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9.7 Self induction
Let’s consider two simple circuits. A coil, which is connected to the terminals of the power supply, a battery, a switch, and let’s say that this is our circuit number one. We place a similar coil across from this, and in this circuit we do not have any power supply. We can place a galvanometer over here to detect the current, and let’s call this one as our circuit number two.
Let’s say the power supply, the battery, is generating ε volts of electromotive force. Of course, when the switch is off there will not be any current flowing through this circuit. When we turn the switch in on position, then we are going to end up certain current flowing through this circuit from positive end towards the negative end, so once the switch is closed.
Well, during this process, once we turn this switch in on position, the current is going to start from 0 and will start to build up towards its maximum value. Then it will reach its maximum value and, therefore, flow through this circuit at that constant value. But during the rise of the current, in other words, as it goes from 0 to its maximum value, then the associated magnetic field that it generates also will start from 0 and increase towards its maximum value because as you recall, the magnitude of current is directly proportional to the strength of the magnetic field that it generates.
Well, therefore during that process, as the magnetic filed rises with the increase in current, and it will generate magnetic field lines through this first loop, first coil lets say, and those lines will go through the area surround by the second coil. As the current increases, this magnetic field will increase. Therefore the flux through the area surrounded by second coil will increase, and as a result of this increase in flux, from Faraday’s law, we are going to end up with an induced EMF and hence, induced current. That current will show up from Lenz law such it will oppose its cause, therefore its going to generate a magnetic field in opposite direction to this one. That’s the magnetic field of the induced current. To be able to generate a magnetic field in this direction, using right-hand, rule the induced current therefore has to be flowing in counterclockwise direction throughout this circuit.
Of course this current will be detected by the galvanometer over here and the needle of the galvanometer will move towards one direction. It will deflect from its original 0, whatever point or location. Well, if we follow this case then the magnetic field of this induced current along this second loop will go through the area surrounded by the first loop like this. As the current builds up in the first one, therefore i induced is going to be generating the second one. The magnetic field of that second one will go through the area surrounded by the first coil. Therefore it’s going to generate a change in flux through the first coil and that flux is going to induce an electromotive force along the first coil and therefore also current, and that current will show up such that it will oppose its cause.
If we trace the coils, the original coils, and it will go back to the fact that i induced in the second coil to show up and that is occurring that the original current i is showing up as increasing from its maximum value. Therefore the current showing up induced for the first coil will show up to oppose that cause. In other words, it will try to oppose the increase of that original current. In order to do that, it has to be generating a magnetic field in opposing the direction of flow of the original current. So it has to be running, therefore, in opposite direction of flow of the original current. And because of this reason we are not going to be able to end up with the case that the current will go from 0 to its maximum value right away. It will take some time to reach that value.
Well, we can state over here that by saying, an induced electromotive force also appears in a coil if we change the current in that same coil. This is called self-induction and the EMF, electromotive force, produced is called a “self-induced EMF”. Now, therefore, once we change the current in the first coil, we are going to induce current through the second coil and the magnetic field of this current is going to cause a change in flux through the first coil. Therefore, we are going to end up with i induced along the first coil. And if the current is increasing in the first coil, that induced current will flow in opposite direction to the direction of original current.
Therefore it will cause the original current to reach its maximum value right away. Or if i is decreasing, if the original current is decreasing, from its maximum value towards 0, then the decrease in magnetic field is going to generate or induce electric current along the second coil. That current will flow in such a direction that it will oppose its cause and therefore it is going to flow in a direction such that the magnetic field that it generates will be in the same direction with the original magnetic field. That magnetic field, again, is going to generate a change in magnetic flux through the area surrounded by first coil. That will cause an induced current along the first and in this case, again, it will show up along oppose its original cause and that is the decrease in the original current.
Therefore in that case, that current will show up in the same direction with the original current. Therefore, let me turn the switch off, the current will not drop to 0 right away and it will take some time and we call this electromotive force, which appears as a result of a change in current in the same could as the self-induced electromotive force and the associated current as the self-induced current.
All right. If we recall the definition of inductance for any inductor we have L is equal number of flux linkages divided by current. From here if we make cross multiplication, Li will be equal N times ΦB. Well, Faraday’s law was induced electromotive force is equal to –N times number of turns times change in magnetic flux. We can put this N into the derivative operator since it’s a constant and write down this relationship of d of NΦB over dt. But NΦB is, from the definition of inductance, equal to L time i. Therefore ε becomes equal to –d of Li over dt.
Since inductance is constant we can take it outside of the derivative operator. Then induced EMF becomes equal to –L di over dt and this is the expression for self-induced electromotive force. It simply tells us that if the current is changing, then we are going to end up with a self-induced EMF through the same coil. If the current is constant flowing through any inductor, then self-induced EMF is going to be equal to 0.
So therefore we can summarize by saying that, thus, in any inductor — this can be a simple coil solenoid or a toroid — a self-induced EMF appears whenever the current changes with time. The magnitude of electromotive force has no influence on the induced electromotive force. Only the rate of change of the current counts. In other words, as we generate self-induced electromotive force in the first coil itself, the self-induced EMF and as well as the associated induced current does not have anything to be with the magnitude with the original current. It is directly dependent how fast or how slow this original current is changing.
Well, the direction of the self-induced electromotive force is, again, determined from the Lenz law, in other words, self-induced acts to oppose the change that brings it about. So if we add, therefore, we can say direction of self-induced electromotive force is determined from the Lenz law. That is, it opposes its cause.
In that sense, if you look at a couple of interesting cases assuming that we have an inductor, can be solenoid or toroid or a simple coil, and lets assume that the current is flowing from left to right, and assume that i is increasing. Therefore we are going to end up with a self-induced electromotive force and it will show up such that it will oppose its cause. Obviously to be able to have current flowing in this direction, we have to have our original EMF arrow. It’s pointing to the right for the current to flow from left to right. So if i is increasing, we are going to end up with a self-induced EMF along this inductor such that it will oppose its cause. In other words, it will behave as if we have another power supply which is opposing to this current. In other words, generating an induced current in opposite direction to this original current.
On the other hand if we consider he same inductor, in this case the current is decreasing, again, in the same direction. So again, the EMF arrow is pointing in the direction of flow of current but now the current is getting lesser and lesser so we are going to end up with a self-induced EMF through this inductor, but that EMF will show up in a way that it will try to oppose its cause. To do that it will generate an induced current that is going to be flowing in the same direction with the original current. Therefore it will behave as if we have an ε′ of electromotive force induced and it is generating a current in the same direction with this original one. Whereas in the previous case, that induced current will be in opposed direction to the direction of flow of original current.
Although this is not a very good representation but it will help you to understand because, remember, we can not really use EMF arrows due to the electromotive force generated as a result of induction. Well, it is because of these reasons for the first case we can look at that case as the instant that we turn the switch on so the current builds up from its, from 0 towards its maximum value. Whereas the second case we can visualize this as the moment we turn the switch off so the current decreases from its maximum value towards 0. In both cases, neither the maximum value, nor the 0 value of the current will not be achieved immediately. It will take some time because of the self-induced electromotive force.