Chapter 28 Physics

Biot-Savart Law

Describes the differential magnetic field created by a current element ($ids$) as: $dB = \frac{\mu_0}{4\pi}\cdot \frac{ids\sin\theta}{r^2}$

Magnetic Permeability of Free Space ($\mu_0$)

A fundamental constant with a numerical value of $4\pi \times 10^{-7} \text{ Tm/A}$.

Magnetic Field from a Long, Straight Wire (Magnitude)

The equation for the magnitude of the magnetic field at a distance $d$ from a straight wire carrying current $i$ is: $B = \frac{\mu_0 i}{2\pi d}$

Right-Hand Rule (for magnetic field direction)

If you point the thumb of your right hand in the direction of current, the curled fingers will point in the direction of the magnetic field.

Magnetic Force between Two Parallel Wires

The force $F$ on a length $L$ of wire 2 due to wire 1 (carrying currents $i<em>1$ and $i</em>2$ separated by distance $d$) is: $F = i<em>2 L B</em>1 = i<em>2 L \left(\frac{\mu</em>0 i_1}{2\pi d}\right)$

Ampere (SI Definition)

The constant current that maintains a force of $2 \times 10^{-7} \text{ N}$ per meter of length between two parallel conductors when both carry $1\text{ A}$ of current and are separated by $1\text{ m}$.

Magnetic Field at the Center of a Circular Loop

For a circular loop with current $i$ and radius $R$, the magnitude of the magnetic field at its center is: $B = \frac{\mu_0 i}{2R}$

Ampere's Law

States that the line integral of the magnetic field over a closed loop is proportional to the total current enclosed by the loop: $\oint B \cdot ds = \mu<em>0 i</em>{\text{enc}}$.

Magnetic Field Inside a Solenoid

For a solenoid with $n$ turns per unit length carrying current $i$, the uniform magnetic field inside is: $B = \mu_0 n i$.

Magnetic Field Inside a Toroid

For a toroid with $N$ turns carrying current $i$ at a radius $r$ within the toroid, the magnetic field is: $B = \frac{\mu_0 N i}{2\pi r}$.

Diamagnetism

A property of materials with negative magnetic susceptibility (\chi_m < 0) that oppose external magnetic fields.

Paramagnetism

A property of materials with positive magnetic susceptibility (\chi_m > 0) that show alignment under external magnetic fields.

Ferromagnetism

A property of materials that maintain magnetic alignment even after external magnetic fields are removed.

Curie Temperature

The critical temperature above which heating disrupts ferromagnetic alignment.