Animated Solution for Physics - Magnetic Effects of Current: A loop carrying current $I$ lies in the $x$-$y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is

Welcome to a fascinating journey into the world of electromagnetism! Today, we are going to tackle a visually intriguing problem from JEE Advanced 2012. At first glance, the current-carrying loop in the figure might look like a complex, four-leaf clover. But as we will soon discover, the beauty of physics lies in breaking down complex structures into simple, manageable components.

Imagine you are looking at the $x$-$y$ plane. We have a closed wire loop carrying a steady current $I$.

The shape of this loop is what makes the problem interesting. It is not a simple circle or a standard rectangle. Instead, it consists of four outward-bulging semi-circles.

If we look closely at the dimensions provided in the figure, we can see two labels, both marked as $a$. These labels represent the distance between the opposite inner edges of the semi-circles.

This gives us a massive clue! We can visualize a hidden, central square of side length $a$ perfectly nestled inside the loop. The four semi-circles are seamlessly attached to the four sides of this central square.

Our objective is to find the magnetic moment of this current loop.

In physics, the magnetic dipole moment $\mathbf{M}$ of a planar current loop is a fundamental vector quantity. It dictates how the loop will interact with external magnetic fields.

The master equation for the magnetic moment is elegantly simple:

Here, $I$ represents the magnitude of the steady current flowing through the loop, and $\mathbf{A}$ is the area vector.

The magnitude of the area vector is simply the total area enclosed by the loop.

The direction of the area vector—and consequently the magnetic moment—is determined by the Right-Hand Thumb Rule.

To find the total area $A$ enclosed by our "clover" loop, we don't need any complex calculus. We just need basic geometry.

We can decompose the total area into two distinct parts: the area of the central square and the combined area of the four semi-circles.

First, let's consider the central square. Since its side length is $a$, its area is straightforward:

Next, let's look at the semi-circles. There are four of them, one on each side of the square.

Since the side of the square is $a$, the diameter of each semi-circle is also $a$. This means the radius $r$ of each semi-circle is $a/2$.

The area of a single full circle of radius $a/2$ would be $\pi (a/2{)}^2$.

Since we have four semi-circles, their combined area is equivalent to the area of two full circles:

Let's simplify this expression. Squaring the radius gives us $a^2/4$.

Now, we simply add the area of the square and the area of the semi-circles to get the total enclosed area:

We can factor out $a^2$ to make the expression cleaner:

With the magnitude of the area calculated, we must now find the direction of the magnetic moment vector.

This is where the Right-Hand Thumb Rule comes into play.

Look at the arrows on the loop in the figure. They indicate that the current $I$ is flowing in a counter-clockwise direction.

Take your right hand and curl your fingers in the direction of this counter-clockwise current.

You will notice that your thumb naturally points straight out of the screen, towards you.

In our coordinate system, the plane of the screen is the $x$-$y$ plane. The direction pointing out of the page is the positive $z$-direction.

The unit vector for the positive $z$-direction is $\hat{k}$. Therefore, the direction of our magnetic moment is $+\hat{k}$.

We now have all the pieces of the puzzle. We have the total enclosed area, the current, and the direction.

Let's substitute these into our master equation:

Rearranging the terms to match the standard format of the options, we get our final, elegant result:

This perfectly matches option (b).

This problem is a beautiful demonstration of how complex shapes can be broken down into simple, fundamental geometries.

By identifying the hidden square and the four semi-circles, a potentially intimidating integration problem was transformed into a simple addition of areas.

Always remember to pay close attention to the direction of the current. If the current in this problem had been flowing clockwise, our right-hand thumb would have pointed into the page, and the magnetic moment would have been in the $-\hat{k}$ direction.

Keep visualizing, keep breaking down problems, and the physics will always reveal its underlying simplicity!