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# Part 4: Magnetism And Electromagnetic Induction

Sections:

### 4.1 Magnetic Fields and Permanent Magnets

The region around a permanent magnet where magnetic effects can be experienced is called the magnetic field. The field extends outwards from the magnet and gets weaker as the distance from the magnet increases. We can not see the magnetic fields but we can imagine them as containing ‘lines of magnetic flux’ The study of magnetic fields can be quite complex however to understand generators, motors and AC power supplies a rudimentary knowledge of magnetism and electromagnetic induction is useful. The following sections present a simplified view.which have the following properties:

1. they always form complete closed loops
2. they never cross one another
3. they have a definite direction
4. they try to contract as if they were stretched elastic threads
5. they repel each one another when lying side by side and have the same direction.

It is important to remember that these lines of flux are imaginary although they do help us imagine and quantify the magnetic field. However, mathematically magnetic fields are quantified by considering charged particles moving through them. Just as electricity can be described as the flow of electrons around wires, magnetism can be described in terms of a charged particle moving through a magnetic field. These charged particles can be electrons in a wire and the combination of a current, a magnetic field and movement is very important. To keep the maths simple we will keep to imagining magnetic fields as lines of flux.

A freely suspended bar magnet will rotate so that one end will face north, this is the north pole of the magnet. The other end will face south and this is the south pole of the magnet (note that this means that the Earth’s magnetic field actually has a north magnetic pole at the south pole). Magnets obey two simple laws of attraction and repulsion:

1. like poles repel
2. unlike poles attract.

The magnetic flux lines of a magnet (figure 4.1a) obey the 5 rules quoted above and are said to run from the north pole to the south pole, giving the magnetic field a direction. The fields from two north poles repel each other (figure 4.1b), whereas the fields from a south pole and a north pole attract each other (figure 4.1c). The path taken by magnetic flux lines, from north to south, is often called the magnetic circuit. Figure 4.1: (a) the magnetic field from a bar magnet; (b) the magnetic filed due to two bar magnets with unlike poles adjacent; (c) the magnetic field due to two bar magnets with like poles adjacent.

Figure 4.2 shows how the concept of magnetic flux lines can be used to quantify the magnitude and direction of a magnetic field. Magnetic flux (symbol Φ, ‘phi’) has the weber (Wb) as its unit and is a measure of the total quantity of magnetic flux. Magnetic flux density (B) is the amount of magnetic flux concentrated in a given cross-sectional area and has the units of webers per square meter (Wb/m2) or tesla (T). In terms of flux lines this means that the flux density is the number of flux lines passing through a cross-sectional area 1m2. The direction of the flux density at any point is a tangent to to the flux lines at that point.

(It should be noted that strictly the magnetic flux density is a vector quantity which acts perpendicularly to both the force on a charged particle travelling through the magnetic field and the velocity of the particle. If the flux lines are not parallel (as in figure 4.2c) the direction of the flux density will act at a tangent to the field line at any point. In a uniform field, where the field lines are parallel, the direction of the flux density will be parallel to the field lines.) Figure 2: (a) a magnetic field with a uniform and high flux density; (b) a magnetic field with a uniform but lower flux density; (a) an uneven magnetic field with which has a decreasing flux density, although the total amount of flux remains constant (the flux density acts at a tangent to the flux line at any point P).

One weber of magnetic flux spread evenly throughout a cross-sectional area (a) of one square meter results in a flux density of one tesla. Thus: $\mathbf{B=\dfrac {\Phi}{a} }$

### 4.2 Electromagnetism

Whenever an electrical current flows in a conductor a magnetic field is produced. If a current flows down a long straight conductor the magnetic flux lines can be imagined as concentric circles, as shown in figure 4.3. These circles continue for the entire length of the conductor for as long as the current flows and the direction of the magnetic field can be obtained using the right-hand grip rule:

Hold your right hand as if you are going to shake hands but with your thumb pointing straight up. Curl your fingers to grip the wire while pointing your thumb in the direction of the current. Your fingers are now pointing in the direction of the field. Figure 4.3: The magnetic field created when a current flows through a wire. Note that the field is circular so that there are no poles.

When drawn from above, as in figure 4.4, diagrams show current travelling through the wires as if it were a dart; a cross indicates current travelling into the page and a dot indicates that the current moving out of the page. Figure 4.4: (a) wires carrying current into and out of the page and the magnetic fields they create; (b) adjacent wires carrying current in the same direction; (c) adjacent wires carrying current in opposite directions.

Figure 4.4b and 4.4c shows the magnetic field produced if two wires sit next to each other carrying current in the same and opposite directions, the directions of the fields can be checked using the right-hand grip rule. Figure 4.4b shows that two wires carrying current in the same direction produce no field in between the wire but a large field in the duct that holds them, whereas the two wires carrying current in opposite directions in figure 4.4c produce an intense field between the wires but opposing fields in opposite sides of the duct. Since the configuration in figure 4.4b can lead to large heating losses the feed and return cables should be in the same duct.

