Question

A wire carries a current I_1 = 5.0 A, a second wire carries a current of I_2 = 5.0 A as shown in the figure. Find the magnetic field at point 'P' shown in the figure.

Answer 1

The magnetic field due to a straight wire carrying current i is
B = $\mu$ ₀ i / 2 $\pi$ r
Where r is the distance from the wire to the point where the magnetic field is measured
The magnetic field due to the wire carrying the current I₁ at the point p is
B₁ =   $\mu$ ₀ I₁ / 2 $\pi$ r
Where r = sqrt (20²  + 10²)
r = 22.36 cm = 0.2236 m
B₁ = 4 $\pi$ x 10^-7 x 5 / 2 $\pi$ x 0.2236
B₁ = 44.72 x 10^-7 T
The magnetic field due to the wire carrying the current I₂ at the point P is
B₂  = $\mu$ ₀ I₂ / 2 $\pi$ r
Where r = 10 cm = 0.1 m
B₂  = 4 $\pi$ x 10^-7 x 5 / 2 $\pi$ x 0.1 = 100 x 10^-7 T
These two magnetic fields are oriented in different directions
The angle between these two fields should be found ,from the figure
tan $\theta$   = 20 cm / 10 cm = 2
$\theta$   = 63.43⁰
The angle between the fields is $\theta$   = 180⁰ - 63.43⁰  = 116.57⁰
The resultant field is
B = sqrt (B₁²  + B₂²  + 2 B₁ B₂ cos $\theta$ )
B = sqrt ( 44.72² + 100²  + 2 x 44.72 x 100 x cos 116.57⁰)
B = 89.51 x 10^-7 T