Thread: divergence

Hi I have this problem that I cannot find the solution:

2 students propose the following 2 different flux densities as solutions to an electrostatics problem:
D1= ε* ρ^2 ρ (ρ, φ unit vector)

D2= ε*ρ^2 *(sin(φ))^2 ρ+ 3*ε*ρ*z*(cos(φ))^2φ

Show that the volume charge densities that correspond to these two fields are identical.

I tried to apply the divergence to D1 and D2 but I don't find the same response.
So I dont know what else I can do here.
Please can I have some help?

Originally Posted by braddy

Hi I have this problem that I cannot find the solution:

2 students propose the following 2 different flux densities as solutions to an electrostatics problem:
D1= ε* ρ^2 ρ (ρ, φ unit vector)

D2= ε*ρ^2 *(sin(φ))^2 ρ+ 3*ε*ρ*z*(cos(φ))^2φ

Show that the volume charge densities that correspond to these two fields are identical.

I tried to apply the divergence to D1 and D2 but I don't find the same response.
So I dont know what else I can do here.
Please can I have some help?

These are both in cylindrical coordinates, right?

I'll call epsilon e, rho p, phi f, and let d/dp stand for the partial derivative with respect to p, etc.

Since D1 is entirely radial:
(grad)(dot)D1 = 1/p * d/dp[p*D1]

= e/p * d/dp[p^3] = 3pe

(grad)(dot)D2 = 1/p * d/dp[p*D2(p)] + 1/p * dD2(f)/df
where D2(p) is the p component of D2, etc.

= e/p * sin^2(f) * d/dp[p^3] + 3epz/p * d/df[cos^2(f)]

= 3ep*sin^2(f) + 3ez*sin(2f)

I'm assuming D is the displacement vector, so
(grad)(dot)D = (charge density)

Thus I would agree with you. The two charge densities are not the same.

-Dan