Animated Solution for Physics - Magnetic Effects of Current: Find the magnetic field at point $P$ due to a straight line segment $AB$ of length $6\text{ cm}$ carrying a current of $5\text{ A}$ (See figure). (Take, ${\mu }_0=4\pi \times 10^{-7}{\text{ N-A}}^{-2}$)

Imagine you are looking at a straight wire segment $AB$ of length $6\text{ cm}$, carrying a steady current of $5\text{ A}$. We are tasked with finding the magnetic field at a specific point $P$, which is located exactly $5\text{ cm}$ away from both ends $A$ and $B$.

Because the distances $PA$ and $PB$ are equal, the triangle formed by $P$, $A$, and $B$ is an isosceles triangle. To apply our standard physics formulas, we need the shortest distance from the point to the wire. Let's drop a perpendicular from $P$ to the wire segment $AB$, meeting it at point $D$.

In an isosceles triangle, the altitude to the base also bisects the base. Therefore, the perpendicular divides the $6\text{ cm}$ wire into two equal segments of $3\text{ cm}$ each. So, $AD=DB=3\text{ cm}$.

This is a classic scenario for the Biot-Savart Law applied to a finite straight wire. The magnetic field $B$ at a perpendicular distance $d$ from a finite wire is given by the elegant formula:

Here, ${\theta }_1$ and ${\theta }_2$ are the angles subtended by the ends of the wire at point $P$, measured from the perpendicular line $PD$.

To use this formula, we first need to find the perpendicular distance $d$. Looking at the right-angled triangle $PDA$, we can use the Pythagorean theorem:

Next, we need the sine of the angles. From the same right-angled triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse.

By symmetry, $\sin {\theta }_2$ is also $\frac{3}{5}$.

Now, let's bring all these pieces together and substitute them into our master equation. Remember to convert all distances into standard SI units (meters) to avoid any silly mistakes. The distance $d=4\text{ cm}=4\times 10^{-2}\text{ m}$.

Notice how beautifully the $4\pi$ terms cancel out immediately.

The $5$ in the numerator cancels perfectly with the $5$ in the denominator of our sine sum fraction.

Finally, what about the direction? Using the Right Hand Thumb Rule, if you point your right thumb in the direction of the current (from $A$ to $B$), your fingers will naturally curl into the page at point $P$. Thus, the magnetic field is directed perpendicularly into the plane of the paper.