### 4.3 Solenoids

The magnetic field around a straight wire will be weak even when large currents are carried. A much stronger field is created if an insulated wire is wound to make a tight coil called a solenoid (figure 4.5a). Figure 4.5b shows the direction of the current through the solenoid and the resulting magnetic field. The direction of the magnetic field can be found using the right-hand grip rule as follows:

Hold the hand as before and grip the solenoid as that the figures curl in the direction of current. The thumb is now pointing to the north pole. Figure 4.5: (a) a solenoid; (b) the direction of current and the resultant magnetic field.

If a long solenoid is wound, the ends can be bent round to form a doughnut shape (a toroid, figure 4.6). In such solenoids a circular magnetic field is created and the direction of the flux density is at right angles to this at any given point. If the toroid solenoid is perfectly wound there will be no magnetic field outside the coil.

The magnetomotive force (MMF) of a solenoid is measured in ampere-turns (At) (or more strictly just amperes because the number of turns is a dimensionless quantity i.e. has no units) and is found by multiplying the current flowing by the number of turns of wire (N). Thus: $MMF=I\times N$

Magnetic fields and circuits are often considered to be analogous to electric circuits although no charged particles are flowing. In such an analogy the MMF is analogous to the EMF and the magnetic flux is analogous to the current. The same MMF can be obtained from a solenoid of 1000 turns carrying 1A or a solenoid of 100 turns carrying 10A; 100At in both cases. In the latter case the MMF is produced over a shorter distance and if a third solenoid carried 10A and containing 10 layers of 100 turns each, a much lager MMF of 10 000At can be produced in the shorter distance. Thus the MMF per unit length is an important quantity called the magnetising force or magnetic field strength (H) and given units of ampere-turns per unit (At/m). Therefore: $H=\dfrac{\text{amperes}\times \text{turns}}{\text{length of the flux path}}$ $\mathbf{H=\dfrac {IN}{\mathit{l}}}$

Where:
H = magnetising force (At/m or A/m)
I = solenoid current (A)
N = number of turns of conductor on solenoid
l = mean length of magnetic flux path (m)

Since the value of l is difficult to ascertain for a straight solenoid (figure 4.5b), toroid solenoids are often used in simple examples because the flux path is a circle. In such cases the mean path length is the circumferences around the middle of the coil shown by the dotted line in figure 4.6.

The magnetising force caused by a current flowing in a coil sets up the magnetic flux in and around the coil (figure 4.5b), the magnitude of this flux depends on how easily the material in the centre of the coil is magnetised. If there is a vacuum in the centre of the coil the amount of flux created depends directly on the applied magnetising force and will always produce the ratio: $\mathbf{\dfrac{B}{H}=4\pi \times 10^7=\boldsymbol{\mu} _0}$

Where μ0 is a constant known as the permeability of free space and has the units of Wb/A.m. The ratio of B/H for air and non-ferromagnetic materials also approximates to this value. Thus the flux density in an air-cored coil is given by: $\mathbf{B=H\boldsymbol{\mu} _0=\dfrac{IN\boldsymbol{\mu} _0}{\mathit{l}}}$

For example a large magnetising force of 1×106At/m would produce a flux density of 1.257T within an air-cored solenoid and this is quite a small flux density. To dramatically increase the flux density produced by the same solenoid it can be wrapped around a ferromagnetic material rather than air. There are not may such materials however iron, nickel and cobalt are ferromagnetic. Oddly in some cases a material such as pure iron can be made more strongly ferromagnetic by adding a non-magnetic material such as carbon and certain steels are more magnetic than pure iron.

A solenoid with a ferromagnetic core is effectively an electromagnet and when a current flows the core becomes magnetised thus greatly increasing the magnitude of the flux density. The ratio of the flux density produced with a ferromagnetic core to the flux density produced with an air core is called relative permeabilityr). Air cores of course have a μr = 1 and the values for ferromagnetic cores ranges from 150 to 1200. μr is not a material constant (as density and resistivity are for example) since it is dependent on the magnetising force applied (see part 5), therefore: $\mathbf{\dfrac{B}{H}=\boldsymbol{\mu} _r\boldsymbol{\mu} _0}$

Sometimes the product μrμ0 is called the absolute permeability.

Example

An air-cored solenoid in the form of a closed ring of mean length 0.2m (i.e. the circumference through the middle of the coil) and cross-sectional area 1000mm2. It is wound with 1000 turns and carries a current of 2A. Find the magnetic flux density and the total magnetic flux produced within the solenoid. $H=\dfrac{IN}{l}=\dfrac{2\times 1000}{0.2}=10\; 000At/m$ $B=\mu _oH=4\pi \times 10^{-7}\times 10\; 000=0.01257T=12.5mT$ $\text{Total Flux }\Phi =B\times a$ $=0.01257\times 1000\times 10^-6=0.000\; 01257Wb=0.12.57\mu Wb$

If the same solenoid was wound around a wrought iron core with μr = 500 and a current 0.2A flowing, calculate the flux density and total flux. $H=\dfrac{IN}{l}=\dfrac{0.2\times 1000}{0.2}=1000At/m$ $B=\mu _r\mu _oH=500\times 10\; 000=0.628T$ $\text{Total Flux }\Phi =B\times a$ $=0.628\times 1000\times 10^-6=0.000\; 628Wb=0.628mWb$

This example illustrates that the introduction of a metal core has increased the flux and flux density 50 times even though the current is 10 times smaller.