Magnetism and Magnetic Effects: Diagram-Based Questions (5)

Question

A long straight wire carries current $I = 5\text{ A}$ along the $+x$-axis. Point $P$ is at $(0, 0.1\text{ m}, 0)$. Find the magnitude and direction of the magnetic field at $P$.

Solution — Step by Step

Why This Works

The Biot-Savart law integrated along an infinite wire gives the simple $1/r$ falloff. The field forms concentric circles around the wire — the right-hand rule tells us which way they go. At any point in space, $\vec{B}$ is tangent to the circle through that point, perpendicular to both the wire and the radial vector from the wire to the point.

For finite wires, the formula gets a $(\sin\theta_1 + \sin\theta_2)/2$ correction factor where $\theta_1, \theta_2$ are angles to the wire ends. For “long” wires, both angles approach $90°$ and we get the formula above.

Alternative Method

Vector form of Biot-Savart:

$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\, d\vec{l} \times \hat{r}}{r^2}$

For an element $d\vec{l} = dx\,\hat{x}$ and $\hat{r}$ from the element to $P$, the cross product points in $+z$ when $P$ is in the $+y$ direction. Integrating from $-\infty$ to $\infty$ gives the same magnitude.

Common Mistake

Confusing the right-hand rule for a straight wire (thumb along current, fingers curl) with the rule for a loop (fingers curl with current, thumb gives north). Both are right-hand rules, but applied differently. Practice both on simple cases until they’re automatic.