To begin the study of electromagnetism it is important to study the concept of magnetic field associated with an electric charge.

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Through experiments, it was noted that the charge carriers were influenced by another force, apart from that resulting from the action of the electric field. Such a force depended not only on the position of the particle but also on the speed of its displacement and this force was called Magnetic Force . Thus, at every point in space there are two vector quantities that determine the resultant force acting on a charge: one of them is the electrical force , being independent of the movement of the charge and it is possible to describe it in terms the electric field (as already studied); the other is an additional component called Magnetic Force . When we talk about the magnetic field, we can see some characteristics of our own when we have    moving loads. It can be summarized as follows:

To better understand this behavior we will introduce the variable called vector magnetic induction , also known as magnetic flux density , represented by the letter B^→. Thus, we can define magnetic force by the following equation:

As we have a cross product between two vectors, then we have a third vector that is orthogonal (angle of 90°) to the other two vectors. In Figure 74-01 we see a spatial representation of what this product looks like. Note that the vectors v^→ and B^→ can form an angle different from 90° to each other. However, the angle between them and the magnetic force will always be 90° . To determine the direction of the magnetic force we use the right hand rule , as shown in Figure 74-01 . Note that the fingers of the palm point in the direction of the velocity vector. When closing the hand towards the magnetic field vector, the thumb points in the direction of the magnetic force. Note that this is valid when the particle has a positive charge . For negative charge , simply invert the sense of force.

The eq. 74-01 is known as the vectorial form of the magnetic force. When we are working with scalar quantities we can write this equation as follows:

Where θ represents the angle between the vectors v^→ and B^→ and varies from 0 < θ < π. A conclusion we get from eq 74-02 , is that if an electric charge moves parallel to the magnetic field it does not suffer magnetic force, because the angle between the velocity vector and the magnetic field vector is equal to zero , because as we know sin 0° = 0.

Therefore, we can affirm that the resulting force to which an electric charge is subjected, will be the sum of electrical force with magnetic force . Thus, we can write that:

The eq. 74-03 represents the so-called Lorentz Force , an extremely important equation of electromagnetism.

It should be noted that the unit of measurement of the vector magnetic induction is the Tesla , equivalent to 10 000 gauss or ,Weber/ m².

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    2.1   Particle Charged in a Magnetic Field

If a particle moves along a circumference with constant speed, we can say, with certainty, that there is a constant modulus force acting on the particle and pointing towards the center of the circumference, remaining perpendicular to the particle speed. We can think of a charge moving in a region where there is a magnetic field, as represented in Figure 74-02.

The direction and sense of the magnetic force is perfectly defined by the right hand rule, as explained above.

Naturally, there is an interest in determining the parameters that characterize the movement circular of this particle that has charge |q|, mass m and that moves with velocity v^→ perpendicular to a field magnetic B^→. We are aware of the magnetic force, given by eq 74-02 , and we know that Newton's second law ( F = ma ) applied to motion circular represents centripetal force . In this case, for the particle to remain in a circular motion, the two forces must have the same module. Therefore, we can write the following equality:

Working with this equation, we easily determine the radius of the charge's orbit, or:

It is also possible to calculate the period , T, of the movement, which is the time needed to complete a revolution. Since this time is given by the length of the circumference divided by the speed, this period can be expressed by eq. 74-06 , after algebraic work from the eq. 74-05.

On the other hand, we know that the frequency of the movement, measured in Hertz , is the inverse of the period. So, just invert the eq. 74-06 and we get the frequency. For the case of angular frequency, it can be calculated using the eq. 74-07.

The quantities T , f and ω do not depend on the particle speed, as long as its speed is much less than the speed of light. Thus, fast particles move in large circles, while slow particles move in small circles. It should be point out that all particles that have the same charge-to-mass ratio (|q| / m) complete a revolution at the same time, that is, they have equal periods . Based on Figure 74-02 and looking at the direction of the magnetic field B, the direction of rotation for a particle with a positive charge is the counterclockwise and for a particle with a negative charge , the direction of rotation is clockwise .

As additional information, we can state that when the velocity of a charged particle has a parallel component to a uniform magnetic field the particle describes a helical path , whose axis is the direction of the magnetic field.

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    2.1.1   Helical Trajectories

In the previous item we studied that when a particle enters a region that has a magnetic induction field and its speed is orthogonal to it, it performs a circular motion.

But, what happens if the direction of the velocity vector is not orthogonal to the induction field magnetic?

