Electromagnetic energy formula, equations, uses, examples

The electromagnetic energy It is the one that propagates through electromagnetic waves (EM). Examples of this are the sunlight that radiates heat, the current that is extracted from the electrical outlet and that which X-rays possess 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..

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 oldest known and primordial source of electromagnetic energy, but using electricity is somewhat more recent.

It was only in 1891 when 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 regarding their use..

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

Article index

• 1 Formula and equations
  • 1.1 Instantaneous power per unit area
• 2 Uses
  • 2.1 Dipole antenna
  • 2.2 Electric field energy: the capacitor
• 3 Examples
  • 3.1 Example 1: Intensity of an electromagnetic wave
  • 3.2 Example 2: Application to a transmitting antenna
• 4 References

Formula and equations

Electromagnetic waves are transverse waves, in which the electric field AND and the magnetic field B are perpendicular to each other, the direction of propagation of the wave being 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 AND in blue, oscillates in the plane zy, magnetic field B in red it does it on the plane xy, while the velocity of the wave is directed along the axis +Y, according to the coordinate system shown.

If a surface is interposed in the path of both waves, say a plane of area TO 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 poynting vector:

S = (1 / μ_or) AND × B

μ_or is the permeability of the vacuum (μ_or = 4π .10^-7 Tesla. meter / ampere), a constant related to the ease that the medium gives the electromagnetic wave to move.

The Poynting vector was introduced by the English astrophysicist John Henry Poynting in 1884, a pioneer in the study of the energy of electric and magnetic fields..

Instantaneous power per unit area

Now, it must be taken into account that energy is a scalar, while S it's a vector.

Remembering that power is the energy delivered per unit of time, then the modulus of S indicates the instantaneous power per unit area in the direction of electromagnetic wave propagation (energy transfer rate).

Since AND Y B are perpendicular to each other, the modulus of AND x B it's just EB and the instantaneous power (a scalar) is as:

S = (1 / μ_or) EB

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

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

E = cB

Substituting this relation in S, we obtain:

S = (1 / μ_or.EC^two

The Poynting vector varies with time in a sinusoidal way, so the previous expression is its maximum value, because the energy delivered by the electromagnetic wave also oscillates, as do the fields. Of course, the frequency of the oscillation is very large, that is why 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 or microwave waves, 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 AND when the + load is up (left figure). AND points down at the point shown.

In figure 3 right, the dipole changed position and now AND points up. Let's repeat this change many times and very quickly, let's say with a frequency F. This creates a field AND variable in time gives rise to a magnetic field B, also variable and whose shape is sinusoidal (see figure 4 and example 1 below).

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

I feel that B points in or out of the screen alternately (always perpendicular to AND).

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 the two 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 capacitance 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^two/ C) = ½ QV = ½ CV^two

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.EC^two.

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 it is calculated in this way:

I = S_half = S = (1 / μ_or.EC^two_half

An electromagnetic wave is represented by a sine function:

E = E_or sin (kx - ωt)

Where AND_or is the amplitude of the wave, k the wave number and ω the angular frequency. Then:

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 incident on a square sheet with a side of 10 cm in a period of 5 minutes.

The data are:

Speed ​​of light in vacuum: c = 300,000 km / s

Vacuum permeability: μ_or = 4π .10^-7 T.m / A (Tesla. Meter / ampere)

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:

10 kW = 10000 W

100 MHz = 100 x 10⁶ Hz

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

It was previously said that the magnitudes of AND Y B they were related by the speed of light:

E = cB

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

Solution b

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

5 minutes = 300 seconds

Area = (10/100)^two  m^two = 0.01 m^two.

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.