# Electrostatics test. 3 questions

1) Only the x component of the vector E varies. At a boundary the planar components of the E vector need to be constant (otherwise the field would move virtual charges in the boundary plane). This means the boundary plane is parallel to (0yz) plane. The equation of the boundary is thus

$x = x0 (= constant)$

or in parametric form

$(x,y,z) = (x0,y0,z0) + s<0,1,0>+u<0,0,1>$

where $(x0,y0,z0)$ is a point in the boundary plane and $s$,$t$ are parameters of the plane ($<0,1,0>$ and $<0,0,1>$ are directional vectors of the plane)

2. Outside a conducting sphere that has charge Q the electric field is

$E = Q/((4piepsilon)*R^2)$

$epsilon = er*e0$ and $er =1+chi _e$

$E =Q/(4pi(1+chi _e)*e0)*R^2$

Polarisation is

$P = e0chi _eE = chi _e/(1+chi _e) Q/(4pi R^2)$

Displacement is

$D =e_0(1+chi _e)E = Q/(4pi R^2)$

3.

$sigma(s) =sigma0*s^2/R^2$

$d A = 2pi*s*d s$

$Q = int (from 0 to R) sigma(s)*d s = 2pisigma0/R^2 *int s^3*d s =$

$= 2pi R^4/(4R^2)sigma_0 =pi R^2/2 sigma_0$