ELECTROMAGNETIC ENERGY: FORMULA, EQUATIONS, USES, EXAMPLES - PHYSICAL - 2025

The electromagnetic energy is one that spreads via electromagnetic waves (EM). Examples of this are the solar light that radiates heat, the current that is extracted from the electrical outlet and that of X-rays to produce X-rays.

Like sound waves when they vibrate the eardrum, electromagnetic waves are capable of transferring energy that can later be converted into heat, electrical currents, or various signals.

Figure 1. Antennas are necessary in telecommunications. The signals they work with have electromagnetic energy. Source: Pixabay.

Electromagnetic energy propagates both in a material medium and in a vacuum, always in the form of a transverse wave and making use of it is not something new. Sunlight is the primordial source of electromagnetic energy and the oldest known, but using electricity is somewhat more recent.

It was only in 1891 that the Edison Company put into operation the first electrical installation in the White House in Washington DC. And that as a complement to the gas-based lights that were used at the time, because at first there was a lot of skepticism about their use.

The truth is that even in the most remote places and lacking power lines, the electromagnetic energy that constantly arrives from space continues to maintain the dynamics of what we call our home in the universe.

Formula and equations

Electromagnetic waves are transverse waves, in which the electric field E and the magnetic field B are perpendicular to each other, and the direction of propagation of the wave is perpendicular to the fields.

All waves are characterized by their frequency. It is the wide range of frequencies of EM waves, which gives them versatility when transforming their energy, which is proportional to the frequency.

Figure 2 shows an electromagnetic wave, in it the electric field E in blue oscillates in the zy plane, the magnetic field B in red does so in the xy plane, while the speed of the wave is directed along the axis + y, according to the coordinate system shown.

Figure 2. An electromagnetic wave incident on a surface delivers energy according to the Poynting vector. Source: F. Zapata.

If a surface is interposed in the path of both waves, say a plane of area A and thickness dy, such that it is perpendicular to the speed of the wave, the flux of electromagnetic energy per unit area, denoted S, is described through from the Poynting vector:

It is easy to check that the units of S are Watt / m ² in the International System.

There's still more. The magnitudes of the E and B fields are related to each other by the speed of light c. In fact, electromagnetic waves in a vacuum propagate that fast. This relationship is:

Substituting this relation in S we obtain:

The Poynting vector varies with time in a sinusoidal way, so the above expression is its maximum value, because the energy delivered by the electromagnetic wave also oscillates, just like the fields do. Of course, the frequency of the oscillation is very large, so it is not possible to detect it in visible light, for example.

Applications

Among the many uses that we have already mentioned for electromagnetic energy, here are mentioned two that are used continuously in numerous applications:

Dipole antenna

Antennas are everywhere filling space with electromagnetic waves. There are transmitters, which transform electrical signals into radio waves or microwave, for example. And there are receivers, which do the reverse work: they collect the waves and convert them into electrical signals.

Let's see how to create an electromagnetic signal that propagates in space, from an electric dipole. The dipole consists of two electric charges of equal magnitude and opposite signs, separated by a small distance.

In the following figure is the electric field E when the charge + is above (left figure). E points down at the point shown.

Figure 3. Electric field of a dipole in two different positions. Source: Randall Knight. Physics for Scientists and Engineers.

In figure 3 right, the dipole changed position and now E is pointing up. Let's repeat this change many times and very quickly, say with a frequency f. In this way, a field E variable in time is created, giving rise to a magnetic field B, also variable and whose shape is sinusoidal (see figure 4 and example 1 below).

And as Faraday's law ensures that a time-varying magnetic field B gives rise to an electric field, it turns out that by oscillating the dipole, one already has an electromagnetic field capable of propagating in the medium.

Figure 4. A dipole antenna generates a signal that carries electromagnetic energy. Source: F. Zapata.

Notice that B points in or out of the screen alternately (it is always perpendicular to E).

Electric field energy: the capacitor

Capacitors have the virtue of storing electrical charge and therefore electrical energy. They are part of many devices: motors, radio and television circuits, car lighting systems, and much more.

Capacitors consist of two conductors separated by a small distance. Each one is given a charge of equal magnitude and opposite sign, thus creating an electric field in the space between both conductors. The geometry can vary, being a well known one that of the flat-parallel plate condenser.

The energy stored in a capacitor comes from the work that was done to charge it, which served to create the electric field inside it. By introducing a dielectric material between the plates, the capacity of the capacitor increases and therefore the energy it can store.

A capacitor of capacity C and initially discharged, which is charged by a battery that supplies a voltage V, until reaching a charge Q, stores an energy U given by:

U = ½ (Q ² / C) = ½ QV = ½ CV ²

Figure 5. A flat parallel plate capacitor stores electromagnetic energy. Source: Wikimedia Commons. Geek3.

Examples

Example 1: Intensity of an electromagnetic wave

Previously, it was said that the magnitude of the Poynting vector is equivalent to the power that the wave delivers for each square meter of surface, and that also, as the vector is time dependent, its value oscillated up to a maximum of S = S = (1 / μ _or.c) E ².

The average value of S in one cycle of the wave is easy to measure and indicative of the energy of the wave. This value is known as wave intensity and is calculated in this way:

An electromagnetic wave is represented by a sine function:

Where E _o is the amplitude of the wave, k the wave number and ω the angular frequency. So:

Figure 5. The antenna radiates the signal in a spherical shape. Source: F. Zapata.

Example 2: Application to a transmitting antenna

There is a radio station that transmits a signal of 10 kW of power and a frequency of 100 MHz, which spreads in a spherical way, as in the figure above.

Find: a) the amplitude of the electric and magnetic fields at a point located 1 km from the antenna and b) the total electromagnetic energy that falls on a square sheet of side 10 cm in a period of 5 minutes.

The data are:

Solution to

The equation given in example 1 is used to find the intensity of the electromagnetic wave, but first the values ​​must be expressed in the International System:

These values ​​are immediately substituted in the equation for the intensity, since it is a source that emits the same everywhere (isotropic source):

Earlier it was said that the magnitudes of E and B were related by the speed of light:

B = (0.775 /300.000.000) T = 2.58 x 10 ^-9 T

Solution b

S _means is power per unit area and in turn power is energy per unit time. Multiplying _mean S by the area of ​​the plate and by the exposure time, the requested result is obtained:

U = 0.775 x 300 x 0.01 Joules = 2.325 Joules.

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 6. Electromagnetism. Edited by Douglas Figueroa (USB). 307-314.
2. ICES (International Committee on Electromagnetic Safety). Electromagnetic Energy Facts, and a Qualitative View. Retrieved from: ices-emfsafety.org.
3. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson. 893-896.
4. Portland State University. EM waves transport energy. Retrieved from: pdx.edu
5. What is Electromagnetic Energy and Why is it Important ?. Recovered from: sciencestruck.com.