Chapter nine, magnetism. In the last chapter we discussed and developed what was known as an electric field that was set up by an electric charge on two plates and the field he existed in between the plates. Now we, we hypothesized that that was the field but it really you can't see it, you can't touch it, you can't feel it. But we know how it works and we studied the characteristics of it, we can develop formulas and, and do some predictions based on those on that modeling. The same thing happens in magnetic fields. And there are two types of magnetic fields.

There's a permanent magnet that sets up a field nice. kids play with magnets all the time and, and there are magnets that are used in, in industry and in tools. etc, etc. And there's also electric fields, magnetic fields, which we'll, we'll discuss in a in a few more slides. But I'm going to look at permanent magnets just for now, because it demonstrates what a magnetic field or I should say what the characteristics of a magnetic field are. And then we will transpose those over to electric magnetic fields, because they act in the same way.

A magnet is surrounded by a magnetic field and magnetic field exists. A force on other magnets and objects that are made of metal. The magnetic field is strongest, close to the magnet and weaker The farther away you get. The magnetic field can be represented by lines of force or magnetic field lines. a magnetic field also has a direction. The direction of the magnetic field magnetic field around a bar magnet is no is shown with arrows and the magnetic poles are at either end of the bar magnet.

Now, what you see here is a couple of magnets. The top one is referred to as a horseshoe magnet for obvious reasons. The two ends of a horseshoe magnet are north and south are designated north and south poles that are connected by magnetic lines of flux. The magnetic field lines always connect the North Pole and the South Pole glow magnet. If we were to place a compass close to a bar magnet, you would find that the north pole of the compass points in the direction of the magnetic field. This direction is always away from the north magnetic pole towards the south magnetic pole.

In electromagnetism, there is a magnetic field that is produced by an electric current flowing in a wire. Now remember we can't see these fields when we can feel them but we can observe their effect on other magnetic fields and other things. And the magnetic field of a produced by a current flowing in a wire is perpendicular to the wire. Is imagined to form rings around the the wire itself as current is flowing in the direction of the magnetic lines of flux are given by what we call the right hand rule. If we place our right hand on the conductor and their thumb is pointed in the direction of the current flow, then the magnetic field forms concentric circles in the direction of your fingers as demonstrated in the diagram here. And as I said the magnetic field in circles the wire and the flux lines have no north or south pole.

They are continuous lines of flux that formed around the wire. So far, we've been Banting about the terms of flux lines of flux and the force fields. And that's been okay up to now when we're only dealing with the singularity things and having talked about formulas, but we need to define them in quantitative terms, which we're going to do here, so that we can go forward and develop formulas for use in electromagnetic fields. When analyzing magnetic fields, we need to further define them as being made up of magnetic lines of flux. And a flux line is demonstrated here. And it's, again, we can't see them, but we can imagine them going through a particular surface.

Group together. These magnetic lines of flux form in what is known as a magnetic flux field, and Greek symbol phi is usually used to symbolize magnetic flux field. And it's a measure of quantity of magnetism. It takes into account the strength and the extent of the magnetic field. If we consider a flat area such as what is in yellow there, that the magnetic flux passes through, then we can define magnetic flux density as the amount of magnetic flux in an area that is perpendicular to the direction of the magnetic flux. Or in other words, magnetic flux density is the amount of magnetic flux perpendicular to a given flat surface area divided by that area and it's usually defined by The capital Greek letter beta, but beta is very close to our capital letter B.

So, quite often, you'll just see the letter B as designated designating flux density. And the international symbol used for for measuring the quantity of flux density is the Tesla. And one Tesla is equal to one Webber per meter per square meter. And quite often we don't use the term test load quite often we're just going to use the term Webber per square meter and you'll see that happening often. So to summarize, here is what we will be using going forward for discussing magnetic flux and magnetic flux density magnetic flux is usually symbolized by the Greek letter phi It's usually measured in Weber's magnetic flux density is B or beta. And it's measured in Tesla, which is equivalent to Weber's per square meter.

