Finding flux through ellipsoid in Cylindrical Coordinates

[BrianA.]

Homework Statement

Using Cylindrical coordinates, find the total flux through the surface of the ellipsoid defined by x² + y² + ¼z² = 1 due to an electric field E = xx + yy + zz (bold denoting vectors | x,y,z being the unit vectors)

Calculate ∇⋅E and then confirm the Gauss's Law

Homework Equations

Cylindrical Coordinates being used: (s,φ,z)
Conversion to Cylindrical Coordinates:
x = scosφ
y = ssinφ
z= z

Surface Element of a Cylinder:
da = sdφdz

The Attempt at a Solution

I converted the ellipsoids equation into cylindrical, so it looks like:
s²cos²φ + s²sin²φ + ¼z² = 1

solving for s looks like s = √(1-¼z²)

I solved for both the volume and surface area of the ellipse through integration. Surface Area was as Follows:
∫ (from -2 to 2) ∫ (from 0 to 2π) sdφdz = ∫∫ √(1-¼z²) dφdz = 2π²

How to use this to find Flux, I am unsure of.
I know to convert E to cylindrical so E = scosφx + ssinφy + zz, but don't know what to do about the unit vectors.

Is the flux just ∫E⋅da ? and if so, how do I take the dot product and what do I do about the unit vectors?

 

[Chestermiller]

Are you allowed to use the divergence theorem?

Chet

 

[BrianA.]

I think that's expected.
The two are related right?

 

[Chestermiller]

I think that's expected.
The two are related right?

Yes. Please state the divergence theorem in terms of E. In your problem, the divergence of E is a constant. What is that constant?

Chet

 

[paranormal]

I would recommend approaching this problem in the following steps:

1. Calculate ∇⋅E, also known as the divergence of E, using the given electric field in cylindrical coordinates. This will give you a scalar function that represents the flux density at any point in space.

2. Use Gauss's Law, which states that the total flux through a closed surface is equal to the enclosed charge, to confirm your result from step 1. In this case, the enclosed charge would be zero, as there is no mention of any charge present in the problem.

3. To find the total flux through the surface of the ellipsoid, you need to integrate the flux density over the surface. This can be done by breaking the surface into small elements, calculating the flux density at each element, and then summing them all up. The integral you have mentioned, ∫E⋅da, is the correct way to do this, where E is the flux density and da is the surface element.

4. To take the dot product, you need to multiply the vector components of E with the corresponding components of the surface element da. In this case, since the surface element is given in cylindrical coordinates, you will need to express E in terms of cylindrical unit vectors, which are s, φ, and z. Remember that s, φ, and z are just unit vectors, so you can replace them with their corresponding components (scosφ, ssinφ, and z) in the dot product.

I hope this helps guide you in solving the problem. It is important to always follow the necessary steps and use the correct equations to arrive at a solution.