Biot Savart law (vector form) | Moving charges & magnetism | Khan Academy

Biot-Savart Law: Fundamentals
📌 The strength of the magnetic field produced by a current-carrying wire is calculated using the Biot-Savart Law (pronounced Bo-Savart).
📐 The law requires considering an infinitesimal segment of the wire, $dl$, carrying current $I$, to find the magnetic field contribution, $dB$, at a distance $r$.
⚛️ The vector form of the magnetic field contribution is given by $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \ d\vec{l} \times \hat{r}}{r^2}$.

Magnitude and Dependencies of the Magnetic Field
📏 The magnitude of the magnetic field contribution is $|dB| = \frac{\mu_0}{4\pi} \frac{I \ dl \sin\theta}{r^2}$, where $\theta$ is the angle between $d\vec{l}$ and $\hat{r}$.
📈 The field strength is directly proportional to the current ($I$) and the length of the current element ($dl$).
📉 The field strength is inversely proportional to the square of the distance ($r^2$), similar to gravitational and Coulomb's laws.
📐 A crucial difference from electric fields is the $\sin\theta$ term: $dB$ is maximum when $\theta = 90^\circ$ (perpendicular to the wire segment) and zero when $\theta = 0^\circ$ or $180^\circ$ (on the axis of the current element).

Permeability of Free Space ($\mu_0$) and Field Direction
⚡ The constant $\mu_0$ (permeability of vacuum) has a value of $4\pi \times 10^{-7}$ in SI units.
🖐️ The direction of the magnetic field can be determined using the Right-Hand Clasp Rule (thumb in the direction of current, fingers curl in the direction of $B$).
✖️ The direction can also be confirmed using the right-hand rule for the cross product ($\vec{dl} \times \hat{r}$), where the thumb points in the direction of $d\vec{B}$.

Key Points & Insights
➡️ To find the total magnetic field from a complete wire, one must perform an integral summing the contributions from all infinitesimal elements ($d\vec{B}$).
➡️ Maximum magnetic field occurs only when the observation point is perpendicular to the direction of the current element ($\theta = 90^\circ$).
➡️ The magnetic field is zero along the axis of the current element ($\theta = 0^\circ$ or $180^\circ$), regardless of distance.

📸 Video summarized with SummaryTube.com on Nov 20, 2025, 03:45 UTC