Calculating Electric Flux of Displacement Vector w/ Cylindrical Co-ords

[michael2812]

Using cylindrical co-ordinates (r, theta, z), calculate the electric flux
phi=integral D . dS
when the displacement vector D is given by:
D=10ezcos(theta)/r2 . kr + z2sin(theta)/r2 . Kz and the surface S is part of a cylinder of radius a=0.3, defined by 0 is less than or equal to z which is less than or equal to 1 and 0 is less than equal to theta which is less than or equal to pi/4


I’ve put the equation in the cylindrical co-ordinates:

(10ezcos(theta)/r2, 0, z2sin(theta)/r2)

From there I’ve written out S as S(Sr, Stheta, Sz)

I know that the perpendicular lines define the surface. In this case Sz is perpendicular which means than Sr is the main component.
Then:

D . DS = DrdSr + DthetadStheta + DzdSz

But I don’t know where to go from here... Any help?

 

[mark726]

Hello there,

To calculate the electric flux, we can use the formula:

phi = integral D . dS

First, we can rewrite the displacement vector D in cylindrical coordinates as:

D = (10ezcos(theta)/r^2)kr + (z^2sin(theta)/r^2)kz

Next, we need to find the surface area element dS in cylindrical coordinates. Since we are dealing with a cylindrical surface, the surface area element can be written as:

dS = r dtheta dz

Now, we can plug in the values for D and dS into the electric flux formula:

phi = integral (10ezcos(theta)/r^2)kr + (z^2sin(theta)/r^2)kz . (r dtheta dz)

Note that the dot product between kr and kz is zero, so we can simplify the equation to:

phi = integral (10ezcos(theta)/r) dtheta dz + integral (z^2sin(theta)/r) dtheta dz

Next, we can plug in the limits of integration for theta and z, which are 0 to pi/4 and 0 to 1, respectively:

phi = integral (10ezcos(theta)/r) dtheta dz + integral (z^2sin(theta)/r) dtheta dz

= integral (10ezcos(theta)/r) dtheta from 0 to pi/4 + integral (z^2sin(theta)/r) dtheta from 0 to pi/4

= 10e/r * integral cos(theta) dtheta from 0 to pi/4 + 1/r * integral z^2sin(theta) dtheta from 0 to pi/4

= 10e/r * sin(pi/4) - sin(0) + 1/r * (1^2 - 0^2) * (cos(pi/4) - cos(0))

= (10e/r * (sqrt(2)/2) + 1/r * (1 * (sqrt(2)/2 - 1))

= (5sqrt(2)e/r + sqrt(2)/2r - 1/r)

= sqrt(2)(5e + 1)/2r

Therefore, the electric flux for the given displacement vector and cylindrical surface is sqrt(2)(5e + 1)/2r. I hope this helps! Let me know if you have any