### Maxwell's equations for electrostatics

#### Problem:

Consider the vector field

E(x,y,z) = a(x2 - y2 + z2, z2 - 2xy, 2zy + 2zx),

where a is a constant expressed in the appropriate units.
(a)  Is this field irrotational?
(b)  What is the corresponding charge density?

Solution:

• Concepts:
Maxwell's equations
• Reasoning:
× E = 0?  If yes, we have a static field.  E = ρ/ε0.
• Details of the calculation:
× E = (∂Ez/∂y - ∂Ey/∂z)i + (∂Ex/∂z - ∂Ez/∂x)j + (∂Ey/∂x - ∂Ex/∂y)k
∂Ez/∂y -∂Ey/∂z = a(2z - 2z) = 0,  ∂Ex/∂z - ∂Ez/∂x = a(2z - 2z) = 0,
∂Ey/∂x - ∂Ex/∂y = a(-2y + 2y) = 0.
× E = 0.
The field is irrotational.
(b) E = ρ/ε0,
E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z = a(2x - 2x + 2y + 2x) = 2a(x + y)
ρ = 2aε0(x + y)

#### Problem:

In a volume of space V the electric field is

E = c(2x2 -2xy - 2y2)i + c(y2 - 4xy -x2)j,

where c is a constant.
(a)  Verify that this field represents an electrostatic field.
(b)  Determine the charge density ρ in the volume V consistent with this field.

Solution:

• Concepts:
Maxwell's equations, conservative fields
• Reasoning:
Conservative electrostatic fields are irrotational, ×E = 0.
Details of the calculation:
(×E)x = ∂Ez/∂y - ∂Ey/∂z = 0,  (×E)y = ∂Ex/∂z - ∂Ez/∂x = 0.
E has no z-component and the x- and y-components do not depend on z.
(∇×E)z = ∂Ey/∂x - ∂Ex/∂y = c(-4y - 4y) - c(-2x - 4y) = 0.
E is irrotational and therefore an electrostatic, conservative field.
(b)  ρ/ε0  = ∇·E  =  ∂Ex/∂x + ∂Ey/∂y = c(4x - 2y) + c(2y - 4x) = 0.
Charges outside the volume V or on its surface must produce this field.

#### Problem:

Can the following vector functions represent static electric fields?   If yes, determine the charge density.  This is not a yes/no question.  You must justify your answer mathematically.
(a)  E = r×(c×r)   (Here c is a constant vector.)
(b)  E = c r r   (Here c is a constant and r = |r|.)

Solution:

• Concepts:
Maxwell's equations
• Reasoning:
In electrostatics ∇·E = ρ/ε0,    ∇×E = 0, (SI units).
• Details of the calculation:
(a)  Er×(c×r)  = c(r·r) - r(c·r) =  cr2 - r(c·r).
×E = ×(cr2 - r(c·r))  = r2× c - ×r(c·r) = 2r  c - ×r(c·r).
r = r/r.  ×r(c·r) = (c·r)×r  -  r×(c·r) = ∇(c·rr  since ×r   = 0.
(c·r) = icx + jcy + kcz = c.
×E = 2r×c - c×r = 3r×≠ 0.
E = r×(c×r)   cannot represent a static electric field.

(b)  E = c r r×E = (∇·cr)×r + cr ×r = (c/r) r×r = 0.
E = c r r can represent a static field.
∇·E = ∇·c r r   = cr·r + cr ∇·r = (c/r)r·r + 3cr = cr + 3cr = 4cr = ρ/ε0.
ρ = ε04cr.

#### Problem:

The electric field in some volume V of space is  E = [2A/(x3y2) - By]i + [2A/(x2y3) - Bx]j.
(a)  Show that the line integral of this field over a close path is zero.
(b)  Which charge density ρ in the volume V is consistent with this field?