Electromagnetic Induction theory and study

This chapter is included in the items below. If you want to go directly to an item, click on it.

2 - Principles of the Faraday Induction Law click here!

3 - Induced Electromotive Force (fem) click here!

3.1 - HALL Effect click here!

4 - Induced Current in a Circuit click here!

5 - Energy Conservation click here!

6 - The Magnetic Flow click here!

7 - Lenz's Law click here!

8 - Faraday's Law click here!

9 - Inductance click here!

In the previous chapter we studied the fact that an electric current produces a magnetic field. Now we are going to study the fact that a magnetic field can generate an electric field capable of produce an electric current. One of the scientists who dedicated himself to studying this phenomenon was Michael Faraday (1791 - 1867), a British physicist with great influence throughout the development of science in its day. In your honor, the phenomenon we are going to study was called Faraday's Induction Law.

2. Principles of the Faraday Induction Law

Faraday found that an electromotive force (emf) and an electric current can be induced in a loop just vary the amount of magnetic field that goes through the loop. On this site, we understand the loop as a conducting wire, preferably in circular shape.

In the Figure 75-01the experience that Faraday carried out to prove his discovery is shown. He used two coils wrapped around an iron ring and hoped the magnetic field generated in the iron ring by the electric current flowing through the left coil, would induce an electric current in the right coil. So what Faraday did was to close the switch and, at first, he realized that nothing happened with the current meter. However, Faraday managed realize, after a few attempts, that as soon as he closed the switch the needle the current meter made a small jump and immediately returned to rest. In a second moment, when he opened the switch, he again noticed the jump of the needle the meter, but in the opposite direction to the previous one. And, immediately after opening, returned to rest. The movement of the needle indicated that there was a small current flowing by the coil on the right only when the switch was closed or opened.

Faraday's observation that the current meter needle bounced only when the switch was open or closed led Faraday to conclude that the current was generated only while the magnetic field was changing as it passed through the coil. That would explain why all previous attempts to generate a current from of magnetism: in them, only static and immutable magnetic fields were used.

According to what we studied in chapter 23 (circuit RL), we saw that when a voltage is applied suddenly on the inductor it behaves like a circuit open. After this initial instant, the current in the inductor increases exponentially until it stabilizes at its final value. The same happens when we remove this voltage (opening the switch), decreasing exponentially to zero. Only in these two moments does the current vary in the inductor and, therefore, only in these two moments there is variation in the magnetic field generated by it.

With that observation, Faraday concluded that there should be variation in the magnetic field, then there would be no need for the iron ring. Then, he set up an experiment, as indicated in Figure 75-02 , placing one coil above the other and he observed, once again, that current was only induced in the second coil when the switch was opened or closed. In the closed switch situation there was no any movement of the electric current meter needle.

But Faraday wanted to explore this phenomenon further. Then he wondered if moving a permanent magnet inside a coil, if it would not induce an electric current in the coil. So, he set up the experiment as shown in Figure 75-03 . In this case, he noticed that when he pushed the magnet into the coil, there was a displacement of the meter needle and, if he pulled the magnet, the meter needle would it was moving in the opposite direction. But if you stayed with the magnet inside the coil, without moving it, nothing would happen.

The next step was to ask yourself what would happen if the magnet remained immobile and the coil was moved quickly? Faraday managed to observe a deflection moment of the current meter needle when pushing the coil towards the magnet. And as he moved the coil away from the magnet's magnetic field, he watched the needle on the meter pointer move in the opposite direction. The setup for this experiment is shown in Figure 75-04 . And so, Faraday found that the effect was the same in all cases.

3. Induced Electromotive Force (fem)

As we studied in Item 2, we can generate induced electric current in two different ways:

By varying the size or orientation of a circuit in a stationary magnetic field.

Varying the magnetic field that passes through a stationary circuit.

Although the effects are the same in both cases, the causes turn out to be different. We will start our study of electromagnetic induction, analyzing the case in that the magnetic field is fixed, while the circuit moves or varies.

