Animated Solution for Physics - Magnetic Effects of Current: A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point $P$ which lies on the centre of the semicircle?

Imagine you are holding a long, straight wire and you bend it right in the middle to form a hairpin shape.

This creates a fascinating geometry that we can break down into three distinct, manageable pieces.

First, we have the top straight wire coming from infinity and ending exactly above our point of interest. Second, we have the semicircular arc that connects the top and bottom. Finally, we have the bottom straight wire heading back out to infinity.

Our goal is to find the total magnetic field at point $P$, which sits perfectly at the center of the semicircle.

Before we dive into the math, we must understand the direction of the magnetic field created by each segment. Magnetic fields are vectors, so direction is everything!

Let's apply the Right-Hand Grip Rule. Point your right thumb in the direction of the current for the top wire (to the right). Your fingers will curl into the screen at point $P$.

Now, follow the current around the semicircular arc (clockwise). Again, your fingers curl into the screen.

Finally, check the bottom wire where the current flows to the left. Since point $P$ is above this wire, your fingers once again curl into the screen.

Because all three magnetic field contributions point in the exact same direction (into the page, denoted by $\otimes$), we can simply add their magnitudes together.

Now, let's bring in our standard Biot-Savart Law results.

For a semi-infinite straight wire, the magnetic field at a perpendicular distance $r$ from its end is given by:

Since we have two identical semi-infinite wires (top and bottom), their combined contribution will be exactly twice this amount.

Next, we look at the semicircular arc. The magnetic field at the center of a full circular loop is $\frac{{\mu }_{0}I}{2r}$. Because we only have half of a circle, we divide this by two:

We are ready to assemble the final piece of the puzzle. We sum the magnetic fields from all three segments:

Substituting our expressions, we get:

Let's combine the terms for the straight wires:

To make this expression elegant and match our options, we factor out $\frac{{\mu }_{0}I}{4\pi r}$. Notice that factoring out $\frac{1}{\pi }$ from the second term leaves a $\pi$ in the numerator:

This is our final answer! The magnetic field at the center of the hairpin is a beautiful combination of linear and circular geometries.