Magnetic Fields

A magnetic field is a region of space where a moving electric charge, electric current, or magnetic material experiences a force. These fields are represented by magnetic field lines, which indicate the direction of the force on a north pole. The density of these lines represents the field's strength.

The strength of a magnetic field is quantified by magnetic flux density (B), defined as the force acting per unit current per unit length on a wire placed at right angles to the field. The SI unit for magnetic flux density is the Tesla (T).

### Force on Moving Charges

A fundamental principle is that a magnetic field exerts a force on an individual moving charge. This force (F) is given by the equation:

F = Bqv sin θ

Here, B is the magnetic flux density, q is the magnitude of the charge, v is its velocity, and θ is the angle between the velocity vector and the magnetic field vector. The force is maximum when the charge moves perpendicular to the field (θ = 90°) and is zero when it moves parallel to it (θ = 0°). The direction of this force is determined by Fleming's Left-Hand Rule: with the thumb, forefinger, and middle finger held mutually perpendicular, if the Forefinger represents the Field and the Centre finger represents the conventional Current (direction of positive charge flow), then the Thumb points in the direction of the Force (Thrust).

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force is always directed towards the centre of its path, acting as a centripetal force. This causes the particle to follow a circular path. By equating the magnetic force to the centripetal force, we get:

Bqv = mv²/r

This relationship is crucial in applications like mass spectrometers and particle accelerators.

### Force on a Current-Carrying Conductor

Since an electric current is a flow of charge, a current-carrying conductor placed in a magnetic field will also experience a force. This force is the sum of the forces on all the individual charge carriers. The formula for the force on a straight conductor of length L carrying a current I is:

F = BIL sin θ

Here, θ is the angle between the conductor and the magnetic field lines. The direction of the force is again found using Fleming's Left-Hand Rule.

### Magnetic Flux and Flux Linkage

Magnetic flux (Φ) is a measure of the total number of magnetic field lines passing through a given area. For a uniform magnetic field B passing through a flat area A, the flux is calculated as:

Φ = BA cos θ

Where θ is the angle between the magnetic field vector and the normal to the area. The unit of magnetic flux is the Weber (Wb). When a coil of wire with N turns is placed in a magnetic field, the magnetic flux linkage (NΦ) is the product of the number of turns and the flux passing through each turn:

Flux Linkage = NΦ

This concept is essential for understanding electromagnetic induction (Faraday's Law).

### The Hall Effect

The Hall Effect provides direct evidence for the existence of charge carriers and can be used to determine their sign and density. When a thin, flat conductor carrying a current I is placed in a magnetic field B perpendicular to the current's direction, a potential difference, known as the Hall Voltage (V_H), is established across the sides of the conductor.

The process is as follows: The magnetic field exerts a magnetic force (F_B = Bqv) on the moving charge carriers, deflecting them towards one side of the conductor. This accumulation of charge creates a transverse electric field (E) and thus the Hall Voltage. This electric field exerts an opposing electric force (F_E = qE) on the carriers. A steady state is reached when these two forces balance: Bqv = qE. Since E = V_H / w (where w is the width of the conductor) and the current I can be expressed in terms of the charge carrier density n (I = nAvq, where A is the cross-sectional area), we can derive the formula for the Hall Voltage:

V_H = BI / (ntq)

Here, t is the thickness of the conductor. This effect is utilised in a Hall probe, a device calibrated to measure the strength of unknown magnetic fields by measuring the resulting Hall Voltage.