Let's consider a conductor of length L that moves with speed v^→ in the presence of a uniform magnetic field B^→, as shown in Figure 75-05. Also consider that v^→ be perpendicular to B^→. So, the magnitude of the magnetic force is given by F_B = q v B. And this force moves the charges carriers, causing a separation between the charges.

The separation of charges, in positive and negative, creates an electric field inside the conductor, as shown in Figure 75-06 . This electric field generates a potential difference electric (emf) between the ends of the moving conductor. Note that if the movement of the conductor ceases, this potential difference disappears instantly due to the collapse of the electric field, as there will be no more separation between the charges.

The charges flow continuously until the electrical force F_E^→ downwards is big enough to balance the magnetic force F_B^→ oriented upwards. In this case, the resulting force on the charges becomes null and the electrical current within the conductor ceases.

So when the electrical force F_E = q E exactly balances the magnetic force F_B = q v B, then the intensity of the electric field is given by:

eq. 75-01

From the above, we can conclude that:

"The magnetic force on the charge carriers of a conductor in motion under the action of a uniform magnetic field B , creates an electric field E = v B inside the conductor."

The electric field, in turn, gives rise to a difference in electrical potential between the two ends of the moving conductor. So we can write:

eq. 75-02

So, it follows that:

"The movement of a conductor in the presence of a magnetic field induces a potential difference v L B between the ends of the conductor."

3.1 HALL Effect

We will take advantage of what has been studied so far and introduce the study on a very interesting that it was studied by Edwin H. HALL in 1879. In his honor this effect is known as the HALL effect . This effect allows to verify if the carriers have a charges positive or negative. In addition, it allows measuring the number of carriers per unit volume of the conductor.

So, let it be a conductive tape of width L , of sectional area A , covered by an electric current I and placed in the presence of a magnetic field B. In this case, the charges of this conductive tape will undergo a force, as shown in the Figure 75-08 . Consider the case of electric current I moving from left to right. Therefore, the negative charges will be diverted to the bottom of the conductive tape.

And, of course, the positive charges move to the top of the ribbon. Thus, the magnetic force causes an electric current perpendicular to the direction of propagation of the initial current. From this fact, a region will appear with a concentration of positive charges and another with a concentration of negative charges. This will cause an electric field to appear perpendicular to the magnetic field. This current cease when the balance of positive and negative charges creates an electrical force that cancel the magnetic force on the charges. We already saw this in the previous item, where we arrived to a relationship between the magnetic field, the electric field and the speed of the charges through the eq. 75-01.

Therefore, when we reach the stability of the forces, there will no longer be that current, however, there is a potential difference between the sides of the tape. Let's represent this electrical voltage for V_hall . So, based on the electric field generated, we can write the potential difference of Hall by:

eq. 75-03

Based on eq. 73-05 and eq. 73-06 we can find the value of the drift speed, V_d , through an algebraic manipulation in the equations. So:

eq. 75-04

Where J = I/ A is the current density, n is the volumetric density of carriers of charge. On the other hand, we have the area A = L x, and beyond that, considering equations eq. 75-01 , eq. 75-03 and eq. 75-04 we can determine the equations of carrier volumetric density and hall voltage, as a function of the known values of the magnetic field and current I.

eq. 75-05

Da eq. 75-05 it is possible to find the value of the volumetric density of charge carriers, n, passing n to the first member of the equation and passing V _hall for the second member of the equation.

There is a laboratory test where it is possible to measure the intensity of magnetic fields, called Hall test , from the measurement of Hall voltage , using a conductor called "poor" , as long as its volumetric density of carriers is known. The conductor it is considered "poor" when it has lower densities of charge carriers. Normally these types of conductors have the highest Hall voltage . An example of this type of conductor is the bismuth , which presents a n = 1,35 x 10²⁵ carriers m^-3.

