# Moving Charges and Magnetism

## Magnetic Field due to Current

A current-carrying conductor produces a magnetic field around it, as discovered by Hans Christian Ørsted.

The direction of the magnetic field is given by the right-hand rule, where the thumb points in the direction of the current, and the curled fingers indicate the direction of the magnetic field.

## Ampere's Circuital Law

Ampere’s law relates the magnetic field around a closed loop to the total current passing through the loop.

It is expressed as $∮B⋅dl=μ_{0}I_{enc}$, where $B$ is the magnetic field, $dl$ is an infinitesimal element of the loop, $I_{enc}$ is the total enclosed current, and $μ_{0}$ is the permeability of free space.

## Magnetic Force on Current-Carrying Conductor

A current-carrying conductor placed in a magnetic field experiences a magnetic force according to the Lorentz force equation: $F=I⋅L×B$, where $F$ is the force, $I$ is the current, $L$ is the length of the conductor in the magnetic field, and $B$ is the magnetic field.

## Magnetic Field due to a Circular Loop

The magnetic field at the center of a circular loop carrying current is given by $B=μIR$, where $B$ is the magnetic field, $μ_{0}$ is the permeability of free space, $I$ is the current, and $R$ is the radius of the loop.

## Magnetic Dipole

A current loop behaves like a magnetic dipole, having a magnetic moment given by $m=I⋅A$, where $m$ is the magnetic moment, $I$ is the current, and is A the area vector of the loop.

## Torque on a Magnetic Dipole

A magnetic dipole in a magnetic field experiences a torque given by $τ=m×B$, where $τ$ is the torque, $m$ is the magnetic moment, and $B$ is the magnetic field.

## Earth's Magnetism

The Earth itself acts like a giant bar magnet, with its magnetic north located near the geographic south pole and its magnetic south near the geographic north pole.

The angle between the magnetic and geographic meridians is called the magnetic declination.

## Magnetic Field Lines

Magnetic field lines form closed loops, and the density of the lines indicates the strength of the magnetic field.

Field lines never intersect, and their direction is from the north pole to the south pole outside the magnet and from the south pole to the north pole inside the magnet.