Question

Two parallel long (infinite for our purposes) wires are oriented along the z-axis. The figure below shows the (xy)-plane perpendicular to the wires, including the positions where the wires cross this plane. The wires carry some unknown electric currents 11 and 12, which you need to find from a single measurement of the magnetic field B=By i + By j at point A, whose position in the plane is also indicated. We will treat the currents algebraically: the current I is positive when it runs in the direction of the z-axis and negative when it runs in the opposite direction. In the right-hand system of coordinates, the z-axis is directed out of the screen towards us. The geometric parameters are known: a=0.98 cm, b=0.34 cm. b 1 12 a a х Given the measurement of the magnetic field: By=-0.44 G and By=0.76 G, what is the unknown current I1?: 11= A. What is the unknown current 12?: 12= A.

Answer 1

By right hand thumb rule the magnetic field due to the wires at A will be given as-

у В1 Ву BX 9. В2 а

Now we know that the magnetic field due to an infinitely current carrying wire is given as

B = $\mu_o$ I/2$\pi$r

Now the distance of A from both the wires is given as r = (a² + b²)^1/2 = 1.0373 cm

The angle made by the magnetic field from vertical will be $\theta$ for I₂ and I₁.

Now calculating the magnetic field in x-y direction we have;

B_x = -$\mu_o$I₁/2$\pi$r x sin$\theta$ - $\mu_o$ I₂/2$\pi$r x sin$\theta$

=> I₁ + I₂ = -B_x2$\pi$r/$\mu_o$sin$\theta$ .............(i)

B_y = $\mu_o$ I₁/2$\pi$r x cos$\theta$ - $\mu_o$ I₂/2$\pi$r x cos$\theta$

=> I₁ - I₂ = B_y2$\pi$r/$\mu_o$cos$\theta$ ..............(ii)

From the above equations;

I₁ = B_y$\pi$r/$\mu_o$cos$\theta$ -B_x$\pi$r/$\mu_o$sin$\theta$

I₂ =  -B_y$\pi$r/$\mu_o$cos$\theta$ -B_x$\pi$r/$\mu_o$sin$\theta$

Now from the triangle you can see tan$\theta$ = b/a => tan$\theta$ = 0.347

sin$\theta$ = 0.3278

cos$\theta$ = 0.945

Thus,

I₁ = 5.567 A

I₂ = 1.395 A