Another use of the Hall effect is to measure the drift speed V_d of carriers charge, which, as we know, is on the order of a few centimeters per hour. Therefore, we must move the tape, in the presence of a magnetic field, in the opposite direction to that of the drift speed of carriers. The tape speed is adjusted so that the difference Hall's potential is null . In order not to observe the Hall effect, it is necessary that the speed of the carriers in relation to the laboratory is zero . That wants to say that, under these conditions, the speed of charge carriers has the same module that the tape speed and opposite direction.

4. Induced Current in a Circuit

As we saw in the previous item, a moving conductor generates a fem, but as we don't have a closed circuit, the current has nowhere to circulate. Thus, we can build a system in such a way that it is possible to close the circuit. This is shown in Figure 75-09, where we have a U conductive rail fixed on a table, for example, where a conducting wire slides with speed v. The wire and the rail together form a single conductive loop which, in itself, represents a closed circuit.

From the figure to the side we can see that the magnetic field B is entering the page plane and is perpendicular to the circuit plane. The moving wire charges will be displaced to the wire ends by the magnetic force, only now they can continue to flow around the circuit, that is, the moving wire acts like a battery in a circuit. The current in the circuit is an induced current . In this case, the current is counterclockwise.

And we can say that the induced current is due to the magnetic forces exerted on the moving charges. And it is possible to calculate this current I if we consider that the entire circuit has a resistance value R . So, applying the Ohm's law the electric current I will be:

eq. 75-06

When we said that the wire was moving at a constant speed v on the rail, it was implied that there was a F force pulling the wire to the right. However, the wire carries an induced electric current I and, as we know, it will be subject to a magnetic force F_m , pointing to the left (right hand rule), because it is under the action of a uniform magnetic field. That magnetic drag force will cause a deceleration and the final stop of the wire, unless we exercise constantly a force to pull him, F_pull , of the same module, but opposite, in order to keep you moving.

In Figure 75-10 we see the depiction of the situation described. It should be noted that as the wire moves to the right, the magnetic force F_B pushes the carriers charge parallel to the wire. Their movements, as they move through the circuit, form the current induced I . And because of the existence of a current, a second magnetic force, F_m , appears on the wire. This force is perpendicular to the wire and decelerates the moving wire.

As we know, the magnitude of the magnetic force on the current-carrying wire is given by equation F_m = I L B . Using this result in conjunction with the eq. 75-06 for the induced current, we obtain that the tractive force necessary to keeping the wire at a constant speed v is given by:

eq. 75-07

5. Energy Conservation

We know that a force that pulls or pushes a body with speed v , develops a power that can be expressed by P = F v . Thus, the power delivered to the circuit by the force that pulls the wire is:

eq. 75-08

This is the rate at which energy is added to the circuit by the force that pulls it.

But the circuit also dissipates energy by transforming electrical energy into energy heat in the wires and components that heat up. In the circuit, which has a resistance R , the power dissipated is given by P = R I². The eq. 75-06 provides the value of the induced current I and, after algebraic work we can calculate the dissipated power, P_dissi , by the circuit, or:

eq. 75-09

We can easily see that the equations eq. 75-08 and eq. 75-09 are absolutely identical . Thus, we can affirm that:

"The rate at which work is done on the circuit exactly balances the rate at which the energy is dissipated. Therefore, energy is conserved."

This act of pulling, sometimes pushing a conducting wire under the action of a magnetic field, through a mechanical force, causes it to give rise to an induced electric current. In other words, we are transforming mechanical energy into electrical energy and this device that makes this transformation is called generator .

With everything that has been seen so far it is possible to make a summary.

Summary

Pull or push a conductive wire in the presence of a magnetic field with speed v will create an electromotive force, fem , of movement in the wire and induce a current electrical I = fem / R.
To keep the wire moving at a constant speed, a force exerted that pulls it or push must balance the magnetic force on the wire. This force does work on the circuit.
The work done by the pulling or pushing force exactly replenishes the energy that is dissipated current when crossing the circuit resistance.

6. The Magnetic Flow

To understand the concept of magnetic flux, let's imagine a conductive loop rectangular with an area A = a b , where a and b are the dimensions sides of the loop, and that is immersed in a uniform magnetic field, as shown in Figure 75-11.

