Magnetic field energy density and magnetic field strength, calculation formulas

Any electric current is surrounded by a magnetic field. A nontrivial question is the localization of the current's own energy - is it in the conductor where the charges move, or in the magnetic field (the substance that surrounds the current)?

The answer to the question asked is obtained by studying alternating magnetic fields or electromagnetic waves. In an electromagnetic wave, magnetic fields, variable in space and time, can exist in the absence of currents. We know that electromagnetic waves carry energy, therefore we can conclude that energy is localized in a magnetic field.

Volumetric magnetic energy density

The formula for finding the volumetric energy density is as follows:

ω=W/V.

By ω here we mean the actual density being sought, by W the energy of the existing field, and V the volume of space in which the field is active. If we express the value of W in terms of magnetic permeability µ and induction B and substitute it into the formula, it will take the following form:

ω=B2/2* µ0* µ (here µ0 is the magnetic constant).

Conversion using the induction vector is used to eliminate the binding of the active magnetic field to the characteristics of the inductor. The formula for calculating the induction characteristic looks like this:

B= µ0* µ*I*n.

I here is the current force in the coil circuit; n is used to express a value such as the density of the winding. It is equal to the quotient of the number of turns in the solenoid winding and the length of the fragment on which the turns are located. Then the formula for W :

W= B2*V/2* µ0* µ.

By substituting the expression into the basic density formula, you can bring it to the previously designated form.

Flat capacitor.

So, the simplest capacitor consists of two flat conducting plates located parallel to each other and separated by a dielectric layer. Moreover, the distance between the plates should be much smaller than, in fact, the dimensions of the plates:

Such a device is called a flat capacitor , and the plates are called capacitor plates . It is worth clarifying that here we are considering an already charged capacitor (we will study the charging process itself a little later), that is, a certain charge is concentrated on the plates. Moreover, the greatest interest is in the case when the charges of the capacitor plates are identical in magnitude and opposite in sign (as in the figure).

And since the charge is concentrated on the plates, an electric field arises between them. The field of a flat capacitor is mainly concentrated between the plates, however, an electric field also arises in the surrounding space, which is called a leakage field. Very often its influence in tasks is neglected, but it should not be forgotten.

To determine the magnitude of this field, consider another schematic representation of a flat-plate capacitor:

Each of the capacitor plates individually creates an electric field:

• a positively charged plate (+q) creates a field whose strength is E_{+}
• a negatively charged plate (-q) creates a field whose strength is E_{-}

The expression for the field strength of a uniformly charged plate is as follows:

E_{pl} = \frac{\sigma}{2\varepsilon_0\thinspace\varepsilon}

Here \sigma is the surface charge density: \sigma = \frac{q}{S}, and \varepsilon is the dielectric constant of the dielectric located between the capacitor plates. Since the area of ​​the capacitor plates is the same, as is the magnitude of the charge, then the electric field strength modules are equal to each other:

E_+ = E_- = \frac{q}{2\varepsilon_0\thinspace\varepsilon S}

But the directions of the vectors are different - inside the capacitor the vectors are directed in one direction, and outside - in the opposite direction. Thus, inside the plates the resulting field is defined as follows:

E = E_+ + E_- = \frac{q}{2\varepsilon_0\thinspace\varepsilon S} + \frac{q}{2\varepsilon_0\thinspace\varepsilon S} = \frac{q}{\varepsilon_0\thinspace \varepsilon S}

Accordingly, outside the capacitor (to the left and right of the plates) the fields of the plates compensate each other and the resulting voltage is 0.

The presence of a magnetic field around a conductor or coil carrying current

When a solenoid (coil) is connected to an electrical circuit, a field is formed around it. The characteristics of the field depend on a number of parameters: on the environmental features of the environment, current strength (measured in amperes) and the material from which the conductor or coil winding is made. Electromagnetic waves can be generated in field space. Since the field energy potential is, first of all, influenced by the strength of the electric current flowing in the system, we can conclude that the work of the current to generate magnetic space will be equivalent to the energy of the latter. If a coil with a magnetic core is connected to the system, then the energy density will be affected by the field energy in the vacuum and in the material from which the core element is made.

Connecting an inductive coil to a current source

What is the source of the magnetic field

To study the dynamics of the phenomenon, you can consider an electrical circuit that includes a choke, a lamp, a closing key and a source of constant electric current. When the switch is closed, the current path will go from the “positive” terminal of the source through the lamp and inductive coil. At first, the incandescent lamp will light up brighter, which is due to the significant value of the inductor resistance. As the resistance drops and the current passing through the winding increases, the intensity of the light bulb will decrease. This is due to the fact that at first the current supplied to the inductor has a value proportional to the high-frequency current.

In order to practically build a circuit suitable for calculation, it is necessary that the energy resource of the power source be spent on generating a magnetic field. Therefore, it is permissible to neglect the parameters of the internal resistance of the inductor and the feed source.

Important! According to Kirchhoff's second law, the sum of the voltages connected to an electrical circuit equals the sum of the voltage drops for all components of the circuit.

Kirchhoff's second law

Notes[ | ]

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Measuring the energy density of magnetic fields

Magnetic field induction

This value shows the energy contained in a unit volume of the environment subject to the influence of the field. It is denoted by the Greek letter ω. The formula is used for calculation:

ω=W/V, in this case W is the field energy in the volume of space V.

The unit of measurement of field density in the international SI system also looks like a quotient of the units in which these quantities are measured: joules and cubic meters (J/m3). The indicator for batteries (ionic, lead-acid and others) is indicated in the attached documentation.

For a solenoid connected to an electrical circuit, both components of this quotient can be expressed in terms of the following units:

1. The value of the field energy resource will be equal to half the product of the solenoid inductance and the square of the current force in its winding:

W=L*I2/2.

1. The coil itself is considered as the “space”, then V=S*l, where S is the cross-sectional area of ​​the coil element in diameter, and l is its length.

Then the final formula takes the following form:

ω=L*I2/2*S*l.

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Coil inductive reactance formula

Electric field energy

You can calculate the resistance value of the XL throttle using the following formula:

XL=2πfL.

Here the letter L denotes the inductance parameter of the inductor, and f is the current frequency. Based on this expression, at first the current entering the winding will be proportional to the electric current of high purity. At this time, the inductor exhibits behavior similar to a circuit break situation, with a strong increase in inductive reactance. Over time, the latter drops to zero.

The thread built into the lamp has a high resistance value, while the active value of the winding, on the contrary, tends to zero. Because of this, a situation arises where almost all of the circuit current passes through the inductor. When the circuit is opened with a key, the lamp does not go out gradually. On the contrary, it first sharply begins to burn intensely, then slowly fades away. In order for the lamp to burn, an energy resource is required. It comes from the magnetic field generated by the inductive coil. Thus, the choke manifests itself as a source of self-induction.

In the example considered, a coil with windings connected to a circuit acts as a source of a magnetic field. Since in such a situation this field is not uniform, to perform calculations it is necessary to use an indicator characterizing the concentration and distribution of energy in the field. We can conclude that this is precisely the meaning of introducing the field density parameter.