Animated Solution for Physics - Magnetic Effects of Current: The fractional change in the magnetic field intensity at a distance $r$ from centre on the axis of current carrying coil of radius $a$ to the magnetic field intensity at the centre of the same coil is (Take, $r<a$)

The beauty of physics often lies in how complex, intimidating expressions can elegantly collapse into simple, intuitive results when we apply the right mathematical tools. Today, we are going to explore a classic problem from electromagnetism that perfectly demonstrates this: finding the fractional change in the magnetic field along the axis of a current-carrying coil.

Imagine a circular coil of radius $a$ carrying a steady current $i$. This current creates a magnetic field that is strongest right at the center of the coil and gradually weakens as we move away along its central axis.

We are interested in a specific point $P$ on this axis, located at a distance $r$ from the center. The question asks us to find the fractional change in the magnetic field intensity as we move from the center to this point $P$, given a very crucial condition: $r<a$ (which in the context of such approximations usually implies $r\ll a$).

To solve this, we first need to arm ourselves with the fundamental formulas derived from the Biot-Savart Law. The magnetic field at the center of the coil is given by:

And the magnetic field at a distance $r$ on the axis is:

What exactly is "fractional change"? It is simply the change in a quantity divided by its original (or maximum) value. In our case, the field is maximum at the center, so the fractional drop as we move to point $P$ is:

This can be beautifully simplified by splitting the fraction:

Now, let's substitute our master equations into this simplified expression.

Notice how the constants ${\mu }_0$, $i$, and $2$ cancel out perfectly. This is the universe telling us we are on the right track! We are left with:

To make this expression ready for approximation, we need to factor out $a^2$ from the denominator:

Here is where the magic happens. The problem states that $r<a$, which means the ratio $\frac{r}{a}$ is small, and its square $\frac{r^2}{a^2}$ is even smaller.

Whenever a physicist sees an expression of the form $(1+x{)}^{-n}$ where $x$ is very small, they immediately reach for their favorite tool: the Binomial Approximation.

According to the Binomial theorem, for $|x|\ll 1$:

Applying this to our expression, where $x=\frac{r^2}{a^2}$ and $n=\frac{3}{2}$, we get:

Let's plug this approximation back into our fractional change equation:

The $1$ and $-1$ cancel out, and the double negative becomes a positive, leaving us with our final, elegant answer:

This result tells us that near the center of the coil, the magnetic field drops off quadratically with distance. It's a powerful reminder that in physics, understanding the constraints of a problem (like $r<a$) is just as important as knowing the formulas. Keep practicing, and never underestimate the power of a good approximation!