Two parallel wires C and D placed perpendicular to the plane of paper at a separation of `r(=8cm)`, carry electric currents `I_1=6A` and `I_2=3A` in opposite directions as shown in figure. Find the point on the line CD where the resultant magnetic field is zero.

Explanation

The required point on the line CD where the resultant magnetic field will be zero when the magnetic field at that point due to one wire must be equal and opposite to the magnetic field due to another wire. This will be possible when the required point lies on the right side of wire D. Let this point be P at distance x from D.
Magnetic field due to current `I_1` at point P is
`B_1=(mu_0)/(4pi)(2I_1)/((r+x))`
It is directed perpendicular to CP downwards.
Magnetic field due to current `I_2` at poing P is
`B_2=(mu_0)/(4pi)(2I_2)/(x)`
It is directed perpendicular to DP upwards.
As resultant magnetic field at P is zero so
`B_1=B_2` or `(mu_0)/(4pi)(2I_1)/((r+x))=(mu_0)/(4pi)(2I_2)/(x)`
or `(I_1)/((r+x))=(I_2)/(x)`
or `x=(I_2r)/((I_1-I_2))=(3xx(8xx10^-2))/(6-3)`
`=8xx10^-2m=8cm`