The electric potential and the electric field are closely related and mathematically represented differently, and it is possible to calculate the electric potential from the electric field and vice versa.

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The potential energy per charge unit associated with an electric field has a unique value at each point in space and is represented by the letter V according the eq. 72-01.

Thus, we can define the electric potential difference ΔV between two points i and f as the difference between the electric potentials of the two points as shown in the eq. 72-02.

So we can define potential energy in terms of the work done by a force F^→ over the load q at its offset from the starting position i to its ending position f. It is mathematically possible to express this as eq. 72-03.

From the previous item, we know the relation between the force exerted on the charge q by the electric field, that is, F = q E (eq. 71-02). Using this relation with eq. 72-01 and eq. 72-03, we find that the charge q will be canceled, resulting that the potential difference between two given points is:

Integration takes place along the path that q takes when moving from the point i to point f. Since the force exerted on q is conservative, this line integral does not depend on the integration path.

As with potential energy, only differences in electrical potential are significant. In general, we set the electrical potential value to zero at some point convenient in an electric field.

The potential difference should not be confused with that of potential energy . The potential difference between i and f exists only because of a charge source and depends on the distribution of that charge source. For potential energy to exist, we must have a system of two or more charges. The potential energy belongs to the system, and changes only if a charge moves in relation to the rest of the system. Since potential energy is an amount scalar, so is the electrical potential.

Since electrical potential is a measure of potential energy per unit of charge, the SI unit electrical potential and potential difference is the coulomb by joule, defined as volt:

In other words, it means that 1 joule of work must be performed so that a charge of 1 coulomb is displaced by a potential difference of 1 volt.

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The equations eq. 72-02 and eq. 72-04 are valid for all electrical fields, be they uniform or variable. When working with uniform electrical fields, we can simplify the eq. 72-04, eliminating the integral. So, let's suppose that we want to calculate the potential difference between the points i and f, separated by a distance d, where the displacement d s^→ is pointing from i to f and is parallel to the lines of the electric field. Note that as the field is uniform, it can be removed from within the integral in eq. 72-04. Thus, it remains to integrate the path d s^→, that as we know is the distance d between the points i and f. So, when the electric field is uniform, we can write:

The negative sign indicates that the electrical potential at point f is less than at point i. Thus, we can write V_f < V_i . So, we see that the lines of the electric field always point towards the decreasing electrical potential.

Assuming that a charge moves from point i to point f, we can calculate the variation of the potential energy of the charge-field system through eq. 72-07.

This equation shows that if the charge q is positive, then ΔU is negative. Like this, in a system composed of a positive charge and an electric field, the electrical potential energy of the system decreases when the load moves in the same sense as the electric field. In this way, we can say that an electric field applies work to a positive charge when it moves in the same sense of the electric field.

In the case of a negative charge, if it is released from rest in the presence of an electric field, it will accelerate in the opposite sense to that of the electric field. For the negative charge to move in the same sense as the electric field, an external agent must apply a force and perform a positive work on the charge.

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The expression equipotential surface is used to refer to any surface that consists of a continuous distribution of points with the same electrical potential. Thus, the equipotential surfaces of an electric field form a family of parallel planes that are all perpendicular to the electric field.

Suppose we have two parallel plates 1 meter apart and are connected to a 100 volt voltage source. Therefore, the electric field between the plates will have a magnitude of 100 volts/meter. With this we can say that a plane parallel to the negative plate, whose distance is 10 cm from the plate, has a potential of 10 volts in relation to it. If the distance is 50 cm , then the potential is 50 volts . That is, these planes form an equipotential surface, since as long as the fixed distance is maintained, the potential will be the same regardless of the lateral position of the measurement point.

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The effect of a charge q on the space around it can be described in two ways. The charge establishes the electric field vector, E^→, which is related to the force applied to a test charge placed in the field. The charge also creates a scalar potential V , which is related to the potential energy of the two-charge system, when a charge test is placed in the field. To find the potential difference between two points in space, caused by a charge q, we can use the eq. 72-04. Suppose we have two points, called i and f, far from the charge q by r_i and r_f. Substituting in eq. 72-04 the value of the electric field given by eq. 71-05 and performing the integral between the mentioned points, we will find:

It is evident by the eq. 72-08 that the value of the integral is independent of the path between the points i and f. In addition, this equation also informs that the electric field of a point charge fixed is conservative. Another information we get is that the potential difference between any two points i and f in a field created by a point charge depends only on the radial coordinates r_i and r_f. One way to simplify the eq. 72-08 , is consider the potential in r_i = ∞ as V = 0. So, we established a reference electrical potential for a point charge. By choosing this reference, the potential established by a point charge at any distance r from the charge can be expressed as:

Where in this equation we are using the definition made in chapter 71 of the constant K, given by eq. 71-03, reproduced below.