To create a stronger magnetic field force and consequently more field flux with the same amount of electric current, we can wrap the wire into a coil shape where the circling magnetic fields around the wire joined together to create a larger field with a defined magnetic north and south pole polarity. The amount of magnetic field force generated by the coiled wire is proportional to the current through the wire multiplied by the number of turns or wraps of that coil wire, the more raps the more turns, the larger the magnetic field that will be produced. This force field is called Magnum motive force mmf and is very much analogous to electro motive force which is voltage in an electric circuit. mm f is measured in ampair turns. The poles of a magnetic field in a in a coil of wire or a solenoid which is called can be determined again by the right hand rule.

Imagine your right hand gripping the coil of the solenoid such that your fingers are pointing in the same way that the current is flowing. Your thumb then points in the direction of the field since the magnetic field is always coming out From the North Pole, therefore, the thumb points towards the north pole. Here we have wrapped our coil around a block of iron, which I'll refer to as core material. The nature this material is such that the magnetic lines of flux want to travel through it rather than the air. In fact, it will boost the amount of flux that is produced by the current in the coil. In this example, the magnetic flux will still vary as the curtain the coil, but at a greater extent due to the core material.

The magnitude of the magnetic flux will also depend on the number of terms of the coil. As I said, the magnitude of magnetic flux will also depend on the material through which the flux will travel. If an iron core is substituted for an air core, in a given coil the magnitude of magnetic field is greatly increased. All materials have a property defined as permeability, which is the measure of the ability of the material to support the formation of a magnetic field within itself. Hence, it is a degree of magnetization that a material obtains in response to an applied magnetic field. magnetic permeability is typically represented by the Greek letter mew.

Permeability also called magnetic permeability is a constant of proportionality that exists between magnetic induction due to the current flow and the magnetic field intensity of the or the amount of flux that will be produced. This constant is equal to approximately 1.257 times 10 to the negative six Henry's per meter, in free space or a vacuum. In other materials, it can be much different, often substantially greater than in free space, which is symbolized by the Greek letter emu with a subscript zero. materials that cause lines of flux to move farther apart resulting in a decreased amount of magnetic flux density compared to a vacuum are called Daya magnetic materials. materials that concentrate the magnetic flux by a factor of more than one but less than or equal to 10 are called paramagnetic materials and, most importantly, the materials that concentrate the flux by a factor of more than 10 are called ferromagnetic.

The permeability factors of some substances change with the rising and falling of temperatures or the intensity of the applied magnetic field. In engineering applications permeability is often expressed in relative rather than absolute terms. If mew subscript zero or we call it mu not represents the permeability of free space and mew represents the permeability of the substance in question. Then the relative permeability, you mute subscript R is given by this equation, the permeability of material and its capability of either supporting or resisting a magnetic field is called reluctance or magnetic resistance. The reluctance of a magnetic material is proportional to the mean length of that material and inversely proportional to the cross sectional area. And of course, the permeability of the magnetic material itself.

It gives rise to the equation r is equal to L over u A, where r is the reluctance or magnetic resistance of the material. mew is the Mayday permeability coefficient, L is the length of the material in meters and a is the cross sectional area of that material in square meters. So, looking at our coil again on our block of iron the iron is a ferromagnetic material is said to have a reluctance are to the flux that is being tried that is being formed by the current. The flux current in reluctance are related by this equation, and i is equal to the flux times the reluctance. So, if we try if we increase the current, we're going to increase the amount of flux. If we increase the terms we're going to increase the amount of flux.

If the term the reluctance goes up, for the same amount of current and turns, then the flux density would have to go down. So, the reluctance actually provides some resistance to the formation of magnetic flux lines in the iron core. Remember now that the ni is termed Magnum modal force or m m f And it's measured in ampair turns. Sometimes it's helpful to draw the analogy between the electric circuits and the magnetic circuits where the driving force the EMF or voltage is related to the mmf driving force of a magnetic circuit, which would produce in the case of the electric circuit, a current and in the case of magnetic circuit, flux lines of flux. And the thing that would be limiting it would be in the case of the electric circuit the resistance and in the case of a magnetic circuit you have reluctance that would oppose the formation of the magnetic circuit.

