Electromagnetic Fields and WavesYear 1

Energy of a capacitor and an electric field

capacitor_energy

Elementary work of external forces to move charge dq in electric field of a capacitor

dA=dq*(φ1φ2)=dqqC

Total work is

A=0QdqqC=Q22C

this work determines total energy stored in a capacitor, Q is a total capacitor charge.

Q=C(φ1φ2)

and energy of a charged capacitor

W=C(φ1φ2)22

Let’s express these characteristics through the electric field parameters. For flat capacitors

φ1φ1=Ed

and

C=εε0Sd

then

W=εε0Sd (Ed)22=εε0E22Sd=εε0E22V

V – is is the volume between capacitor plates. Therefore, volume energy density can be found for a capacitor

we=WV=εε0E22

Volume energy density has local characteristics, and it corresponds to the piece of a capacitor where the electric field is uniform and equal to E. Let’s consider the term of volume energy density, on the example of non-uniform electric field. Take a piece of space with volume dV, that characterises radius-vector r. Volume density of energy in general is the value expressed by the formula

we=dWedV

Figure 22 dWis the energy of the small piece of electric field. So if we know the electric field E(r), we can calculate the energy of any piece of field with finite dimensions Ω. Therefore,

we=εε0E2(r¯)2,We=ΩwedV=Ωεε0E2(r¯)2dV

These formulas work for the uniform capacitor material with permittivity ε=const.

Figure 22. Piece of space with calculated energy
Figure 22. Piece of space with calculated energy
Electric field in dielectrics. Electric dipole
Dipoles play an important role in describing dielectrics in the electric field. Electric dipole is a system of two equal charges q, with opposite signs, placed on the distance l. A Dipole moment is the main dipole characteristic.

p¯=q*l̅, vector I¯

from negative to positive charge – called the dipole shoulder. Figure 23. For example, molecules

H2O,HCl,NH3

are dipoles. Let’s find out how dipoles behave in different kinds of electric fields.

Figure 23. Schematic picture of a dipole
Figure 23. Schematic picture of a dipole
Uniform field
Electric field is constant in any point of space, forces affecting the charges +q and –q, are equal with an opposite sign. Resulting force is 0. The dipole of these forces is not 0, if the dipole is not oriented parallel to the electric field lines. Figure 24. Force momentum, according to the axis normal to the figure plane: N=F*l*sinα=q*E*l*sinα=p*E*sinα,

N¯=[p¯*E¯]

This means a uniform field turns the dipole along the field lines.

Figure 24. Dipole in the uniform electric field
Figure 24. Dipole in the uniform electric field
Non-uniform field
In non-uniform fields, the dipole is affected by turning the moment and electric force. Assuming that field changes only in one direction, Figure 25. Projection of force on the axis x is

Fx=Fx+Fx+=qEx+qEx+=q*dE=q*E/xdx,dx=l*cosα

Then

Fx=q*Ex*l*cosα=q*l*Ex*cosα=p*Ex*cosα

Dipoles are oriented along the field. Fhas a positive or negative sign depending on the electric field, and if it is increasing or decreasing along the 0X axis. Dipoles are drawn into the stronger part of field. Electric field increment is

dE¯=(Exdx)ex¯+(Eydy)ey¯+(Ezdz)ez¯

then the force affecting the dipole is

F¯=(Expx)ex¯+(Eypy)ey¯+(Ezpz)ez¯

The conclusion is that dipoles are orienting along the electric field lines and drawing into the electric field with a bigger intensity.

Figure 25. Dipole in non-uniform field
Figure 25. Dipole in non-uniform field

Dipole energy in the electric field

W=qφ+qφ.φ+φ=φxdx=φxl*cosα,Ex=gradφ

then

W=p*E*cosα=(p ,E )

Force of electric field

F¯=gradW,Fx=Wx=pExcosα

#7 Dielectrics

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