Magnetic fields of currents Knowledge-Based Study

Study notes

1. Oersted's Discovery

   In 1820, Hans Christian Oersted observed that an electric current passing through a wire deflected a nearby compass needle. This historic experiment demonstrated for the first time that moving electric charges produce a magnetic field in the surrounding space.

   Key Principles

   • A stationary electric charge produces only an electrostatic field.
   • A steady electric current (charges in uniform motion) produces both an electric field inside the conductor and a steady magnetic field around it.
   • Magnetic field lines always form closed continuous loops, reflecting the fundamental fact that magnetic monopoles do not exist.

   Field Pattern and Geometry

   The magnetic field lines around an infinitely long, straight wire carrying a steady current $I$ form concentric circles centered on the wire, lying in planes perpendicular to the wire.

   The Right-Hand Grip Rule

   To determine the direction of the magnetic field:

   1. Point the thumb of your right hand along the direction of conventional current.
   2. Curl your fingers around the wire.
   3. The direction in which your fingers curl gives the direction of the magnetic field lines ($\vec{B}$).

   Biot-Savart and Ampere's Law Result

   The magnitude of the magnetic field at a perpendicular distance $r$ from a long, straight wire in vacuum or air is given by:

   $$B = \frac{\mu_0 I}{2\pi r}$$

   • $B$ is the magnetic field strength in teslas (T).
   • $\mu_0 = 4\pi \times 10^{-7}\text{ T}\cdot\text{m/A}$ is the permeability of free space.
   • $I$ is the electric current in amperes (A).
   • $r$ is the radial distance from the wire in meters (m).

   Field Along the Central Axis

   When a wire is bent into a circular loop of radius $R$ carrying current $I$, the magnetic fields produced by each segment add vectorially. At the exact center of a single flat circular loop, the field magnitude is:

   $$B = \frac{\mu_0 I}{2R}$$

   For a Flat Coil of N Turns

   If the coil consists of $N$ tightly wound turns, the individual magnetic fields superimpose constructively:

   $$B = \frac{\mu_0 N I}{2R}$$

   Determining Polarity

   The direction of $\vec{B}$ at the center can be found using the right-hand rule for loops: curl the fingers of your right hand in the direction of the current circulating the loop; your extended thumb points in the direction of the magnetic field along the axis, pointing toward the magnetic north pole of the loop.

   Ideal Solenoid

   A solenoid consists of a long helical coil of closely spaced turns. When current flows through it, the magnetic field inside is strong, nearly uniform, and parallel to the axis, while the field immediately outside is negligibly weak.

   Magnetic Field Inside a Long Solenoid

   For an ideal solenoid with length $L$ and total turns $N$:

   $$B = \mu_0 n I$$

   where $n = \frac{N}{L}$ represents the turn density (number of turns per unit length).

   The Toroid

   A toroid is essentially a solenoid bent into a circular doughnut shape. Applying Ampere's law reveals that:

   • The magnetic field is completely confined within the interior of the toroid.
   • At a distance $r$ from the central axis inside the core: $$B = \frac{\mu_0 N I}{2\pi r}$$
   • The magnetic field is zero both in the central open hole and outside the outer perimeter of the toroid.

   Summary of Key Formulas

   • Long straight wire: $B = \frac{\mu_0 I}{2\pi r}$ (varies inversely with $r$)
   • Center of circular coil: $B = \frac{\mu_0 N I}{2R}$ (varies inversely with $R$)
   • Interior of an ideal solenoid: $B = \mu_0 n I$ (uniform, independent of radius or cross-sectional area)

   Superposition Principle

   When multiple currents are present, the net magnetic field at any point in space is the vector sum of the individual magnetic fields produced by each current:

   $$\vec{B}_{\text{net}} = \sum \vec{B}_i$$

   Always evaluate vector components carefully, particularly when currents run parallel or antiparallel in neighboring wires.