Animated Solution for Physics - Magnetic Effects of Current: The magnetic field due to a current carrying circular loop of radius 3 cm at a point on the axis at a distance of 4 cm from the centre is $54\mu \text{T}$. What will be its value at the centre of the loop?

Imagine you are looking at a circular current-carrying loop. We are given its radius, $R=3\text{ cm}$, and we are told that at a specific point on its axis, located at a distance $x=4\text{ cm}$ from the center, the magnetic field is $B_{axis}=54\mu \text{T}$. Our ultimate goal is to find the magnetic field exactly at the center of this loop.

At first glance, you might think, "Wait, I don't know the current $I$! How can I solve this?" But physics is often about finding clever ratios and isolating grouped constants. Let's see how we can use the given information to unlock the missing pieces.

Let's recall the Biot-Savart law application for the magnetic field on the axis of a circular loop. The formula is:

This equation connects all our knowns ($B_{axis}$, $R$, and $x$) with our unknowns (${\mu }_0$ and $I$). Let's substitute the values we have:

Now, let's carefully simplify the denominator. Inside the parentheses, we have $3^2+4^2$, which is $9+16=25$. This is a classic Pythagorean triplet! So, the expression becomes:

To evaluate $25^{3/2}$, remember that it means taking the square root of $25$ and then cubing it. The square root of $25$ is $5$, and $5^3=125$.

Instead of trying to find the current $I$ alone, let's isolate the entire term ${\mu }_{0}I$. By cross-multiplying, we get:

Notice how $54$ is perfectly divisible by $9$. $54/9=6$. So the calculation simplifies beautifully:

We keep the units as $\mu \text{T-cm}$ because we didn't convert our lengths to meters. As long as we are consistent, these units will cancel out perfectly in the next step.

Now, let's shift our focus to the center of the loop. At the center, the distance along the axis is zero ($x=0$). If we plug $x=0$ into our master equation, the denominator becomes $2(R^2{)}^{3/2}=2R^3$. The $R^2$ in the numerator cancels with two powers of $R$ in the denominator, leaving us with the familiar formula for the magnetic field at the center of a loop:

We already did the hard work of finding ${\mu }_{0}I=1500$. Let's substitute that in, along with our radius $R=3\text{ cm}$:

And there we have it! By treating ${\mu }_{0}I$ as a single unknown constant, we smoothly transitioned from the axis to the center without ever needing the explicit value of the current.