Question

Two very thin metallic wires placed along X– and Y–axes carry equal currents as shown in figure. AB and CD are lines at ${45}^{\circ }$ with the axes with origin of axes at O. The magnetic fields will be zero on the line

• A. AB
• B. CD
• C. Segment OB only of line AB
• D. Segment OC only of line CD

Answer 1

The correct option is A AB

Magnetic field due to an infinitely long current carrying wire
$B=\frac{{\mu }_{0}i}{2\pi d}$ where $d$ is the perpendicular distance of the point from the wire.


Magnetic field at a point P on the line AB,
$=B_1+B_2$ where $B_1$ is the magnetic field due to the wire along the x-axis and $B_2$ due to the wire along y-axis.

$B_1=\frac{{\mu }_{0}I}{2\pi d_1},B_2=\frac{{\mu }_{0}I}{2\pi d_1}$.

Since AB is inclined at an angle of ${45}^{\circ }$ with OX, $d_1=d_2$
We can say that the magnitudes of $B_1$ and $B_2$ are the same. Therefore, the net magnetic field will depend on the directions of $B_1$ and $B_2$

The direction of magnetic fields due to the individual wires on lines AB and CD are shown in the diagram below (can be found using the Right Hand Rule)


When you add the fields due to the two wires, the net magnetic field along AB and CD will be as follows.


Therefore, net magnetic field ($B_1+B_2$) will be zero along line AB.