ELECTRIC POTENTIAL: FORMULA AND EQUATIONS, CALCULATION, EXAMPLES, EXERCISES - PHYSICAL - 2025

The electrical potential is defined at any point where there electric field, as potential energy of said field charging unit. Point charges and point or continuous charge distributions produce an electric field and therefore have an associated potential.

In the International System of Units (SI), the electric potential is measured in volts (V) and is denoted as V. Mathematically it is expressed as:

Figure 1. Auxiliary cables connected to a battery. Source: Pixabay.

Where U is the potential energy associated with the charge or distribution and q _o is a positive test charge. Since U is a scalar, so is the potential.

From the definition, 1 volt is simply 1 Joule / Coulomb (J / C), where Joule is the SI unit for energy and Coulomb (C) is the unit for electric charge.

Suppose a point charge q. We can check the nature of the field that this charge produces by using a small positive test charge, called q _o, used as a probe.

The work W necessary to move this small charge from point a to point b is the negative of the potential energy difference ΔU between these points:

Dividing everything by q _or:

Here V _b is the potential at point b and V _a is that at point a. The potential difference V _a - V _b is the potential of with respect to b and is called V _ab. The order of the subscripts is important, if it were changed then it would represent the potential of b with respect to a.

Electric potential difference

From the foregoing it follows that:

Thus:

Now, the work is calculated as the integral of the scalar product between the electric force F between q and q _o and the displacement vector d ℓ between points a and b. Since the electric field is force per unit charge:

E = F / q _or

The work to carry the test load from a to b is:

This equation offers the way to directly calculate the potential difference if the electric field of the charge or the distribution that produces it is previously known.

And it is also noted that the potential difference is a scalar quantity, unlike the electric field, which is a vector.

Signs and values ​​for the potential difference

From the previous definition we observe that if E and d ℓ are perpendicular, the potential difference ΔV is zero. This does not mean that the potential at such points is zero, but simply that V _a = V _b, that is, the potential is constant.

The lines and surfaces where this happens are called equipotential. For example, the equipotential lines of the field of a point charge are circumferences concentric to the charge. And the equipotential surfaces are concentric spheres.

If the potential is produced by a positive charge, whose electric field consists of radial lines projecting the charge, as we move away from the field, the potential will become less and less. Since the test charge q _o is positive, it feels less electrostatic repulsion the further away it is from q.

Figure 2. Electric field produced by a positive point charge and its equipotential lines (in red): source: Wikimedia Commons. HyperPhysics / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0).

On the contrary, if the charge q is negative, the test charge q _o (positive) will be at a lower potential as it gets closer to q.

How to calculate the electric potential?

The integral given above serves to find the potential difference, and therefore the potential at a given point b, if the reference potential at another point a is known.

For example, there is the case of a point charge q, whose electric field vector at a point located at a distance r from the charge is:

Where k is the electrostatic constant whose value in International System units is:

k = 9 x 10 ⁹ Nm ² / C ².

And the vector r is the unit vector along the line that joins q with point P.

It is substituted in the definition of ΔV:

Choosing that point b is at a distance r from the charge and that when a → ∞ the potential is worth 0, then V _a = 0 and the previous equation is as:

V = kq / r

Choosing V _a = 0 when a → ∞ makes sense, since at a point very far from the load, it is difficult to perceive that it exists.

Electrical potential for discrete charge distributions

When there are many point charges distributed in a region, the electric potential that they produce at any point P in space is calculated, adding the individual potentials that each one produces. So:

V = V ₁ + V ₂ + V ₃ +… VN = ∑ V _i

The summation extends from i = to N and the potential of each charge is calculated using the equation given in the previous section.

Electric potential in continuous load distributions

Starting from the potential of a point charge, we can find the potential produced by a charged object, with a measurable size, at any point P.

To do this, the body is divided into many small infinitesimal charges dq. Each contributes to the full potential with an infinitesimal dV.

Figure 3. Scheme to find the electric potential of a continuous distribution at point P. Source: Serway, R. Physics for Sciences and Engineering.