In the ideal situation where we energize a coil of wire in air or in a vacuum, the flux produced is linearly related to The input current, hence the flux and the flux density is also the linearity related to the input current. If we plot this fact on a magnetization curve, which we call a bH curve, which plots field intent field density versus field intensity, or flux versus the current, the field intensity being the turns ratio times occurred, but essentially we're looking at flux related to input current, we have a straight line. So if we increase the current, then we increase the flux and its goes up as a straight line relating one to the other. In a real world, the core material is not linear. As you can see here, for three plots of sheet steel, cast steel and cast iron, as we increase the current, it starts to go up.

Nearly but as we go up further and further, then the material becomes what we call saturated and as it becomes saturated you can no longer pump anymore or produce any more flux lines No matter how much current you pump into it. So, we end up with these curved lines which have what they call a knee point, as well as a saturation level as to what we could rise to. Another quirk to confound our analysts of magnetic flux versus force is the phenomenon of magnetic histories this history thesis means a lag between the input and the output in a system upon change a direction. In a magnetic system history says is seen in a ferromagnetic material it tends to stay magnetized. After the applied field force, which is the current has been removed from the circuit. Let's use the same graph again.

Only we'll extend the axis in the negative direction as well, because we're going to reverse the current into the negative direction and see what happens. First we apply an increasing field force or current through the coils of our electromagnet. We should see the flux density increase in other words go as we go up and towards the right, according to a normal magnetization curve for ferromagnetic material. Next, we'll stop the current going through the coil of them of the electromagnet. And what happens is the current returns to zero but the ferromagnetic material maintains some magnetism because that's a characteristic of ferromagnetic material. Now, let's slowly apply the same amount of magnetic field force or current in the opposite direction.

The flux density has now reached a point that's equivalent to what it was in the full positive value of field intensity, except in the negative direction. I stopped the current flow again, going through the coil. And once again, due to the natural retentive ality of the material, it will hold a magnetic flux with no power applied coil, except this time it's in the direction opposite to that of the last time when we stopped, current flowing in the coil. If we reapply power in a positive direction, again, we should see the flux density reach its prior peak in the upper right hand corner of the graph again. So, if we keep doing this back and forth with the current adding or applying it in the positive direction, then taking it off and applying it in the reverse direction, then taking it off and applying in the forward direction again, we will see this S curve develop, which is called the hysteresis curve of magnetic material.

And this doesn't necessarily have we did it with DC current, switching it on and switching it off and reversing it. But this indeed is exactly what happens when we're dealing with alternating current. And I'm not going to go into a lot of details now about alternating Current, because we got another lesson or a lesson in AC circuits, which we'll be following at the end of this. At the end of this lesson plan our modern world would be impossible without electromagnetic induction, the phenomenon that underlies the operation of many devices including power station generators, microphones, tape recorders, car alternators, ignition systems, pedometers. I could go on forever, the list is infinitely long, but they all rely on electromagnetic induction in some way, shape or form. qualitatively speaking, this is how electro magnetic induction works.

What we have here is a circuit, a loop of wire, which is shown in black, they're connected to a volt meter. That is our circuit. We also Have a magnet with magnetic lines of flux emanating from the North Pole going to the South Pole. And they are going through the loop of our circuit, we say that those lines of flux are linked to our circuit because it's going through it. If the person holding that magnet doesn't move the magnet, nothing will happen there'll be zero reading on the volt meter. However, as the person starts to move that magnet, inner route up or down, then the number of lines of flux that are linked by our loop of our circuit loop will change and it is that change that will induce a voltage in that circuit that is electromagnetic induction.

