Magnetic Field and Biot-Savart Law

Magnetic Field Created by an Electric Current

Biot–Savart Law

• Defines the magnetic field $d\overrightarrow{B}$ created by a current element $Id\overrightarrow{s}$ at a point P at distance r.
• $d\overrightarrow{B} = \frac{\mu_0}{4\pi} \frac{Id\overrightarrow{s} \times \hat{r}}{r^2}$
• $\mu_0 = 4\pi \times 10^{-7} Tm/A$ represents the permeability of free space.
• The total magnetic field is the integral of the expression for $d\overrightarrow{B}$: $\overrightarrow{B} = \int d\overrightarrow{B}$.
• $d\overrightarrow{B}$'s direction is perpendicular to both $d\overrightarrow{s}$ and $\hat{r}$.
• The magnitude of $d\overrightarrow{B}$ is inversely proportional to $r^2$ (r = distance from current element to point P).
• The magnitude of $d\overrightarrow{B}$ is proportional to the current I and the length of the element $d\overrightarrow{s}$.
• The magnitude of $d\overrightarrow{B}$ is proportional to $sen\theta$ ($\theta$ is the angle between $d\overrightarrow{s}$ and $\hat{r}$).
• The magnetic field dB is zero at points along the wire's extension ($\theta = 0$).

Ampère's Law

• The line integral of $\overrightarrow{B} \cdot d\overrightarrow{s}$ around any closed path equals $\mu_0 I$.
• I is the total current through any surface bounded by the closed path: $\oint \overrightarrow{B} \cdot d\overrightarrow{s} = \mu_0 I$.

Strategy to Solve Problems

• Ampère's law is suited for calculating magnetic fields from current distributions with high symmetry.
• Select a closed path where the magnetic field is constant where $\overrightarrow{B} \neq 0$.
• Where the magnetic field is tangent to the path, $\overrightarrow{B} \cdot d\overrightarrow{s}$ simplifies to $Bds$.
• Where the magnetic field is perpendicular to the path, $\overrightarrow{B} \cdot d\overrightarrow{s} = 0$.
• Where the magnetic field is zero, the integral of $\overrightarrow{B} \cdot d\overrightarrow{s}$ over that portion of the path is also zero.

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