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Integral quantities in electrostatics

Generally the integral quantities of interest in electrostatic analysis are: electric charge, potential difference, mechanical force and torque, field energy.
The following notations are used in formulas:

E - electric field strength;
D - vector of electric flux density (electric displacement);
U - electric potential.

Name,
ActiveField constant

Formula and Description

Electric charge

qfInt_KGrad_n_ds

q = _s∮ (D·n)ds

According to the Gauss theorem, total electric charge in a particular volume can be calculated as a flux of electric displacement over its closed boundary.

Mechanical force

qfInt_MaxwellForce

F = 1/2·_s∮( E·(n·D) + D·(n·E) - n·(E·D) )ds

Total electric force acting on bodies contained in a particular volume. The integral is evaluated over the volume's boundary.

Mechanical torque

qfInt_MaxwellTorque

T = 1/2·_s∮( [r×E]·(n·D) + [r×D]·(n·E) - [r×n]·(E·D) )ds

Total torque of electric forces acting on bodies contained in a particular volume.
The torque vector is parallel to z-axis in the planar case, and is identically equal to zero in the axisymmetric one. The torque is considered relative to the origin of the coordinate system. The torque relative to any other arbitrary point can be obtained by adding extra term of [F×r₀], where F is the total force and r₀ is the radius vector of the point.

Stored energy

qfInt_ElectrostaticEnergy

W = 1/2·_v∫ (E·D)dv

Electric field energy in a particular volume.

Surface energy

qfInt_GradKGrad_n_ds

W_S = 1/2·_s∫ (E·D)ds

Potential difference

qfInt_Grad_t_dl

ΔU = _L∫ (E·t)dl

The potential difference between the ending and started points of a contour can be calculated as a line integral over the contour of electric field strength.

Average surface potential

qfInt_Potential_ds

U_s = 1/S·_s∫ U·ds

Average volume potential

qfInt_Potential_dv

U_v = 1/V·_v∫ U·dv

Average volume strength

qfInt_Grad_dv

E_a = 1/V·_v∫ E·dv

Average electric filed strength in a particular volume.

Average volume displacement

qfInt_KGrad_dv

D_a = 1/V·_v∫ D·dv

Average electric displacement vector in a particular volume.

Mean square strength

qfInt_Grad2_dv

E_a² = 1/V·_v∫ E²·dv

Mean square displacement

qfInt_KGrad2_dv

D_a² = 1/V·_v∫ D²·dv

Line integral of displacement

qfInt_KGrad_t_dl

x = _L∫ (D·t)dl

Surface integral of strength

qfInt_Grad_n_ds

x = _s∫ (E·n)ds