Question

A long wire carrying a 4.5 A current perpendicular to the xy-plane intersects the x-axis at x=−2.0cm. A second, parallel wire carrying a 3.0 A current intersects the x-axis at x=+2.0cm.

At what point on the x-axis is the 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?

Answer 1

given

current in thefirst wire i = 4.5A

current in the second wire i = 3.0 A

a) for x>2 cm and for x<2cm the magnetic fields due to the currents in the two wires add. the point where the two magnetic fields cancel lies on the x axis between the two wires .
let that point be x distance from the origin
from biot savat law magnetic field due to wire
B = u0*i/ 2*pi*r
then
u0*4.5A/2*pi*( 0.020m+x) = u0*3.0A/2*pi*( 0.020m-x)
4.5*( 0.020m-x) = 3.0*( 0.020m+x)
x = 0.004 m = 0.4 cm
b) the magnetic fields due to the currents in teh wires add in the origion -2cm<x< 2cm
for x< -2cm the magnetic field subtract bu the field due to the current is always larger than the field due to 3A curent
however fo x> 2cm the two fields will cancel at a point on the x axis
let the point be a x away from the origion
u0*4.5A/2*pi*(x+ 0.020m) = u0*3.0A/2*pi*(x- 0.020m)
x = 0.1 m = 10 cm