So qualit quantitatively we will introduce Faraday's law of electromagnetic induction which relates the magnitude of an induced voltage to the rate of change of the magnetic flux linking a circuit. Consider the situation shown here. The straight wire pq is moving at a constant velocity delta x at right angles to a uniform magnetic field directed to the right of the screen as it moves, wires PS and qR are fixed and continue to make contact with the moving wire as pq moves to the right, the flux linkages in the circuit s pq r changes in fact it reduces. We will observe a voltage induced between S and R. A volt meter connected between SNR then measures the induced open circuit voltage induced between the ends of the moving wire, but it does not allow a current to flow in the circuit Because it's the volt meter and it's just measuring voltage.

According to Faraday's law, the flux change in this circuit will induce a voltage in the circuit. Faraday's law states that the voltage will be induced according to the change of flux, q2 minus q1 over a period of time T two minus T one or, in other words, more concise terms, v is equal to minus and delta phi over delta t, where V is the instantaneous induced voltage and is the number of turns of our linked circuit or coil of wire and in this case, we only have one turn phi is the magnetic flux in Weber's delta phi is the change of magnetic flux. And delta t is the change in time that that flux changes. The other thing to note here is the minus sign, which is a significant an easy way to create a magnetic field of changing intensity is to move a permanent magnet next to a wire or a coil of wire.

Remember the magnetic field must increase or decrease in intensity perpendicular to the wire so that the lines of flux cut across the conductor or else no voltage will be induced. In other words, he circuit of wire or coil must experience a change in flux a delta phi over a time delta t. Faraday was able to mathematically relate the rate of change of magnetic field flux with induced voltage. This refers to the instantaneous voltage or the voltage at a specific point in time. The deltas represent rate of change of flux over time, and then stands for the number of turns or wraps of the coil of wire. If two coils of wire are brought into close proximity with each other, so that their magnetic fields from one is linked to the other, a voltage will be generated into the second coil as a result, this is called mutual inductance.

The key here is that the induced coil experienced the change in flux over For a period of time, because of the changing flux in the primary coil. Faraday's law still holds true and can still be calculated if it's indeed indeed able to measure it, because it still involves a changing magnetic flux over a period of time. Whenever electrons flow through a conductor a magnetic field will develop around that conductor. This effect is called inductance. Whereas an electric field flux between two conductors allows for an accumulation of free electron charge within those conductors. a magnetic field flux allows for certain inertia to accumulate in the flow of electrons through the conductor producing the field.

Inductors are components designed to take advantage of this phenomenon by shaping the length of the conductive wire in the form of a coil, this shape creates a stronger magnetic field than would be produced by a straight wire. Some inductors are formed with wires around around self. So a self supporting coil, others are wrapped around a solid core material of some type. Sometimes the core of the inductor will be straight and other times it will be joined in a loop, a square rectangle or a circle to fully contain the magnetic flux. These design options are all have effect on the performance and the characteristic of the inductors. The schematic symbol for an inductor like a capacitor is quite simple, being little more than a coil symbol representing a coiled wire although a simple coil shape The generic symbol for any inductor inductors with cores are sometimes distinguished by additional parallel lines to the axis of the coil.

A newer version of the inductor symbol symbol dispenses with the coil shape in favor of several humps in a row. As the electric current produces a concentrated magnetic field around the coil this field flux equates to a storage of energy represented representing the kinetic motion of electrons through the coil. The more current in the coil the stronger the magnetic field will be, and the more energy the inductor will store. Because inductors store the kinetic energy, of moving electrons in the form of, of a magnetic field. They behave quite differently than resistors which simply dissipate energy in the form of heat. Energy storage in an inductor is a function of the amount of current through it.

And inductors ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, inductors tend to resist changes in current. When current through it and doctor is increased or decrease. The inductor resists the change by producing a voltage between the leads in opposite an opposing polarity to that change. Looking at our simplistic circuit here of a resistor in series with a DC power supply or a battery, the voltage drop across that resistor is given by ohms law, which is the product of the current times the resistor it's pretty simple. The voltage drop across an inductor however, is just a little bit more complex.