Then all these contributions are added through an integral and thus the total potential is obtained:

Examples of electric potential

There is electrical potential in various devices thanks to which it is possible to obtain electrical energy, for example batteries, car batteries and sockets. Electric potentials are also established in nature during electrical storms.

Batteries and batteries

In cells and batteries, electrical energy is stored through chemical reactions inside them. These occur when the circuit closes, allowing direct current to flow and a light bulb to light, or the car's starter motor to operate.

There are different voltages: 1.5 V, 3 V, 9 V and 12 V are the most common.

Outlet

Appliances and appliances that run on commercial AC electricity are connected to a recessed wall outlet. Depending on the location, the voltage can be 120 V or 240 V.

Figure 4. In the wall socket there is a potential difference. Source: Pixabay.

Voltage between charged clouds and the ground

It is the one that occurs during electrical storms, due to the movement of electrical charge through the atmosphere. It can be of the order of 10 ⁸ V.

Figure 5. Electrical storm. Source: Wikimedia Commons. Sebastien D'ARCO, animation by Koba-chan / CC BY-SA (https://creativecommons.org/licenses/by-sa/2.5)

Van Der Graff generator

Thanks to a rubber conveyor belt, frictional charge is produced, which accumulates on a conductive sphere placed on top of an insulating cylinder. This generates a potential difference that can be several million volts.

Figure 6. Van der Graff generator in the Electricity Theater of the Boston Science Museum. Source: Wikimedia. Boston Museum of Science / CC BY-SA (https://creativecommons.org/licenses/by-sa/3.0) Commons.

Electrocardiogram and electroencephalogram

In the heart there are specialized cells that polarize and depolarize, causing potential differences. These can be measured as a function of time using an electrocardiogram.

This simple test is carried out by placing electrodes on the person's chest, capable of measuring small signals.

As they are very low voltages, you have to amplify them conveniently, and then record them on a paper tape or watch them through the computer. The doctor analyzes the pulses for abnormalities and thus detects heart problems.

Figure 7. Printed electrocardiogram. Source: Pxfuel.

The electrical activity of the brain can also be recorded with a similar procedure, called an electroencephalogram.

Exercise resolved

A charge Q = - 50.0 nC is located 0.30 m from point A and 0.50 m from point B, as shown in the following figure. Answer the following questions:

a) What is the potential in A produced by this charge?

b) And what is the potential at B?

c) If a charge q moves from A to B, what is the potential difference through which it moves?

d) According to the previous answer, does its potential increase or decrease?

e) If q = - 1.0 nC, what is the change in its electrostatic potential energy as it moves from A to B?

f) How much work does the electric field produced by Q do as the test charge moves from A to B?

Figure 8. Scheme for the resolved exercise. Source: Giambattista, A. Physics.

Solution to

Q is a point charge, therefore its electric potential in A is calculated by:

V _A = kQ / r _A = 9 x 10 ⁹ x (-50 x 10 ^-9) / 0.3 V = -1500 V

Solution b

Likewise

V _B = kQ / r _B = 9 x 10 ⁹ x (-50 x 10 ^-9) / 0.5 V = -900 V

Solution c

ΔV = V _b - V _a = -900 - (-1500) V = + 600 V

Solution d

If the charge q is positive, its potential increases, but if it is negative, its potential decreases.

Solution e

The negative sign in ΔU indicates that the potential energy in B is less than that of A.

Solution f

Since W = -ΔU the field does +6.0 x 10 ^-7 J of work.

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 5. Electrostatics. Edited by Douglas Figueroa (USB).
2. Giambattista, A. 2010. Physics. 2nd. Ed. McGraw Hill.
3. Resnick, R. (1999). Physical. Vol. 2. 3rd Ed. In Spanish. Compañía Editorial Continental SA de CV
4. Tipler, P. (2006) Physics for Science and Technology. 5th Ed. Volume 2. Editorial Reverté.
5. Serway, R. Physics for Science and Engineering. Volume 2. 7th. Ed. Cengage Learning.