Question

A long wire carrying a 5.8 A current perpendicular to the xy-plane intersects the x-axis at x=−1.7cm. A second, parallel wire carrying a 2.0 A current intersects the x-axis at x=+1.7cm. A. At what point on the x-axis is the magnetic field zero if the two currents are in the same direction? B. At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions?

Answer 1

A)

i₁ = current in the wire located at x = - 1.7 cm , = 5.8 A

i₂ = current in the wire located at x = + 1.7 cm , = 2 A

r = distance between the two wires = 1.7 + 1.7 = 3.4 cm = 0.034 m

Let the magnetic field is zero at point P at distance "d" from wire located at x = - 1.7 cm.

r₁ = distance of point P from wire located at x = - 1.7 cm, = d

r₂ = distance of point P from wire located at x = 1.7 cm, = r - d = 0.034 - d

B₁ = magnetic field by wire located at x = - 1.7 cm, at point P = (/4) (2 i₁/r₁)

B₂ = magnetic field by wire located at x = 1.7 cm, at point P = (/4) (2 i₂/r₂)

The two magnetic field are in opposite direction, hence for the net magnetic field to be zero at P ,

B₁ = B₂

(/4) (2 i₁/r₁) = (/4) (2 i₂/r₂)

i₁/r₁ = i₂/r₂

5.8/d = 2/(r - d)

5.8/d = 2/(0.034 - d)

d = 0.0253 m

d = 2.53 cm

x = location of the point P on x-axis = - 1.7 + 2.53

x = 0.83 cm

B)

When the current are in opposite direction

i₁ = current in the wire located at x = - 1.7 cm , = 5.8 A

i₂ = current in the wire located at x = + 1.7 cm , = 2 A

r = distance between the two wires = 1.7 + 1.7 = 3.4 cm = 0.034 m

Let the magnetic field is zero at point P at distance "d" from wire located at x = 1.7 cm.

r₁ = distance of point P from wire located at x = - 1.7 cm, = r + d = 0.034 + d

r₂ = distance of point P from wire located at x = 1.7 cm, = d

B₁ = magnetic field by wire located at x = - 1.7 cm, at point P = (/4) (2 i₁/r₁)

B₂ = magnetic field by wire located at x = 1.7 cm, at point P = (/4) (2 i₂/r₂)

The two magnetic field are in opposite direction, hence for the net magnetic field to be zero at P ,

B₁ = B₂

(/4) (2 i₁/r₁) = (/4) (2 i₂/r₂)

i₁/r₁ = i₂/r₂

5.8/(0.034 + d) = 2/d

d = 0.0179 m

d = 1.79 cm

x = location of the point P on x-axis = 1.7 + 1.79 = 3.49 cm

x = 3.49 cm