the three wires (Bp, Bonom, Bnidde). Do not simplify your answer. (b) Expand your expression from part (a) in the limit x >> d to obtain an expression for the magnetic t far from the charge distribution. Keep the lowest order, nonzero term (obviously).Explanation / Answer
for the given problem
current = i in top and bottom wire, 2i in the middle wire
distacne between wires = d
a. hence magnetic field at x axis is given by
Btop + Bmid + Bbot
the horizontal components of Btop and Bbot will cancel each other out due to symmetry
hence
the vertical components are
Btop = 2kIx/(d^2 + x^2)
Bbot = 2kIx/(d^2 + x^2)
Bmid = -4kI/x
hence Net field
B = 4kIx/(d^2 + x^2) - 4kI/x
b. when x > > d
B = 4kIx/(d^2 + x^2) - 4kI/x
B = 4kIx/x^2(d^2/x^2 + 1) - 4kI/x
now, d/x = 0 for x > > d
B = 4kI/x - 4kI/x = 0