Animated Solution for Physics - Magnetic Effects of Current: Two very long, straight and insulated wires are kept at $90^{\circ }$ angle from each other in $xy$-plane as shown in the figure. These wires carry currents of equal magnitude $I$, whose directions are shown in the figure. The net magnetic field at point P will be

Let's visualize the physical situation presented in the problem. We have two infinitely long, straight, and insulated wires placed in the $xy$-plane. Wire 1 lies along the $y$-axis and carries a current $I$ in the positive $y$-direction (upwards). Wire 2 lies along the $x$-axis and carries an identical current $I$ in the positive $x$-direction (towards the right).

Our goal is to determine the net magnetic field at a specific point $P$, which is located at coordinates $(d,d)$. This means point $P$ is at a perpendicular distance $d$ from both Wire 1 and Wire 2.

To solve this, we rely on the fundamental formula for the magnetic field produced by an infinitely long straight wire. At a perpendicular distance $r$ from the wire, the magnitude of the magnetic field is given by:

However, the magnetic field is a vector quantity, meaning its direction is just as important as its magnitude. To find the direction, we employ the Right-Hand Grip Rule. If you point your right thumb in the direction of the current, your curling fingers will indicate the circular path of the magnetic field lines around the wire.

Let's apply this to Wire 1 first. The current is flowing upwards along the $y$-axis. If you place your right thumb pointing up, your fingers will curl into the page when they reach point $P$ (which is to the right of the wire). Therefore, the magnetic field ${\vec{B}}_1$ is directed inwards, which we denote mathematically as $-\hat{k}$. Its magnitude is:

Now, let's look at Wire 2. The current is flowing to the right along the $x$-axis. Pointing your right thumb to the right, your fingers will curl out of the page when they reach point $P$ (which is above the wire). Thus, the magnetic field ${\vec{B}}_2$ is directed outwards, denoted as $+\hat{k}$. Its magnitude is identical to that of Wire 1 because the current and distance are the same:

According to the principle of superposition, the net magnetic field at point $P$ is simply the vector sum of the individual magnetic fields produced by each wire:

Substituting our calculated vectors:

Because the two magnetic field vectors have the exact same magnitude but point in perfectly opposite directions (one directly into the page, the other directly out of the page), they completely cancel each other out.

This elegant cancellation is a direct result of the symmetry of the setup and the specific location of point $P$.