Question

Two long insulated straight wires carrying currents of 3 A and 5 A are arranged in XY plane as shown in figure. Find the magnitude and direction of the net magnetic fields at points P₁(2m, 2m) and P₂(−1m, 1m).

Answer 1

For a current-carrying wire, using the Biot-savart's law

Magnetic field at point P₁(2 m, 2 m):

`B_1 = (mu_0 I)/(2 pi r_1)` and

`B_2 = (mu_0 I)/(2 pi r_2)`

Distance `r_1 = sqrt8` and

`r_2 = sqrt8`

So, `B_1 = (mu_0 3)/(2 pi sqrt8)` and 

`B_2 = (mu_0 5)/(2 pi sqrt8)`

`B_"net" = sqrt(B_1^2 + B_2^2)`

= `sqrt(((mu_0 3)/(2 pi sqrt8))^2 + ((mu_0 5)/(2 pi sqrt8))^2)`

= `(mu_0 sqrt17)/(4 pi)`

Magnetic field at point P₂(−1m, 1m):

`B_1 = (mu_0 I)/(2 pi r_1)` and

`B_2 = (mu_0 I)/(2 pi r_2)`

Distance `r_1 = sqrt2` and

`r_2 = sqrt2`

So, `B_1 = (mu_0 3)/(2 pi sqrt2)` and 

`B_2 = (mu_0 5)/(2 pi sqrt2)`

`B_"net" = sqrt(B_1^2 + B_2^2)`

= `sqrt(((mu_0 3)/(2 pi sqrt2))^2 + ((mu_0 5)/(2 pi sqrt2))^2)`

= `(mu_0 sqrt17)/(2 pi)`