Animated Solution for Physics - Magnetic Effects of Current: A horizontal overhead powerline is at a height of 4 m from the ground and carries a current of 100 A from East to West. The magnetic field directly below it on the ground is (take ${\mu }_0=4\pi \times 10^{-7}{\text{ T m A}}^{-1}$)

Imagine standing on the ground, looking up at a horizontal powerline $4\text{ m}$ above you. The current is flowing from East to West. This is a classic electromagnetism problem that tests both your spatial reasoning and your ability to apply Ampere's Law.

Before we jump into the math, let's orient ourselves. We have a long straight wire carrying a steady current $I=100\text{ A}$. Because the length of a typical powerline is vastly greater than its height from the ground, we can safely model this as an infinitely long straight conductor.

To find the direction of the magnetic field on the ground, we use the Right-Hand Thumb Rule. This is a crucial step where many students make silly mistakes, so let's visualize it carefully.

Point your right thumb in the direction of the current, which is West. Now, curl your fingers. Your fingers represent the circular magnetic field lines wrapping around the wire. Directly below the wire (at ground level), your curled fingers will point towards the South.

If you were asked for the magnetic field directly above the wire, your fingers would point North. Always visualize these circular loops to lock in the correct direction!

Now for the magnitude. The magnetic field $B$ at a perpendicular distance $r$ from an infinitely long straight wire is given by the formula:

Let's plug in our known values. The current $I$ is $100\text{ A}$, and the distance $r$ is the height of the wire, which is $4\text{ m}$. We also know the permeability of free space, ${\mu }_0=4\pi \times 10^{-7}{\text{ T m A}}^{-1}$.

We can simplify the expression easily. The $2\pi$ in the denominator cancels out with the $4\pi$ in the numerator, leaving a factor of $2$.

Multiplying $2$ by $100$ gives $200$. Dividing by $4$ gives $50$.

To write this in standard scientific notation, we adjust the decimal point:

Combining our magnitude with the direction we found earlier, the final magnetic field directly below the powerline is $5\times 10^{-6}\text{ T}$, Southward.