A 5 cm length of wire carries a current of 3 A in the positive z direction. The force on this wire due to a magnetic field vector is vector = (-0.2 ihatbold + 0.2 jhatbold) N. If this wire is rotated so that the current flows in the positive x direction, the force on the wire is vector = 0.2 khatbold N. Find the magnetic field vector.

B = 4/3 (i + j + k)

Explanation:

I = 3 k

L = 5 x 10⁻² m

F = - 0.2 i + 0.2 j

Let magnetic field be B

B = B₁ i + B₂ j + B₃ k

F = L (I x B)

= 5 x 10⁻² 3 k x (B₁ i + B₂ j + B₃ k)

= 15 x 10⁻² B₁ j - 15 x 10⁻² B₂ i

Given

F = - 0.2 i + 0.2 j

equating equal terms

15 x 10⁻² B₂ =.2

15 x 10⁻² B₁ =.2

B₁ =.02 / 15 x 10⁻² = 20 / 15 N

B₂ =.02 / 15 x 10⁻² = 20 / 15 N

Now wire is rotated so that current flows in positive x direction

I = 3 i

F =. 2 k

F = L (I x B)

= 5 x 10⁻² x 3 i x (B₁ i + B₂ j + B₃ k)

= 15 x 10⁻² B₃ k

15 x 10⁻² B₃ =.2

B₃ =.2 / 15 x 10⁻²

B₃ = 20 / 15

B = B₁ i + B₂ j + B₃ k

20 / 15 9 (i + j + k)

B = 4/3 (i + j + k)