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Quadrupole charge

# Quadrupole charge

Charges are placed in the vertices of the 1x1 [m] square.

**Problem Type:** electrostatics.

**Geometry:** 3D extrusion.

**Given:**

Relative permittivity of air ε_{r} = 1,

Electron charge *q* = 1.602e-19 C.

Quadrupole charges *Q1* = 1**q*, *Q2* = 2**q*, *Q3* = 3**q*, *Q4* = -6**q*.

**Task:**

Calculate the electric field stress distribution along *z* axis.

**Solution:**

Analytical solution is based on the equation derived from Coulomb's law^{*}:

*E*(*z*) = *k* * *q*/*r*^{2} [V/m], where

*k*=8.988e9 [N·m^{2} / C^{2}] is a Coulomb's constant,

*r* - distance from the charge *q*.

For any arbitrary point on axis *z* we can find the electric field stress components produced by each charge.

Total electric field stress components are:

Ex(z) = *k* / r^{2} * cos(β) * (-Q1*cos(α1) - Q2*cos(α2) - Q3*cos(α3) - Q4*cos(α4)),

Ey(z) = *k* / r^{2} * cos(β) * (-Q1*sin(α1) - Q2*sin(α2) - Q3*sin(α3) - Q4*sin(α4)),

Ez(z) = 0.

where β - elevation angle,

α - angle in the plane *XY* between vectors *O-X* and *O-charge*.

(in our model α1 = 3π/4, α2 = π/4, α3 = -π/4, α4 = -3π/4).

**Results:**

Analytical solution for *z*=0: cos(β) = 1, *r*=0.707 m;

Ex(0) = 8.988e9 / (0.707)^{2} * 1 * ( -*q**cos(3π/4) - 2*q**cos(π/4) - 3*q**cos(-π/4) + 6*q**cos(-3π/4)) = 1.271e10*(-10*q*) = 2.04e-8 V/m;

Ey(0) = 8.988e9 / (0.707)^{2} * 1 * ( -*q**sin(3π/4) - 2*q**sin(π/4) - 3*q**sin(-π/4) + 6*q**sin(-3π/4)) = 1.271e10*(-6*q*) = 1.22e-8 V/m;

Ez(0) = 0 V/m.

Electric field stress calculated in QuickField:

*Reference: Coulomb's law in Wikipedia.

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