"Let's define magnetic flux as the amount of magnetic field that passes through the area effective loop."

Since the effective area of the loop is given by A_ef = A cos θ = a b cos θ. We must pay attention to the fact that the angle θ is the angle of inclination of the axis of the loop in relation to the magnetic field. Mathematically we can write the magnetic flux, Φ_m, as:

eq. 75-10

Thus, the magnetic flux measures the amount of magnetic field that passes through a loop of area A when it is tilted at an angle θ in relation to the field. By definition, the unit of measure S I of the magnetic flux is the weber . We have the following equivalence:

1 Weber = 1 Wb = 1 T m²

By the eq. 75-10 and in reference to vector algebra, we realize that this resembles a scalar product of vectors, where A • B = A B cos θ.

Here we will define the area vector represented by A^→ as a vector perpendicular to the loop area and with modulus equal to area A of the same. And this vector has as a unit the m². The Figure 75-12 shows the area vector for a circular loop of area A.

Then, we can define the magnetic flux as a scalar product given by:

This is the case for when we have a constant magnetic field, that is, at any point considered the field is the same. However, if we are in the presence of a non-uniform magnetic field we cannot use the above equation to calculate the magnetic flux. In this case, we must appeal to the integral calculation and take small areas within the total area of the loop. Calculating the flow in each small area and adding up, we must find the flow in the loop. As the summation can get complicated, we appeal for the integral, as shown in eq. 75-11.

eq. 75-11

7. Lenz's Law

So far we have studied the discovery of Faraday where the current can be induced in a wire conductor or conductive loop as long as there is a variation in the magnetic flux through them. The way you vary the flow does not matter. This discovery by Faraday took another physicist, the German Heirich Lenz , to be interested in the subject. So, after studying and doing several experiments Lenz enunciated a rule to determine the direction of the induced current. Today we know this rule as Lenz's law . We can express this law as follows:

“ There will be an induced current in a closed conductive loop if, and only if, the magnetic flux through it is varying. The direction of the induced current is such that the induced magnetic field is opposed to the flow variation. ”

To understand how this works, let's look at Figure 75-13 , where we have the representation of a solenoid with its respective terminals connected to a galvanometer. The galvanometer is a instrument that allows measuring the module and direction of the electric current. There are two terminals: one marked as positive and another marked as negative. If the current enters through the terminal marked positive (red arrow), the galvanometer needle will deflect to the right, as shown in the figure below. Otherwise, that is, if the current enters through the terminal marked negative (blue arrow), the galvanometer needle will deflect to the left. If there is no current flowing through the galvanometer, then the needle will remain at rest in the center of the display ("zero" mark).

In this presentation we are going to use the motion of a permanent magnet to cause the flow to vary magnetic across the solenoid. Rather, we must understand that a solenoid represents a inductor . So, if we introduce the magnet inside the solenoid, this will cause an increase in the magnetic flux in the solenoid. The reaction of the solenoid is not to allow an increase in flow. To do so, it will generate a flow contrary to that of the magnet. As a consequence, using the hand rule right, the thumb should point to the left of the figure. With the other fingers of the hand, we wrap the solenoid and determine that the current must enter the positive terminal of the galvanometer, indicated by the red arrow in the drawing. This will cause the instrument's needle to deflect to the right. If we stop the magnet moving, the current instantly drops to zero and the needle returns to rest in the center of the instrument. From this moment on, let's assume that we make the decision removing the magnet from inside the solenoid. In this case, it will interpret that the magnetic flux is decreasing. Soon, he will oppose this by generating an opposite flow, that is, pointing to the right of the figure to reinforce the flow. Knowing this and using the right hand rule, we determine that the current, in this situation, must enter through the negative terminal of the galvanometer, indicated by the blue arrow in the drawing. This will make the needle of the instrument deflect to the left.

