How Is Electric Flux Calculated Through a Cylindrical Surface?

[smagf]

Homework Statement

The z-axis carries a constant electric charge density of λ units of charge per unit length with λ > 0. The resulting electric field is \vec{E} = 5 \lambda \frac{x \vec{i} + y \vec{j}}{x^2 + y^2}<br />.

Compute the flux of \vec{E} outward through the cylinder x^2 + y^2 = R^2, for 0 \leq z \leq h.

Homework Equations

d\vec{A} = \vec{n}dA

The Attempt at a Solution

There are three surfaces to compute flux through and I believe that I have to sum those to get the answer. The three surfaces are the two circles that cap the cylinder and the cylindrical face. The unit normal vectors for these are k, -k and \frac{x \vec{i} + y \vec{j}}{\sqrt{x^2 + y^2}}.

My first question:
So for each surface I have an integral over the surface which is the dot product of E and ndA?

My second question:
Because the vector equation of E doesn't involve k and the normal vector equations for the top and bottom don't include i or j, the product E*ndA is 0dA? So those integrals are 0 and I just have to find the integral with the cylindrical surface normal?

 

[HallsofIvy]

Homework Statement

The z-axis carries a constant electric charge density of λ units of charge per unit length with λ > 0. The resulting electric field is \vec{E} = 5 \lambda \frac{x \vec{i} + y \vec{j}}{x^2 + y^2}<br />.

Compute the flux of \vec{E} outward through the cylinder x^2 + y^2 = R^2, for 0 \leq z \leq h.

Homework Equations

d\vec{A} = \vec{n}dA

The Attempt at a Solution

There are three surfaces to compute flux through and I believe that I have to sum those to get the answer. The three surfaces are the two circles that cap the cylinder and the cylindrical face. The unit normal vectors for these are k, -k and \frac{x \vec{i} + y \vec{j}}{\sqrt{x^2 + y^2}}.

My first question:
So for each surface I have an integral over the surface which is the dot product of E and ndA?

Yes.

My second question:
Because the vector equation of E doesn't involve k and the normal vector equations for the top and bottom don't include i or j, the product E*ndA is 0dA? So those integrals are 0 and I just have to find the integral with the cylindrical surface normal?

Yes.

You could also use Gauss's theorem (divergence theorem) to integrate over the volume of the cylinder.

 

[tiny-tim]

welcome to pf!

hi smagf! welcome to pf!

My first question:
So for each surface I have an integral over the surface which is the dot product of E and ndA?

yes

My second question:
Because the vector equation of E doesn't involve k and the normal vector equations for the top and bottom don't include i or j, the product E*ndA is 0dA? So those integrals are 0 and I just have to find the integral with the cylindrical surface normal?

yes

(btw, in this case it's probably easier to change to cylindrical coordinates before integrating, using of course xi + yj = r )