Static situations and electric fields - special relativity

[varth]

Homework Statement

Is it possible to create an electrostatic field E(x) (in 3 spatial dimensions x and E is a vector of course) such that
i) E(x) = a × x (cross product)
ii) E(x) = (a.x) b (dot product between a and x)

where a,b and non-zero vectors that do not depend on time and the spatial coordinates. I also have to stay the condition on the constant vectors a and b.

2. A ball of radius R charged homogeneously throughout its volume is centered at the origin.
i) Find the charge density rho(x) for each point of space time
ii) Find the electric field E(x) created by the charged ball.

Not sure how to approach this question, I'm comfortable with working with point charges and using the delta distribution but not for a charged ball.

Homework Equations

I suspected Maxwell's equations would come in handy for the first question but I have been unable to use them to get any conclusion.

Many thanks.

 

[Nate810]

The Attempt at a Solution For the first part of the question, it is not possible to create an electrostatic field E(x) such that E(x) = a x (cross product). This is because the curl of the electrostatic field E(x) must be zero since electrostatic fields are conservative. Thus, it is not possible to have a non-zero cross product of 2 vectors. For the second part of the question, it is possible to create an electrostatic field E(x) such that E(x) = (a.x) b (dot product between a and x). This can be done by setting E(x) = -grad V(x) where V(x) = (a.x)b. For the second question, the charge density rho(x) for each point of space can be found using Gauss's law. Since the ball is charged homogeneously throughout its volume, the charge density rho(x) can be calculated using the following equation:rho(x) = Q/V where Q is the total charge on the ball and V is the volume of the ball. The electric field E(x) created by the charged ball can be found using Coulomb's law. For a point outside the ball, the electric field E(x) is given by the following equation:E(x) = kQ/r^2 where k is the Coulomb's constant, Q is the total charge on the ball and r is the distance from the center of the ball to the point.