Question

Two wires are perpendicular to each other and form a coordinate axis. The current in the vertical wire is going up (in the positive y direction) and the current in the horizontal wire is going to the right(in the positive x direction). Where is the net magnetic field equal to zero?

Answer 1

Answer:

Magnetic field shall be zero at exactly in between the wires.

Explanation:

We can find the magnetic field by biot Savart law as follows

$\overrightarrow{dB}=\frac{\mu _{0}I}{4\pi }\int \frac{\overrightarrow{dl}\times \widehat{r}}{r^{2}}$

For current carrying wire in positive y direction we have

$\overrightarrow{dB_{1}}=\frac{\mu _{0}Idl}{4\pi }\int \frac{\widehat{j}\times \widehat{r_{1}}}{r_{1}^{2}}$

Similarly for wire carrying current in -y direction we have $\overrightarrow{dB_{2}}=\frac{-\mu _{0}Idl}{4\pi }\int \frac{\widehat{j}\times \widehat{r_{2}}}{r_{2}^{2}}$

Thus the net magnetic field at any point in space is given by

$\overrightarrow{dB_{1}}+\overrightarrow{dB_{2}}$

$\frac{\mu _{0}Idl}{4\pi }\int \frac{\widehat{j}\times \widehat{r_{1}}}{r_{1}^{2}}+\frac{-\mu _{0}Idl}{4\pi }\int \frac{\widehat{j}\times \widehat{r_{2}}}{r_{2}^{2}}=0\\\\\Rightarrow \overrightarrow{r_{1}}=\overrightarrow{r_{2}}$

For points with same position vectors from the 2 wires we have a net zero magnetic field. These points are exactly midway between the 2 wires