Flux through a cylindrical surface enclosing part of a sphere

[palaphys]

Homework Statement

A charged shell of radius R carries a total charge Q. Given phi as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[e0 is the permittivity of free space]

Relevant Equations

E.dA= dphi

Here are the options:

so far, I have solved only option A, which is clearly false, as as per the dimensions mentioned in A, the cylinder completely encloses all the charge of the sphere, hence the flux is $\frac{Q}{\epsilon_0}$

here is my attempt at option B

I'm trying to calculate the plane angle subtended by the top cylindrical surface, and then use the flux per unit solid angle, to calculate the flux through the top surface. By symmetry, the flux through the bottom surface would also be the same. Also, the flux through the curved surface will be zero, as the electric field inside a spherical shell is zero. However, I'm facing great difficulty in calculating the solid angle in the first place. all help is appreciated.

 

[Orodruin]

Hint: You don’t need to compute the flux integral.

Edit: with the dimensions of A, the cylinder does not completely enclose the sphere. The cylinder radius is too small.

 

[palaphys]

Hint: You don’t need to compute the flux integral.

Edit: with the dimensions of A, the cylinder does not completely enclose the sphere. The cylinder radius is too small.

how do I proceed using the solid angle approach? is it even correct?

 

[Steve4Physics]

so far, I have solved only option A, which is clearly false, as as per the dimensions mentioned in A, the cylinder completely encloses all the charge of the sphere,

Check you have not mis-read the question - note that $r = \frac 45 R$ in option A.

[for option B] I'm trying to calculate the plane angle subtended by the top cylindrical surface, and then use the flux per unit solid angle, to calculate the flux through the top surface. By symmetry, the flux through the bottom surface would also be the same. Also, the flux through the curved surface will be zero, as the electric field inside a spherical shell is zero.

That's not correct. If you draw a diagram, including some lines of flux, you will see some lines of flux pass through some of the cylinder's curved surface (as well as through its end faces).

how do I proceed using the solid angle approach? is it even correct?

Using solid angles (if you use the correct ones!) works. You may find it useful to do a search for 'area of a spherical cap' - there is a simple formula which, I guess, you can take as provided (rather than have to prove). Then it's easy to find the solid angle subtended by a cap.

 

[palaphys]

Check you have not mis-read the question - note that $r = \frac 45 R$ in option A.


That's not correct. If you draw a diagram, including some lines of flux, you will see some lines of flux pass through some of the cylinder's curved surface (as well as through its end faces).


Using solid angles (if you use the correct ones!) works. You may find it useful to do a search for 'area of a spherical cap' - there is a simple formula which, I guess, you can take as provided (rather than have to prove). Then it's easy to find the solid angle subtended by a cap.

not able to proceed even after repeated tries. open for the solution.

 

[Steve4Physics]

not able to proceed even after repeated tries. open for the solution.

We don't provide homework solutions! But we can gives help/pointers.

Look at the diagram below (not drawn accurately) which shows a central cross-section of a cylinder intersecting a spherical charged shell.

a) Some lines of flux go through through the cylinder faces (e.g. through CD).

b) Some lines of flux go through part of the curved cylinder side (e.g. through AC and BD).

c) The remaining lines of flux do not go through the upper part of the cylinder.

Curve-AB corresponds to a spherical cap. Lines of flux which start on this cap are the ones that pass through the top part of the cylinder (corresponding to a)+ b) above). Similarly for the cap at the bottom.

The surface area of the sphere is $4 \pi R^2$ (corrected - see Post #7). If you can find the area of the cap (see previous post) then you know the cap’s area as a proportion of the sphere’s area. The flux through the cylinder then follows.

EDIT. Tried to improve wording. And a correction.

 

[Steve4Physics]

I think you mean:

The surface area of the sphere is $4 \pi R^2$

Thanks. That was careless of me. I've changed Post #6.

 

[palaphys]

We don't provide homework solutions! But we can gives help/pointers.

Look at the diagram below (not drawn accurately) which shows a central cross-section of a cylinder intersecting a spherical charged shell.

View attachment 363472

a) Some lines of flux go through through the cylinder faces (e.g. through CD).

b) Some lines of flux go through part of the curved cylinder side (e.g. through AC and BD).

c) The remaining lines of flux do not go through the upper part of the cylinder.

Curve-AB corresponds to a spherical cap. Lines of flux which start on this cap are the ones that pass through the top part of the cylinder (corresponding to a)+ b) above). Similarly for the cap at the bottom.

The surface area of the sphere is $4 \pi R^2$ (corrected - see Post #7). If you can find the area of the cap (see previous post) then you know the cap’s area as a proportion of the sphere’s area. The flux through the cylinder then follows.

EDIT. Tried to improve wording. And a correction.

think I got it. so I'll try to find the charge enclosed by the cylinder, using the fact that the entire spherical surface carrying a charge Q subtends an angle of 4pi str, and so the part of the spherical surface inside the cylinder subtends______ and hence carries a charge with respect to this.

Is this the way to proceed?

 

[Steve4Physics]

think I got it. so I'll try to find the charge enclosed by the cylinder, using the fact that the entire spherical surface carrying a charge Q subtends an angle of 4pi str, and so the part of the spherical surface inside the cylinder subtends______ and hence carries a charge with respect to this.

Is this the way to proceed?

That sounds right. Note that "the part of the spherical surface inside the cylinder" is called a "spherical cap". So the charge inside the cylinder is the charge on the two spherical caps.

 

[palaphys]

That sounds right. Note that "the part of the spherical surface inside the cylinder" is called a "spherical cap". So the charge inside the cylinder is the charge on the two spherical caps.

I got it, thanks