Whenever current flows through a circuit or a coil flux is produced surrounding it and this flux also links the coil to itself. Self induced EMF in a coil is produced due to its own changing flux and changing flux is caused by changing current in the coil. So, it can be concluded that self induced emf is ultimately due to changing current in the coil itself. And self inductance is the property of a coil or a solenoid which causes a self induced EMF to be produced. When the current through it changes. Whenever changing flux links with a circuit and emf is induced in that circuit.

This is Faraday's law we've already seen that it's parodies law of electronics. Data conduction and according to this law, the voltage across the coil will be given by this equation where vl is the instantaneous induced voltage and is the number of terms of the coil delta phi. The change in magnetic flux in Weber's delta t is the change in time and delta phi or delta t is the rate of change of the flux linkages. The negative sign of the equation indicates that the induced EMF opposes the change flux linkage. The flux is changing due to changing the changing current in the circuit itself. The produced flux due to a current in a circuit is always proportional to that current.

That means, that I is proportional to phi which i is the current and phi is the flux, we can make an equation out of this rather than just a proportionality by saying that phi is equal to k times i, where k is just some constant that we can use. Since the flux is changing over a period of time delta t, then I will also be changing over a period of time. Therefore, delta phi over delta t we can say is equal to k times delta i all over delta t sometimes This is referred to in calculus terms, where they've replaced the Delta with D, which just means the same thing as delta. So anytime you see D, you can replace it with delta d in calculus terms is a very small difference in quantity, and are some hoops that you got to jump through, but we're not going to go there right now, just assume every time you see D. in this equation, it's it's it means delta.

So, delta phi over delta t is equal to k Delta II all over delta t. We can replace the the factors delta phi over delta t in the Faraday's equation, and we are left with the L is equal to minus k which by the way is incorporating the terms of the coil V l is equal to dy by dt and as I said l would incorporate the constant plus the terms of the coil. Now, L is known as the inductance of the coil in the circuit or the inductor as we would call it, in this equation of inductance if v l is equal to one volt, and di by dt is one ampere per second, then l is equal to one and its unit is Henry. That means, if the circuit produces an EMF of one volt due to the rate of change of current true of one amp air per second then circuit is said to have one Henry self inductance this Henry is a unit of inductance This is how the voltage across the inductor and the current through it will develop.

When the switch is closed at t equals zero vl is equal to VB the supply voltage. However, the induced voltage drops off to zero with time as the inductor acts like a short circuit. As the induced voltage drops off with time, the current will increase and be limited by the resistor R. After a long period of time, it will equal VB all over r which is almost law This stands to reason because once The current stops changing. As you can see the curve it's rising up to a maximum limit. Once it stops changing, then there's no more voltage developed across the inductor. So the voltage across it is zero hence it would be acting as a short circuit.

At the beginning of the time when the switch is closed, the inductor looks like an open circuit, so that the current would start off at zero but it would slowly build up as the magnetic field starts to build up in the inductor itself. The curves for the current and voltage associated with an inductor are going to be shown here. On the left, the current to the inductor will start off rapidly at first then level off to a steady state value that is is equal to the supply voltage divided by the resistor on slop. After a long period of time the inductor will look like a short circuit. The equation for this curve is given by this. Now you can see that when t is equal to zero and oh by the way, I L is energizing current VB is the supply voltage.

And the values of t r r is just time is the exponential term 2.71828182 Hs. Set number that we use, R is the resistance analysis the inductance. So, at the beginning when t is equal to zero, the value of e to the minus RT over L is actually equal to one. So, one minus one is zero and you multiply that by VB over R, the current will be zero at t equals zero when P is equal to infinity or after a long period of time, the term e to the minus RT over L disappears. So you're left with VB over r times one, which is the supply voltage over the resistance, which is just ohms line. And again, the inductor acts as a short circuit after a long period of time.

The inductor voltage will equal the supply voltage at the start and then fall off to zero. And it's given by this equation. And you can see that at t equals zero, either the term e to the minus RT over l will be one. So vl is equal to VB. And after a long period of time, the term e to the minus RT over L becomes zero. So, The voltage will drop off to zero.

