Animated Solution for Physics - Magnetic Effects of Current: A non-planar loop of conducting wire carrying a current $I$ is placed as shown in the figure. Each of the straight sections of the loop is of length $2a$. The magnetic field due to this loop at the point $P(a,0,a)$ points in the direction

The problem of finding the magnetic field of a complex, non-planar 3D loop can seem incredibly daunting at first glance. If you try to apply the Biot-Savart law brute-force to all six segments, you will find yourself drowning in cross products and vector components. But physics is not about brute force; it is about elegance and symmetry.

Principle of Superposition

Imagine you are looking at this folded loop. It looks like a book opened at a $90^{\circ }$ angle, but the spine of the book is missing! The current flows along the edges of the pages but skips the spine.

What if we just put the spine back?

By introducing a fictitious wire along the $y$-axis (the common edge) that carries current $I$ in both directions simultaneously, we change absolutely nothing about the physical reality. The net current added is zero. However, this brilliant mathematical trick allows us to split the complex 3D loop into two perfectly simple, closed 2D square loops: 1. A square loop in the $xy$-plane. 2. A square loop in the $yz$-plane.

Let's focus entirely on the first loop lying flat in the $xy$-plane. Because the side length is $2a$, its center is located at $(a,0,0)$.

Now, where is our target point $P$? The coordinates are given as $P(a,0,a)$. Notice something beautiful? The $x$ and $y$ coordinates of $P$ exactly match the center of our square loop! This means point $P$ lies perfectly on the central axis of this loop, at a height $a$ above it.

Using the right-hand grip rule, if we curl our fingers in the direction of the current (which is counter-clockwise when viewed from above), our thumb points straight up. Therefore, the magnetic field produced by this loop at point $P$ is directed purely along the positive $z$-axis.

Now, let's turn our attention to the second loop standing vertically in the $yz$-plane. Its center is located at $(0,0,a)$.

Look at point $P(a,0,a)$ again. Its $y$ and $z$ coordinates exactly match the center of this vertical loop! Point $P$ lies perfectly on the central axis of this loop as well, at a distance $a$ in front of it.

Curling our fingers along the counter-clockwise current of this vertical loop, our thumb points straight out along the positive $x$-axis. Therefore, the magnetic field produced by this loop at point $P$ is directed purely along the positive $x$-axis.

Vector Addition

We now have two magnetic field vectors at point $P$. Because both square loops have the exact same dimensions (side $2a$), carry the exact same current $I$, and point $P$ is at the exact same perpendicular distance $a$ from both their centers, the magnitudes of their magnetic fields must be identical!

The net magnetic field is simply the vector sum of these two components:

To find the direction, we calculate the unit vector by dividing the net field by its magnitude:

This elegant result perfectly matches option (d). By leveraging symmetry and the principle of superposition, we bypassed pages of tedious calculus and arrived at the solution with pure physical intuition!