Generating Magnetic Fields: From Current to B-Field

Biot-Savart Law

The principle used to calculate the magnetic field produced by a steady electric current.

Coulomb's Law

A law that calculates the electric field produced by a point charge.

Magnetic field contribution

The differential magnetic field contribution from a differential current element.

Permeability of free space

Denoted as $unsym{\mu_0}$, it is a constant value of $4 ext{\pi} \times 10^{-7} ext{ T} \cdot ext{m/A}$.

dB

The differential magnetic field given by the Biot-Savart Law.

Current element

A differential segment of wire carrying a current, represented as $Id\vec{l}$.

Right-Hand Rule

A method to determine the direction of the magnetic field resulting from current.

Ampere’s Law

Relates the integrated magnetic field around a closed loop to the electric current passing through that loop.

Amperian Loop

A closed path used in applying Ampere's Law.

Line integral of magnetic field

The integral $ extstyle rac{B}{L}$ around a closed path in Ampere’s Law.

Net current enclosed

The total current passing through the area defined by the Amperian Loop.

Ideal Solenoid

A long coil of wire with a uniform magnetic field inside and approximately zero field outside.

Toroid

A solenoid bent into a circular shape (like a donut).

Cross product

A mathematical operation used in the Biot-Savart Law which results in a vector perpendicular to two input vectors.

Magnetic field of a toroid

Given by the formula $B = \frac{\mu_0 N I}{2\pi r}$.

Force between parallel wires

The interaction between two parallel wires carrying currents, leading to attraction or repulsion.

Field outside a thick wire

Decreases in magnitude as inversely proportional to $1/r$.

Field inside a thick wire

Increases linearly with $r$ where $r < R$.

Number of turns per unit length

Denoted as $n$, it is equal to total turns $N$ divided by length $L$.

Trigonometric substitution

A method used to simplify integrals in physics calculations.

In enclosed current calculation

Remember to compute current density $J=\frac{I{total}}{A{total}}$ for thick wires.

Angle in Ampere's Law

Must be either $0^ ext{o}$ or $90^ ext{o}$ to simplify calculations.

Sin(θ) relation

In Biot-Savart Law, relates the angle between the wire current and the position vector.

Integration in Biot-Savart

The process of summing the contributions to the magnetic field from all current elements.

Distance from current element

Denoted as $r$, it refers to the distance to the observation point in space.

Units of current

Measured in Amperes (A), indicating the flow of electric charge.