We can say that when this happens the velocity vector can be decomposed into two components: one that is in the same direction as the magnetic induction field; and another that is orthogonal to it. The effect is that, in addition to the particle making a circular motion, due to the orthogonal component of the speed, v_ort , it also moves in the same direction as the magnetic induction field, due to the parallel component, v_par , generating a helical movement . The Figure 74-02.1 shows this effect.

It should be noted that all the equations studied in the previous item are also valid for helical movement. Here we must add a new variable that is called the helix pitch . We can determine its value by knowing the period, T , of the movement and the parallel component of the speed, v_par . Thus, the equation eq. 74-07a determines the pitch of the propeller.

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    2.2   Magnetic Field Produced by Moving Charges

For this topic, let's start from the idea that:

We must remember that the magnetic field created by a moving charge is added to the field electrical charge. The charge always produces an electric field, whether in motion or not. Here, to calculate the magnetic field created by the moving electric charge we are going to use the Biot-Savart law for point charges.

This law is similar to the law of Coulomb , where both depend on the inverse of the square of the distance to the point where you want to calculate the field. However the law of Biot-Savart is a little more complex because the magnetic field depends on the angle θ between the charge speed and the line that goes to the point where the field is measured, as shown the Figure 74-03.

Thus, the equation that we will use to calculate the magnetic field at point P is shown in eq. 74-08.

Note that the eq. 74-08 introduces a new constant called magnetic permeability in a vacuum and represented by μ_o. The value of this constant is equal to:

Keep in mind that the moving charge produces a magnetic field. If the load is at rest, there will be no magnetic field, but there will be an electric field. We can summarize it as follows:

"Charges create electric fields, but only charges in motion create magnetic fields."

The Figure 74-04 shows a graphical representation of how the magnetic field is positioned in space in relation to the path of a positive charge. For a negative charge, the direction of the magnetic field must be reversed, although the module is the same.

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    2.3   Magnetic Field Produced by Electric Current

We are interested in calculating the magnetic field produced by an electric current, i , near a conductor when it travels through that conductor. We assume a point close to the conductor, called P₁. If the conductor has a length s, we can define and represent an element infinitesimal length of the conductor by ds^→, whose module is ds and whose direction is the direction of the current in the element ds. We can define a current element as i ds^→ and calculate the field dB^→ produced at the point P₁ by a typical current element, as shown in Figure 74-05.

To determine this field we will use the so-called Biot-Savart Law , which is defined according to eq. 74-09 in its vector form.

Therefore, by making the vector product, we obtain:

Therefore, we can find the value of the magnetic field by integrating the eq. 74-08 . Considering that a total current equal to I circulates in the conductor, we obtain:

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2.4   Magnetic Field Produced by the Electric

After studying the magnetic field generated by charges and currents, we will study how a electric current running through a long straight conductive wire produces a magnetic field in its vicinity. To this end, we will base ourselves on the scheme presented in Figure 74-06.

First we’ll select a small segment of the wire and designate it by Δx. As a whole, the wire is electrically neutral, but the current I represents the movement of positive charges along the wire. We want to calculate the magnetic field dB_k^→  at point P, which in turn is r_k of the segment Δx.

Solving in a simple way, we can say that the vector product in eq. 74-09 is the same, in module, the sin θ_k. So we can write:

As we want to make an integration in relation to x, we must find the values of r_k and sin θ in function of x_k. Note that by applying the Pythagorean theorem, we can relate the variables of          interest, that is, r_k² = x_k² + x². On the other hand, we can write:

Thus, making the substitutions in eq. 74-12 , we get:

Now, at all points, your magnetic fields are perfectly defined. To achieve the total result we must add all of them. For this, we will use the integral between the limits less and more infinity , as shown in eq. 74-15.

As a final result of the integration we obtain eq. 74-17 which calculates the field at the point P.

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2.5   Magnetic Field Produced by a Loop

We saw the result of a magnetic field produced by a long straight wire. Now let's see what happens if we bend the wire in the shape of a loop. So, let's approach the loop through a conductive ring and calculate the field produced by a current flowing through the ring counterclockwise, as shown in Figure 74-07.

We use the z axis orthogonal to the plane formed by the loop area. The other two axes cut the plane formed by the loop area, as we see in Figure 74-07 . Let's take an infinitesimal element ds^→ of the loop and analyze the field at point P . So, using the right hand rule, we clearly perceive that the ds element produces a field element dB orthogonal to versor rˆ.