There are four basic factors of inductor construction determining the amount of inductance created. These factors all dictate inductance by affecting how much magnetic field flux will develop for a given amount of magnetic field force that is current through the inductors wire coil. Number of wire wraps or turns in a coil. All other factors being equal, a greater number of turns of while you're in a coil results in a greater inductance. Fewer turns of wire in a coil result in less inductance. More turns of wires mean that the coil will generate a greater amount of magnetic field force measured in amp turns for a given amount of coil current.

Coil area. Again, all other factors being equal, greater coil area, as measured looking lengthwise through the coil. At the cross section area of the core results in greater inductance less coil area results in less inductance greater coil area presents less opposition to the formation of magnetic field flux for a given amount of field force amp turns. The longer the coils length, the less inductance the shorter the coils length, the greater the inductance a longer path for the magnetic field flux to take results in more opposition to the formation of that flux for any given amount of field force. And lastly, the core material Greater the magnetic permeability of the core, which the coil is wrapped around the greater inductance. The less permeability of the core, the less the inductance.

A core material with greater magnetic magnetic permeability results in a greater magnetic flux field for any given amount of force field. In other words, amp turns when inductors are connected in series that total inductance is the sum of the individual inductor impedances. To understand why this is so, consider the following. The Definitive measure of inductance is the amount of voltage drop across an inductor for a given rate of current change through it. If inductors are connected together in series, thus sharing the same current than seeing the same rate of change of current than the total voltage drop As a result of the change of current will be additive with each inductor, creating a greater total voltage than either the individual inductors alone, greater voltage for the same rate of change of current means greater inductance. Thus, the total inductance for series inductors is more than any one of the individual inductors.

That's the formula for calculating the series total inductance is the same as calculating for series resistance series inductance l total is equal to L one plus L two plus any number of inductance inductors that you have connected in series. When inductors are connected in parallel that total inductance is less than any one of the parallel and Dr. inductive says again, remember that The Definitive measure of inductance is the amount of voltage drop across an inductor for a given rate of current change through it. Since the current through each parallel inductor will be a fraction of the total current and the voltage across each parallel inductor will be equal, a change in the total current will result in less voltage drop across the parallel array than any one of the inductors considered separately. In other words, there will be less voltage drop across the parallel inductors for a given rate of change in current than for any of those inductors considered separately, because the total current divides among parallel branches, less voltage for the same rate of change in current means less inductance.

That's the total inductance is less than any one of the individual inductances the formula for calculating parallel total induction inductance is the same form as calculating parallel resistances one over L total is equal to one over L one plus one over L two plus one over L three. Unlike capacitors, which are relatively easy to manufacture with negligible stray effects inductors are difficult to find in pure form. In certain applications, these undesirable characteristics may present significant engineering problems. inductor size inductors tend to be much larger physically than capacitors are for storing equivalent amount of it equivalent amounts of energy. This is especially true considering the recent advances in electrolytic, capacitor technology allowing in incredibly large capacitance values to be packed into a small package. If a circuit designer needs to store a large amount of energy in a small volume, and has the freedom to choose either a capacitor or an inductor for the task, they will most likely choose a capacitor.

A notable exception to this rule is in the applications requiring huge amounts of either capacitance or inductance. To store electrical energy inductors made made a superconducting wire zero resistance are more practical to build and safely operate than capacitors of the equivalent value and are probably smaller to interference inductors may affect nearby components on a circuit board with their magnetic fields, which can extend significant distances beyond the inductor This is especially true if there are other inductors nearby on the circuit board. If the magnetic fields of two or more inductors are able to link with each other, there will be a mutual inductance present in the circuit as well as self inductance, which could very well caused unwanted effects. This is another reason why circuit designers tend to choose capacitors over inductors to perform similar tasks. capacitors inherently contain their respective electric fields neatly within component package and therefore do not typically generate any mutual effects with other components.

This ends chapter nine