Compute the divergence and curl of (x+y)i+(y+z)j +(x−2z)k?
a) Compute the divergence and curl of (x+y)i+(y+z)j +(x−2z)k.
b) Could this field be a physical electrostatic field in some limited region of space near the origin? Explain?
c) Given the following mathematical form for a volume charge density ρ(r), describe the charge distribution in words, and also draw a little sketch showing where the charges are: ρ(r)=aδ^3(r−R)+bδ(r−R), where a, b are given constants with appropriate units, R=⟨2,0,0⟩, and as usual r=|r|.
d) What are the (SI metric) units of a
and b in the charge density expression above?
e) What is the total charge, in terms of the given constants in the equation?
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(a) div E = 1+1-2 = 0
curl E = -i + j + k
(b) in electrostatics curl E has to vanish everywhere, so this cannot be an electrostatic field. (div E may vanish, it just means the charge density is zero there)
(c) a point charge q1=a at position R=(2, 0, 0) and a spherical uniform surface charge density b on a sphere of radius 2 centered on the origin.
(d) the one dimensional delta distribution has dimension 1/length. So the three-dimensional delta distribution in the first term has dimension 1/length^3. This implies the SI unit of a is Coulomb (C). The one-dimensional delta distribution in the second term has dimension 1/length, so b has to have SI unit C/m^2.
(e) integrating the density over space obviously gives a+b * 4 pi R^2
= a + 16 pi b