A long wire carrying I A = 4.5 A of current perpendicular to the xy -plane inter

Question

A long wire carrying IA= 4.5 A of current perpendicular to the xy-plane intersects the x-axis at x=?2.0cm . A second, parallel wire carrying IB= 3.5 A of current intersects the x-axis at x=+2.0cm . (As illustrated in the figure, the two currents are either into or out of the page.) (Figure 1)

At what point on the x-axis is the net magnetic field zero if the two currents are in the same direction?

At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions? (Figure 1)

Explanation / Answer

A) I is in same direction.

The point must be in between the wires.

Let the point is "a" cm distance from I1 means from I2 it is (4 - a) cm

B1 = B2

mo I1 / 2 pi a x 10^-2 = mo I2 / 2 pi (4 - a) x 10^-2

I1 / a x 10^-2 = I2 / (4 -a ) x 10^-2

4.5 / (a x 10^-2) = 3.5 / ((4 - a) x 10^-2)

(18 - 4.5 a) x 10^-2 = 3.5 a x 3.5 x 10^-2

(18 - 4.5 a) = 12.25 a

a = 1.07 cm

B) I is in opposite direction

The point where B = 0 must be present towards left or towards right to IA and IB respectively.

Let towards right of IB "a" cm from IB means from IA (4 + a ) cm

B1 = B2

mo I1 / 2 pi (4 + a) = mo I2 / 2 pi a

4.5 / (4 + a) = 3.5 / a

4.5 a = 14 + 3.5 a

a = 14 cm (+x side) (right to IB)

Now

Let towards left of IA "a" cm from IA means from IB (4 + a ) cm

B1 = B2

mo I1 / 2 pi (a) = mo I2 / 2 pi (4 + a)

4.5 / a = 3.5 / (4 + a)

18 + 4.5 a = 3.5 a

a = 18 cm

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