# A-level Physics/Fields, particles and frontiers of physics/Magnetic Fields

In order to answer the relevant questions in the exam, it is necessary to have an understanding of magnetic fields and associated concepts such as magnetic flux and magnetic flux density.

## Magnetic field lines

Like electric fields, magnetic fields can be represented by field lines. The direction of the lines shows the direction of the magnetic field, while the density of the lines shows the magnetic field strength (known as the 'magnetic flux density').

The magnetic field lines for a current carrying wire appear as concentric circles around the wire, with the direction of the field being shown by the right hand grip rule. According to this rule, if you point the thumb of your right hand in the direction of conventional current in the wire, the direction in which your fingers grip around the thumb is the direction of the magnetic field.

The magnetic field inside a solenoid (coil) is uniform inside the coil, with the the field directed from its south pole to its north pole. There are also curved field lines outside the coil, which are directed from the north to the south pole.

For any magnetic field, lines start from the north pole to the south pole.Lines are smooth curves which never touch nor cross.Strength is indicated by the distance between the lines-closer the lines means stronger field

## Fleming's Left Hand Rule

The relationship between the direction of magnetic field, the direction of the current in a current carrying wire and the force experienced by the wire is shown by Fleming's left hand rule. This states that if you hold the thumb, first finger and second finger of your left hand at right angles to each other, the thumb will point in the direction of the force experienced by the wire if the first finger follows the direction of the magnetic field and the second finger shows the direction of conventional current inside the conductor.

## Magnitude of magnetic force

Using Fleming's left hand rule on its own is rarely sufficient in physics: it is also important to know the magnitude of the magnetic force. This is calculated using $F=BIL sin( \theta )$, where F represents force experienced by the wire, I = the current in the wire and L is the length of the wire. The Greek letter theta is used to denote the angle between the magnetic field and the direction of current in the wire.

## Magnetic flux density

Magnetic flux density (B) is the term used to describe the strength of a magnetic field, and its SI unit is Tesla (T). Magnetic flux density is defined as B=F/IL, where F = the force experienced by the current carrying conductor, I = the current in the conductor and L = the length of the conductor that is within the magnetic field. The Tesla is the value for B when F = 1N, I = 1A and L = 1m.

## F=BQv

The magnetic force acting on a charged particle travelling through a magnetic field at right angles to the field is given by F=BQV, where B = magnetic flux density, Q = charge, v= speed of particle.

The derivation for this equation is as follows:

• Imagine the path of the charged particle as a wire.
• The current flowing through this path in a period of time (t) would be Q/t.
• The speed would be equal to the length of this path (l) per unit time. Therefore, v=l/t, which can also be expressed as l=tv.
• Substitute I=Q/t and l=tv into F=BIL to get F=B*(Q/t)*(tv). This can be simplified to F=BQv.

## Motion of Charged Particles in a Uniform Magnetic Field

Unlike in an electric field, in a magnetic field the force on a charged particle passing through it is always at right angles to the direction of motion. This means that the charge particle would experience circular motion, with the magnetic force acting as a centripetal force.

Consequently, problems involving magnetic fields often require the use of the equations for circular motion learnt in module G484:
$F = \frac{mv^2}{r}; a = \frac{v^2}{r}; v = \frac{2 \pi r}{T}$

### Mass Spectrometry

One key use of measuring the path of charged particles in a uniform magnetic field is in mass spectrometry, an important technique in chemistry, physics and materials science that is used to measure the mass of ions and their relative abundance in a sample.

In mass spectrometry, a sample of atoms is ionised before being deflected in a vacuum by a uniform magnetic field. The radius of the circular path followed by the ion is directly proportional to its mass, and so the mass of the ion can easily be found after using a photographic film to detect the path of the ion.

## Magnetic flux

The magnetic flux (Φ) through a surface is defined as the magnetic flux density multiplied by the component of the cross sectional area of the surface which is at right angles to the magnetic field. The unit for magnetic flux is the Weber (Wb), where 1 Wb is the magnetic flux produced when a when a magnetic field with a flux density of 1T, acting perpendicularly to the surface, passes through a surface of cross sectional area 1 m2 at right.

While the above definition would be expected in an exam, the official definition of the Weber, as defined by the International Bureau of Weights and Measures is:

The weber is the magnetic flux which, linking a circuit of one turn, would produce in it an electromotive force of 1 volt if it were reduced to zero at a uniform rate in 1 second.

The equation for magnetic flux is: $\phi = BAcos(\theta)$, where theta is the angle the surface and the normal to the magnetic field.

## Electromagnetic induction

In order to understand electromagnetic induction, it is necessary to appreciate what is meant by the magnetic flux linkage of a coil. Flux linkage can be calculated by magnetic flux * number of terms on coil, and, according to Faraday's law of electromagnetic induction, a changing flux linkage results in the induction of an e.m.f. across the coil equal in magnitude to the rate of change of flux linkage.

While Faraday's law gives the magnitude of the e.m.f., it is important to remember that e.m.f. is a vector. Consequently, it is also important to know the direction, something that can be calculated using Lenzs' law. This says that the direction of the induced e.m.f. will be such as to produce a magnetic field in the direction that resists the change in magnetic flux. For example, if magnetic flux is decreasing, an e.m.f. will be induced in the direction that leads to it increasing.

Brought together, these two laws can be represented in a simple equation: induced e.m.f. = - rate of change of flux linkage.

### AC Generator

AC generators apply the above principles in order to generate electricity. In these generators, a coil of wire is rotated inside a magnetic field, changing the angle between the magnetic field and the coil and so changing magnetic flux. According to Faraday's law, this change will lead to the induction of an alternating e.m.f.

### Transformers

A transformer

Transformers use the principles of electromagnetic induction to either increase (in step-up transformers) or decrease (in step-down transformers) the voltage. Its key use is in electricity transmission, where the voltage is increased significantly before being transmitted into a power line. This reduces the current (as P=VI and power is constant) and so increases the efficiency of this transmission process.

Transformers are made of two coils of wire, the primary (input) and the secondary (output) coil, wrapped around an iron core. The voltage across the two coils is shown by:

$\frac{V_s}{V_p} = \frac{n_s}{n_p}$

Vs = Voltage across secondary coil; Vp = Voltage across primary coil; ns = Number of turns on secondary coil; np = Number of turns on primary coil

Transformers work by the following process:

• An alternating current is passed through the primary coil
• The alternating current in the primary coil leads the primary coil to produce a magnetic field of alternating direction (as the direction of the magnetic field and of the current are related by Fleming's left hand rule)
• The iron core ensures that the magnetic field is significantly stronger inside this core. This ensures that the as much of the changing magnetic flux produced by the primary coil as possible links the secondary coil.
• The changing magnetic flux in the secondary coil induces an e.m.f. across its ends.