12.12 Magnetic Field of Electric Currents

An electric current is a flow of electric charge. A fundamental principle of electromagnetism states that moving charges create magnetic fields. The shape and strength of the magnetic field depend on the geometry of the current-carrying conductor.

1. Magnetic Field Due to a Long, Straight Conductor

Concept: An electric current flowing through a long, straight wire produces a magnetic field in the form of concentric circles centered on the wire.

Strength of the Field: The magnetic field strength ($B$) is directly proportional to the current ($I$) and inversely proportional to the distance ($r$) from the wire:

$B=\frac{{\mu }_{0}I}{2\pi r}$

• ${\mu }_0$ is the permeability of free space (${\mu }_0=4\pi \times 10^{-7}$ T·m/A)
• $I$ is the current in amperes
• $r$ is the perpendicular distance from the wire

Direction — Right-Hand Grip Rule: If the thumb of the right hand points in the direction of the conventional current, the curled fingers indicate the direction of the magnetic field lines.

2. Magnetic Field Due to a Flat Circular Coil

Concept: A circular loop of wire carrying a current produces a magnetic field that is strongest and most concentrated at its center.

Strength of the Field at the center of a single loop ($N=1$):

$B=\frac{{\mu }_{0}I}{2r}$

For a coil with $N$ turns, each turn contributes equally, so the total field is:

$B=\frac{{\mu }_{0}NI}{2r}$

• $N$ = number of turns
• $I$ = current through the coil
• $r$ = radius of the coil

3. Magnetic Field Due to a Solenoid

Concept: A solenoid is a long coil of wire wound around a cylindrical core. When current flows through it, it creates a strong, nearly uniform magnetic field inside the coil — similar to the field of a bar magnet. The field outside is comparatively very weak.

Strength of the Field inside the solenoid:

$B={\mu }_{0}nI$

• $n=N/L$ is the number of turns per unit length (where $N$ = total turns, $L$ = length of solenoid)
• $I$ = current through the solenoid
• ${\mu }_0=4\pi \times 10^{-7}$ T·m/A

Direction — Right-Hand Grip Rule for Solenoids: If the fingers of the right hand curl in the direction of the current flowing through the coils, the thumb points toward the North pole of the solenoid.

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