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?
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.
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|.)
(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.
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?