Question

The figure shows, in cross section, four thin wires that are parallel, straight, and very long. They carry identical currents of i = 2.45 A in the directions indicated. Initially all four wires are at distance d = 15.7 cm from the origin of the coordinate system, where they create a net magnetic field . (a) To what value of x must you move wire 1 along the x axis in order to rotate counterclockwise by 30°? (b) With wire 1 in that new position, to what value of x must you move wire 3 along the x axis to rotate B by 30° back to its initial orientation?

Answer 1

d = 15.7 cm,

current in each wire, i = 2.45 A

θ = 30^o

Part a)

initially, the magnitude of the magnetic field at the origin due to each current is

B = μ₀I/(2πd)

the net field is B_x = 2B = 2*μ₀I/(2πd) in +ve-x-direction (due to current 2 and 4, fields due to 1 and 3 cancel each other)

move 1 to new position which is distance x from the origin,

the net field along y direction is B_y = μ₀I/(2πx) - μ₀I/(2πd)

tanθ = B_y/B_x

tanθ = (1/x - 1/d)/(2/d)

1/x - 1/d = 2tanθ/d

x = d/(1 + 2tanθ)

x = 7.29 cm

new position is x = -7.29 cm

Part b)

move 3 to x = +7.29 cm, then B_y = 0