The Biot-Savart law is a key tool for understanding magnetic fields created by electric currents. It lets us calculate the magnetic field at any point near a current-carrying wire, connecting the current's shape to the resulting field's strength and direction.

This law is crucial for grasping how magnetic fields work in various setups. We'll see how it applies to simple shapes like straight wires and circular loops, and how we can use it to find the total field from complex current distributions.

Biot-Savart Law and Magnetic Fields

Calculating Magnetic Fields from Current Distributions

• Biot-Savart law relates the magnetic field to the current that produces it
  • Allows calculation of the magnetic field $\vec{B}$ at any point in space near a current-carrying wire
  • Expressed mathematically as $d\vec{B} = \frac{\mu_0}{4\pi} \frac{Id\vec{l} \times \hat{r}}{r^2}$
    • $\mu_0$ represents the magnetic permeability of free space, a constant equal to $4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}$
    • $I$ represents the current flowing through the wire
    • $d\vec{l}$ represents an infinitesimal current element (small segment of the wire)
    • $\hat{r}$ represents the unit vector pointing from the current element to the point where the magnetic field is being calculated
    • $r$ represents the distance between the current element and the point where the magnetic field is being calculated
• The magnetic field $\vec{B}$ is a vector field that describes the force experienced by a moving charge or current in the presence of other currents or magnetic fields
  • Measured in units of teslas (T) or webers per square meter (Wb/m²)
  • Represented by field lines, which form closed loops and never cross each other
    • Field lines start at the north pole of a magnet and end at the south pole
    • Density of field lines indicates the strength of the magnetic field

Vector Operations in the Biot-Savart Law

• The Biot-Savart law involves a vector cross product between the infinitesimal current element $d\vec{l}$ and the unit vector $\hat{r}$
  • The vector cross product $\vec{A} \times \vec{B}$ of two vectors $\vec{A}$ and $\vec{B}$ results in a third vector $\vec{C}$ that is perpendicular to both $\vec{A}$ and $\vec{B}$
    • The magnitude of $\vec{C}$ is given by $|\vec{A}| |\vec{B}| \sin \theta$, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$
    • The direction of $\vec{C}$ is determined by the right-hand rule
• The infinitesimal current element $d\vec{l}$ represents a small segment of the current-carrying wire
  • The direction of $d\vec{l}$ is along the direction of the current flow
  • The magnitude of $d\vec{l}$ is the length of the segment
• The unit vector $\hat{r}$ points from the current element to the point where the magnetic field is being calculated
  • Ensures that the magnetic field contribution from each current element points in the correct direction

Applications to Simple Geometries

Superposition Principle

• The superposition principle states that the total magnetic field at a point is the vector sum of the magnetic fields produced by each current element
  • Allows the calculation of the magnetic field from complex current distributions by breaking them down into simpler components
  • Mathematically, $\vec{B}_{\text{total}} = \sum_i \vec{B}_i$, where $\vec{B}_i$ is the magnetic field contribution from the $i$-th current element
• To find the total magnetic field, integrate the Biot-Savart law over the entire current distribution
  • For continuous current distributions, replace the summation with an integral: $\vec{B}_{\text{total}} = \int d\vec{B}$

Magnetic Field of a Long Straight Wire

• The magnetic field around a long straight wire carrying a steady current $I$ can be found using the Biot-Savart law
  • Due to symmetry, the magnetic field lines form concentric circles around the wire
  • The magnitude of the magnetic field at a distance $r$ from the wire is given by $B = \frac{\mu_0 I}{2\pi r}$
    • The field decreases with increasing distance from the wire
  • The direction of the magnetic field is determined by the right-hand rule
    • Point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field
• The long straight wire approximation is valid when the length of the wire is much greater than the distance at which the magnetic field is being calculated

Magnetic Field of a Circular Current Loop

• The magnetic field produced by a circular current loop can be found using the Biot-Savart law
  • Due to symmetry, the magnetic field at the center of the loop is particularly simple to calculate
  • The magnitude of the magnetic field at the center of a loop with radius $R$ carrying a current $I$ is given by $B = \frac{\mu_0 I}{2R}$
    • The field is directly proportional to the current and inversely proportional to the loop radius
  • The direction of the magnetic field at the center of the loop is perpendicular to the plane of the loop
    • Determined by the right-hand rule: point your fingers in the direction of the current, and your thumb will point in the direction of the magnetic field
• The magnetic field at points away from the center of the loop is more complex and requires integration of the Biot-Savart law
  • The field lines resemble those of a bar magnet, with the loop acting as a magnetic dipole