Magnetism

Just as with electric fields, a region of space can have a magnetic field which will bring a force to bear on a charge or magnet. When an electric current moves through a wire, a magnetic field is created around the wire. The strength of a magnetic field is given by:

$$B = {μ_0}/{2π}⋅I/r$$

where B is the strength of the magnetic field (tesla T) around a straight line conductor, $μ_0$ is the magnetic permeability of vacuum ($4π ⋅ 10^{-7} $ N $A^{-2}$), I the current (A), and r the distance (m). Notice that the strength of the field decreases linearly with distance.

The force F on a moving charge is:

$$F = qvBsinθ$$

where q is the charge (coulombs C), v the velocity of the charge or charged particle ($m s^{-1}$), B the magnetic field strength (tesla T), θ is the angle to the magnetic field lines.

The force on a length of wire (m) with current I (amperes A) is given by:

$$F = BILsinθ$$

where θ is the angle between the wire and the magnetic field.

The force between two lengths of parallel wire (m) with currents $I_1$ and $I_2$ (amperes A) is given by:

$$F = μ_0 ⋅ {I_2}/{2πr}I_1⋅L$$

Forces on a Current

An important characteristic of electric and magnetic fields is that they are fundamentally the same. This was discovered by Hans Christian Oersted and described mathematically by Michael Faraday in the 1820s and 1830s.

When an electric current passes through a wire, it creates a symmetrical magnetic field around it. Similarly, if a wire is moved through a magnetic field, or a magneitc field is caused to move around a static wire, a current is induced in the wire. This is the principle behind the electric generator and the electric motor.

Right-Hand Rules

Right-hand rule: finding the magnetic field around a wire with a current

If we know two of three factors, current direction, field direction or force on a charge or magnet, we may calculate the third. There are two rules involving the right hand which are useful in this:

1. The magnetic field around a single wire can be found by gripping the wire with a loosely-fisted right hand, so that the extended thumb indicates the direction of current flow (opposite to the direction of electron flow). The curled fingers indicate the direction of the circles of equal intensity of the magnetic field generated around the wire.

Right-hand palm rule: fingers are the magnetic field, thumb along the current direction, and force is perpendicular to the palm

2. The direction of force can be determined by placing the outstretched right hand so that the fingers align with the magnetic field lines, the salient thumb indicates the direction of current or charged particle, and the force is perpendicular to the palm of the hand.

The illustration demonstrates the principle of the spinning loop. Since the current is moving in opposite directions on either side, the force on the left is into the screen, and the force on the right is out of the screen. This will cause the loop to rotate 90°, till the forces on the two sides of the loop are mutually opposed, resulting in zero net force.

If momentum causes the loop to continue to rotate, the currents in the two sides are now reversed, pushing the loop back the way it came. To prevent this oscillation, a commutator brush is used to reverse the polarity of the current in the loop each half turn, ensuring the forces always push the loop in the same direction.

This is the principle of the electric motor and generator.

The magnetic field strength $B$ at the centre of a circular loop of radius $R$ carrying current $I$ is:

$$B = {{μ_0}⋅I}/{2R}$$

Uniform Magnetic Field

Solenoid: induced magnetic field from a current in a coil

A solenoid is is a coil of wire. When a current is passed through the device, it generates a uniform magnetic field - that is, a field with the same magnitude and direction in a region of space. If the solenoid has an iron core, the magnetic field it generates is much stronger. To find the direction of the magnetic field through a solenoid, use the right-hand thumb rule: wrap the fingers of the right hand around the solenoid in the direction the current flows, and the thumb will be 'hitch-hiking' in the direction of the field.

The magnetic field generated inside a solenoid with current $I$ has a strength B, where:

$$B = {μ_0}⋅{NI}/L$$

where ${μ_0}$ is the permittivity of free space, N is the number of turns in the solenoid coil, and L is the length of the solenoid.

Force between two parallel wires

If two parallel wires, of lengths L, currents $I_1$ and $I_2$, are separated by distance r, then the force between them is equal for both wires and given by:

$$F = {μ_0}⋅{I_2}/{2πr}I_1L$$

If the currents are in the same direction (parallel), the forces are equal and attractive. If the currents are in opposite directions (anti-parallel), the forces are equal and repulsive.

Ampere

The S.I. base unit for current is the ampere (A). It is defined by the magnetic force between two parallel wires: if two wires 1m long, separated by 1m, generate a force of $2 × 10^{-7}$ N, then they each carry a current of exactly 1 ampere (A).

An ampere (amp) equals one coulomb of charge passing a point in a circuit per second: $1 A = {1 C}/s$

Demonstration of the electric motor principle. A cork with a copper wire wrapped tightly around it, on a brass rod, suspended between two flat magnets.

Electromagnets induce a magnetic field by passing a current through a coil of wire

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