Finding electric flux through the circular cap of a sphere

[quaticle]

Homework Statement

A sphere of radius R = 1.40 m surrounds a particle with charge Q = 42.0 μC located at its center as shown in the figure below. Find the electric flux through a circular cap of half-angle θ = 28.0°.

Homework Equations

Φ = ∫E⋅dA
E = (kq)/r²
A = πa² where a = rsin(θ) --> from Archimedes's formula for area of a spherical cap [excluding the h term since it is along the surface]).

The Attempt at a Solution

First thing I did was calculate the electric field, using E = (kq)/r² obtaining 1.928x10⁵ N/C. Then using A = πa² I found the area to be 1.357 m². Finally using the flux equation I obtained Φ = 2.62x10⁵ Nm²/C. This answer is incorrect and the correct answer is given as 2.78x10⁵ Nm²/C. I am unsure my method is correct. Any suggestions?

 

[Doc Al]

[excluding the h term since it is along the surface]

Not sure what you mean, but h is not zero.

 

[quaticle]

Not sure what you mean, but h is not zero.

I wasn't sure if the h (height) variable was needed since the portion I am supposed to calculate the flux through is a flat area on the surface of the sphere. I wasn't sure how to find the area of that, and through searches found Archimedes' formula shown in the op. I just re-calculated and when taking the height to be r*cos(θ) I am further from the correct answer. We haven't used the formula before, I had found it an assumed it suitable for these circumstances but do not think it is useful anymore...

 

[Doc Al]

I wasn't sure if the h (height) variable was needed since the portion I am supposed to calculate the flux through is a flat area on the surface of the sphere.

Realize that you need the component of the field perpendicular to whatever area you choose. Using the spherical cap makes that easy.

I just re-calculated and when taking the height to be r*cos(θ) I am further from the correct answer.

That's not the height.

Even better than all this: What fraction of the total flux goes through that cone of angle θ? (Think in terms of solid angles.)

 

[ehild]

Homework Statement

A sphere of radius R = 1.40 m surrounds a particle with charge Q = 42.0 μC located at its center as shown in the figure below. Find the electric flux through a circular cap of half-angle θ = 28.0°.

Homework Equations

Φ = ∫E⋅dA
E = (kq)/r²
A = πa² where a = rsin(θ) --> from Archimedes's formula for area of a spherical cap [excluding the h term since it is along the surface]).
?

You have used wrong formula for the surface area of the cap. It is A=pi(h²+a²) where a is the radius of the cap and h is its high. a= Rsin(θ) and h = R(1-cos(θ))

 

[quaticle]

Even better than all this: What fraction of the total flux goes through that cone of angle θ? (Think in terms of solid angles.)

The fraction of flux we are interested in goes through θ°/360°, right? So find the total flux like one normally would then find that fraction of it?

 

[Doc Al]

The fraction of flux we are interested in goes through θ°/360°, right? So find the total flux like one normally would then find that fraction of it?

Almost. But it's a sphere, not a circle. Hint: How many steradians make up a sphere?

 

[quaticle]

Almost. But it's a sphere, not a circle. Hint: How many steradians make up a sphere?

I am not sure what a steradian is, but a quick google search tells me 4π steradians are in a sphere. Do I then multiple the fraction by 4π?

 

[Doc Al]

I am not sure what a steradian is, but a quick google search tells me 4π steradians are in a sphere.

Good.

Do I then multiple the fraction by 4π?

First you need to know how many steradians are in that cone of angle θ. Then find what fraction that is of the whole sphere.

 

[quaticle]

First you need to know how many steradians are in that cone of angle θ. Then find what fraction that is of the whole sphere.

And this can be done using Ω = 2π(1-cos(θ)) ? This will give me the solid angle of the cone and then I divide that by the total sr of a sphere (4π) to obtain the fraction of the whole through which the flux I need flows...

 

[Doc Al]

And this can be done using Ω = 2π(1-cos(θ)) ? This will give me the solid angle of the cone and then I divide that by the total sr of a sphere (4π) to obtain the fraction of the whole through which the flux I need flows...

Yes.

 

[quaticle]

Yes.

Awesome thanks a lot, using these ideas I was able to obtain the correct answer!

 

[Doc Al]

Cool.