In other words, the inductor (or solenoid) is always that of the "against" . That is, if we want to increase the flow, it will do everything to reduce it. If, on the other hand, we want to decrease the flow, it will try to increase the flow. And it achieves that goal, generating a current with the proper direction. This is what is written in Lenz's law , when it says "it opposes the variation of the flow".

It must be clear that in order to have induced electric current we must have a variation of magnetic flux . For this, we can list three situations:

The magnetic field through the loop varies (increases or decreases).

The loop varies in area or angle of inclination.

The loop moves in or out of a magnetic field.

So it must have been very clear that an induced current generates its own magnetic field.

8. Faraday's Law

So far we've seen that Faraday has discovered that an electric current will be induced in a conductive loop whenever there is a variation in the magnetic flux over it. And Lenz's law allows we can determine the direction of the induced electric current. As the charges do not come into contact movement spontaneously, you need a fem that provides energy.

Therefore, we can summarize that the induced fem is the rate of change of flow through the loop. In this way, we can express the modulus value of the fem induced in the loop as

eq. 75-12

fem

eq. 75-13

As we saw in item 4 , we studied the case of a conductor wire moving through a field magnetic over a U-shaped conductive rail. We repeat the figure here, with minor changes as shown in Figure 75-14.

Note that the magnetic field is perpendicular to the plane of the conductive loop, so we conclude that the angle θ = 0°. As a result, the magnetic flux is reduced to the product between the area, A , of the loop and the magnetic field, B . Considering that the sliding wire is at a distance x from the end, the area is A = x L and the flow in that instant of time is given by

eq. 75-14

As the wire slides over the rail to the right, its area A is increasing and, consequently, the flow through the loop is also increasing. So, according to the law of Faraday , deriving the flow through the loop, in relation to time, the induced fem is

eq. 75-15

Where the wire speed is v = dx / dt . Making use of eq. 75-15 and applying the Ohm's law to the circuit, we establish the equation that determines the current induced in the wire, or

eq. 75-16

In this example that we used to develop this theory, we remember that we assumed that the wire slid to the right, increasing the loop area and, consequently, increasing the magnetic flux through her. By Lenz's law , if the flow is increasing, then the induced current on the wire must oppose this growth. That is, an opposite flow must be generated through the loop so that there is a reduction in the total flow through the loop. For this, we must have the current induced electrical flow counterclockwise.

In short, Faraday realized that all induced currents are associated with variations in the magnetic flux. There are two different and fundamental ways of varying the magnetic flux through conductive loop:

The loop can move, expand or rotate, creating a fem of movement.

The magnetic field may vary.

These two situations can be better understood if we write Faraday's law in the following way:

eq. 75-17

The first term on the right side of the equality represents the fem due to the position variation of the loop itself, causing a variation in the magnetic flux through it. This term includes situations like the circuit with a sliding wire, where the area A varies, as well as the case of turns that rotate in the presence of a magnetic field. The physical area of a loop that rotates does not vary, but the vector A^→ , yes. This movement of the loop causes magnetic forces to act on the charge carriers of the loop.

The second portion on the right side of the equality represents the variation of the magnetic field, even if nothing is in motion. That is, the rate of change of the magnetic flux through the loop, regardless of what causes the flow variation, generates an fem induced in the loop.

9. Inductance

It is a quantity that, associated with a given inductor, characterizes its greater or lesser capacity to produce flux for a given current. We already know that in order to create an induced electromotive force in a conductor, it is necessary for it to be subjected to a variable magnetic field. As we can see, the inductance of a body is a property that only manifests itself when the current passing through the body varies in value, which produces a variable magnetic field, to which the body itself or another conductor is subjected.

When a body induces an electromotive force in itself, we call the phenomenon self-induction and say that the body exhibits self-inductance. The induced electromotive force in this case is known as self-induction electromotive force or counter-electromotive force. The other case of inductance is known as mutual inductance and the phenomenon is known as mutual induction. Whenever two conductors are placed close to each other, but without a connection between them, an induced voltage appears in one of them when the current passing through the other is variable.

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