If we have more than one charge, we can find the potential resulting from the configuration using the principle of superposition. Thus, for a group of point charges, we can express the total electrical potential at a given point P, through eq. 72-10.

Where, in this case, once again we consider the potential equal to zero at infinity and, r_i represents the distance from point P to the charge q_i. We must note that the sum indicated in eq. 72-10 represents an algebraic sum of scalar values. That is, it is not a vectorial type.

Let's look at the potential energy, U, of a system composed of two charged particles. Considering V₂ as the electrical potential at a point P originating from a charge q₂, the work that must be done by an agent to move a second charge, q₁, from infinite to the point P, without acceleration, is given by q₁ V₂, according to the eq. 72-07. This work represents a transfer of energy into the system, and the energy appears in the system as potential U, when the particles are separated by a distance r₁₂. Thus, we can express the potential energy of the system by eq. 72-11.

If the charges have the same sign, then U is positive. This positive work must be carried out by an external agent on the system in order to place the two charges close to each other, since loads with the same signal repel each other. If the signs are opposite, then U is negative. In this case, the negative work done by an external agent is against the force of attraction between the two particles of opposite signs, when they are placed close to one of the another. That is, this force must have felt opposite to the displacement to prevent q₁ accelerate towards q₂.

If the system consists of more than two charged particles, we can obtain the total potential energy of the U system for each pair of charges, adding the terms algebraically. This means that the potential energy of a point charge system it is equal to the work required to bring the charges, one at a time, from an infinite separation to their final positions.

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The electric field, E^→, and the electric potential, V, are related according to eq. 72-04, reproduced below.

This equation shows that the value of ΔV can be calculated if the electric field is known. Starting from this equation, we will consider the potential difference dV between two points separated by an infinitesimal distance ds. So, we can write that

Considering that the electric field may have, in Cartesian coordinates, only the component x, it is possible to write:

This equation shows the mathematical formulation of the electric field as a measure of the ratio of variation of the electrical potential in relation to its position. Naturally, the same statement can components y and z will be applied. If we equate eq. 72-04 to zero, this means that the electric field must be perpendicular to the displacement along the surface equipotential. Thus, this demonstrates that equipotential surfaces must always be perpendicular to the lines of the electric field that cross them. Therefore, we can say that the equipotential surfaces associated with a uniform electric field consist of a family of plans perpendicular to the field lines.

On the other hand, if the charge distribution that creates an electric field has spherical symmetry so that the volumetric charge density depends only on the radial distance r, the electric field will be radial. In that case, we can express dV as dV = - E_r . dr and, then

Since V is a function of r only, the potential function, in this case, has a spherical symmetry. Note that the potential varies only in the radial position, not in any direction perpendicular to r, and this justifies the idea that equipotential surfaces are perpendicular to the field lines. Thus, equipotential surfaces are a family of concentric spheres with the distribution of spherically symmetrical charge.

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As we know, an electrical dipole consists of two charges of the same module and opposite signals, separated by a certain distance that we will call 2a. A Figure 72-01 shows a dipole positioned along the x axis and centered at the origin. Initially, we will calculate the electrical potential at point P, which is positioned on the axis y.

As we have two charges separated by a distance 2a, to find the electrical potential we can employ the principle of superposition, using eq. 72-10. In this way, both the positive and the negative charge, differ from the point P the value given by the theorem of Pythagoras , that is, d = (a² + y² ) ^(1/2). So, we find:

Note that the potential produced by the positive charge at point P is completely canceled out by the potential produced by the negative charge. So, we get

Now we will find the potential at the point R, located on the x axis and at a distance x from the source. In this case, we obtain the eq. 72-17 when using the superposition principle.

In eq. 72-17, when calculating the m.m.c in the denominator we obtain x² - a² and in the numerator the variable x is canceled, because q (x-a) + (-q)(x+a) = - 2 q a. Then the eq. 72-18 shows the resulting potential at the point R.