We are interested in calculating the field on the loop axis, which coincides with the z axis . From the figure above we can see that we can decompose the elementary vector dB in two components: one on the y axis, represented by dB_y, and another on the z axis, represented by dB_z. Notice that, by symmetry, the component dB_y does not contribute to the calculation of field at point P, because for each ds element that we consider there will always be another diametrically opposite point that will produce a field of the same module, but in the opposite direction, canceling the contribution from that point. On the other hand, considering the same reasoning above, we found that the component dB_z, for each element ds that we consider, there will be another diametrically opposite point, contributing to the production of the magnetic field at point P. The resulting vector is the sum of the two. Therefore, we can express the result through an integral, or:

For the calculation of the field we will use Biot-Savart's law . This law, already studied in item 2.3 , was presented in eq. 74-11 . From Figure 74-06 we see that dB_z = dB sinθ. So, from trigonometry we know that sinθ = R / r. Furthermore, from trigonometry, we can obtain another relation, given per r² = R² + z² . And in the vector product we have ds^→ x rˆ = ds sen 90° = ds. Substituting these values in eq. 74-11 , we get:

We must pay attention that in this equation the values of I , R and z are constant, then they can get out of the integral. Therefore, we should only integrate ds . The equation looks like shown in eq. 74-20.

Since the integral of ds is the full circle, this means that its result is the circumference length, that is, 2 π R . So, the final equation that determines the field at point P , located on the axis of the loop, is given by eq. 74-21.

This equation allows you to calculate the field anywhere on the z axis . From this equation we can determine the field exactly at the center of the loop, just making z = 0 . With this, we obtain:

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   2.6   Magnetic Field Produced by a Solenoid

As we saw in the previous item, the magnetic field in the center of a loop can be determined by eq. 74-22 . A solenoid, also known in the technical field as coil , can be understood as a series of juxtaposed turns (one next to the other), generating an element that when an electric current circulates through the conductive wire that forms it, establishes a magnetic field. In the Figure 74-08 we can see the schematic of a solenoid. Using the right hand rule, we determine that the flux points to the left of the figure, because the current flows counterclockwise. Where does the flux get out, we call north and where the flow enters we call south (an analogy with the poles of a permanent magnet), as can be seen in the figure below.

A solenoid has a certain number of turns represented by the letter N . So, based on what we studied in the previous item, where the magnetic field in the center of a turn can be determined by eq. 74-22 , we can extend this concept to determine the magnetic field within a solenoid. For that, we realized that the flow will go through N turns, then the equation that determines the field inside of the solenoid can be expressed by

It should be noted that some authors in their books prefer to replace the relationship N/L_S with the letter n. Thus, the letter n represents the number of turns of the solenoid per unit length. Then the eq. 74-23 gives the eq. 74-23a, below.

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As we studied earlier, if the charge distribution has a planar, cylindrical or spherical symmetry, we can use Gauss's law to determine the total electric field, making calculations easier. In the case of the total magnetic field, associated with any current distribution, even if it is complicated, but if there is some kind of symmetry we can use Ampère's law to determine the total magnetic field. Ampère's law is based on a mathematical procedure called line integral. This law was developed by the physicist André-Marie Ampère (1775-1836) and later perfected by the physicist Clerk Maxwell (1831-1879).

In Figure 74-08 we see the representation of a straight conductive wire through which I current flows. At a distance r from the center of the wire we have the magnetic field tangent to the circle. Note that the field lines are circumferences centered on the conductor.

As we already saw in item 2.4, the magnetic field produced by a long straight wire is given by eq. 74-17. We will reproduce it here, adapting it to Figure 74-08, or

As stated earlier, Ampére's law uses a line integral along a path, as shown in Figure 74-08. From the figure we see that the radius is constant around the circumference, so the magnetic field, B, is also. Therefore, we can remove B from the integral and write:

It is a line integral along an entire path until the chosen curve is closed. This path is known as amperian. This means that we must start from a certain point i and return to the same point. This integral is the scalar product between the magnetic field associated with the bounded curve and an infinitesimal element of the integration line. From this scalar product, we can learn two lessons:

Thus, simplifying the eq. 74-24 we find the equation that defines Ampére's law, or

The current signal depends on the direction of integration in relation to the current. The right hand rule indicates whether, in the direction of the line integral adopted, the current is positive or negative. If the integration path does not include a conductor, then the result of the line integral will be null.