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 = (∂E_z/∂y - ∂Ey/∂z)i + (∂E_x/∂z - ∂E_z/∂x)j + (∂E_y/∂x - ∂E_x/∂y)k
  ∂E_z/∂y -∂E_y/∂z = a(2z - 2z) = 0,  ∂E_x/∂z - ∂E_z/∂x = a(2z - 2z) = 0,
  ∂E_y/∂x - ∂Ex/∂y = a(-2y + 2y) = 0.
  ∇× E = 0.
  The field is irrotational.
  (b) ∇∙E = ρ/ε₀,     
  ∇∙E = ∂E_x/∂x + ∂Ey/∂y + ∂E_z/∂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 = ∂E_z/∂y - ∂E_y/∂z = 0,  (∇×E)_y = ∂E_x/∂z - ∂E_z/∂x = 0.
  E has no z-component and the x- and y-components do not depend on z.
  (∇×E)_z = ∂E_y/∂x - ∂E_x/∂y = c(-4y - 4y) - c(-2x - 4y) = 0.
  E is irrotational and therefore an electrostatic, conservative field.
  (b)  ρ/ε₀  = ∇·E  =  ∂E_x/∂x + ∂E_y/∂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)  E =  r×(c×r)  = c(r·r) - r(c·r) =  cr² - r(c·r).
  ∇×E = ∇×(cr² - r(c·r))  = ∇r²× c - ∇×r(c·r) = 2r ∇r× c - ∇×r(c·r).
  ∇r = r/r.  ∇×r(c·r) = (c·r) ∇×r  -  r×∇(c·r) = ∇(c·r)×r  since ∇×r   = 0.
  ∇(c·r) = ic_x + jc_y + kc_z = c.
  ∇×E = 2r×c - c×r = 3r×c  ≠ 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   = c∇r·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?