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To determine the potential generated by a continuous load distribution, we continue to use the principle of superposition, according to eq. 72-10 . We must consider that the object is evenly charged, that is, the charges are equally spaced object. If the distribution of charges is known, we can consider the potential generated by a small charge element dq, treating it as a point charge. Thus, according to eq. 72-09 , the electrical potential dV at a given point P established by the charge element dq is

Where r is the distance from the charge element to point P. So, to obtain the full potential at point P, we must integrate the eq. 72-19, as this way we include the contribution of all elements of the load distribution. Then, the eq. 72-20 calculates the electrical potential produced by a charge distribution.

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Suppose an electrically charged ring of uniform shape and radius R, as shown in Figure 72-02. The charge distribution being represented by λ and knowing that the length of the ring is given by 2 π R , then the total charge is given by Q = 2 π R λ. In the adopted coordinate system, the ring is contained in the x y plane and the point P is on the z axis.

Dividing the ring into several segments and choosing a i segment, we can determine the distance r_i of this segment to the point P using the Pythagorean theorem. That way, we have to r_i = √ (R² + z²). So, considering a charge element dq_i and eq. 72-19, we can find the potential generated by the charged ring by integrating this equation. Soon

Note that for the point P, both R and z are constant, so we must integrate the load dq_i, obtaining the total load, Q, of the ring. Therefore, the potential in point P is given by eq. 72-22, or

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Let's consider a uniformly loaded disk with radius R and surface charge density σ. The disk is in the x y plane and we want to calculate the electrical potential at a point P located on the z axis. Let's take an element dq at a distance r from center of the disc. Using polar coordinates we can write that dq = σ r dr dθ. So, using Pythagoras, we can calculate the distance d from the dq element to the point P. With that, we get d = (r² + z²)^1/2.

Using this data and doing a double integration, one in θ and the other in r, we will obtain the potential at point P. Note that 0 ≤ θ ≤ 2 π e 0 ≤ r ≤ R. In this way, we can write:

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We know that a solid conductor, which has a net charge in equilibrium, the charge is located on the outer surface of the conductor. In addition, the electric field outside, close to the conductor, is perpendicular to the surface, and the electric field inside is null.

At this point, we are going to study another property of an electrically charged conductor in relation to the electrical potential. Let's consider two points, X and Y, on the surface of that conductor. Considering a path along the conductor's surface between these two points, the electric field is always perpendicular to the surface and therefore forms an angle of 90° with the displacement vector ds^→. In this case, we have E^→. ds^→ = 0 .This means, according to eq. 72-04, that the potential difference between these two points is null. This result applies to any two points on the conductor's surface. Thus, the potential V is a constant at all points on the surface of a balanced conductor. Therefore, it can be concluded that:

Let's consider a solid conducting metal sphere, with radius R and positive charge Q, as shown by Figura72-04.

We know that the electric field outside the sphere is given by E = K Q / r² and points radially out of the sphere. Since the field outside a spherically symmetrical charge distribution is identical to that of a punctual charge, we should expect that the electrical potential is also identical to that of a punctual charge, or V = KQ / r.

As we have seen, within a conductor the potential is the same as that of the surface. This is what the graph shows (in red) of the figure to the side, where we have a constant value in all the conductor. When we move away from the conductor, the potential decreases in inverse distance.

Due to the fact that the potential is constant inside the conductor, using eq. 72-14 , we conclude that the electric field inside the conductor must be NULL, as shown in the graph (in blue) in the Figura 72-04.

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Whenever a spherical electrical conductor has a net charge, the surface charge density it will be uniform. However, if the conductor is not spherical, the load density will be high where the radius of curvature of the conductor is small and low in places where the radius of curvature is large. In other words, the charge density is not uniform. And as we know, the electric field immediately outside the conductor is proportional to the density of the surface charge and, therefore, the electric field is large close to convex points with a small radius of curvature and reaches very expressive values in thin points.

In Figure 72-05 we see a photograph of a lightning rod, a device that uses the properties of fine-tipped conductors. Observe the conductor that comes out from under the arrester and will be used for the connection between the arrester and the ground. As the surge arrester is installed in the most the top of a building, there must be an electrical cable connecting these two points.

In Figure 72-06, we can see the efficiency of a lightning rod installed on top of a building.