# Find Flux Across Portion of Sphere in 3 Ways

• zcd
In summary, Stoke's theorem does not apply to the situation given. The flux of field across a surface is found using divergence theorem.
zcd

## Homework Statement

Find the flux of field $$\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$$ across the portion of the sphere $$x^{2}+y^{2}+z^{2}=4$$ in the first octant in the direction away from the origin in three ways:

a. using formula for flux when sphere is a level surface
b. using formula for flux when sphere is a parametric surface
c. using Stoke's theorem

## Homework Equations

$$S=\iint \limits_{R}\frac{|\nabla f|}{|\nabla f\cdot\hat{p}|}dA$$

$$Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV$$

Stoke's theorem (I thought Stoke's theorem involved circulation and curl)?
$$\oint \limits_{C} \mathbf{F}\cdot d\mathbf{r}=\iint \limits_{S}\nabla\times\mathbf{F}\cdot\mathbf{n}d\sigma$$

Divergence theorem (we haven't learned this yet but I think it will apply better than stoke's theorem for part c)
$$\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV$$

## The Attempt at a Solution

for part a:
$$Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV$$
the gradient vector is always perpendicular to a surface f(x,y,z)=c, so $$\mathbf{n}=\frac{\nabla f}{|\nabla f|}$$
this should lead to $$\iint \limits_{S} \frac{(\mathbf{F}\cdot\nabla f)|\nabla f|}{|\nabla f||\nabla f\cdot\hat{p}|}dA$$
after some simplifying and conversion to polar coordinates...
$$\int_{0}^{2\pi}\int_{0}^{2} r^{3}+r\sqrt{4-r^{2}}dr d\theta=...=\frac{40\pi}{3}$$

I temporarily skipped part b cause of the work involved for finding $$d\sigma=|\frac{\partial{\mathbf{r}}}{\partial{\phi}}\times\frac{\partial{\mathbf{r}}}{\partial{\theta}}|d\phi d\theta$$

for part c:
using divergence theorem to find flux-
$$\nabla\cdot\mathbf{F}=\frac{\partial{M}}{\partial{x}}+\frac{\partial{N}}{\partial{y}}+\frac{\partial{P}}{\partial{z}}=3$$
$$\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV$$
$$=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2}3\rho^{2}\sin\phi}d\rho d\phi d\theta=...=32\pi$$

and here's where I have the problem. The same flux from two different methods should not be different. Where in my work did I go wrong?

Last edited:
You are given just a portion of a sphere which is in the first octant. There is no volume enclosed by it so the divergence theorem is irrelevant. The sphere is cut off by the 3 coordinate planes.

I overlooked the part about being in first octant, but how would I apply Stoke's theorem to find flux of the field across a surface? I thought Stoke's theorem was flux of the curl of the field?

Reworking part a:
$$\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{2}\frac{4r}{\sqrt{4-r^{2}}}dr=4\pi$$

Last edited:
I agree Stokes' theorem doesn't seem appropriate since your $\vec F$ is not the curl of a field. Your answer or $4\pi$ is correct.

Thanks for the verification. Is there another theorem I can use to finish the problem? Or should I just solve for circulation and give 0 as an answer?

## What is the concept of finding flux across a portion of a sphere?

The concept of finding flux across a portion of a sphere involves calculating the amount of fluid or electromagnetic energy that passes through a given area on the surface of a sphere. This involves understanding the surface area of a sphere and the rate of flow of the fluid or energy.

## Why is it important to find flux across a portion of a sphere?

Finding flux across a portion of a sphere is important because it allows us to understand and quantify the flow of fluids or energy in various systems. This information can be used in a variety of fields such as fluid dynamics, electromagnetism, and thermodynamics to make predictions and solve problems.

## What are the three different methods for finding flux across a portion of a sphere?

The three methods for finding flux across a portion of a sphere are surface integral, Gaussian surface, and divergence theorem. Each method involves a different approach and may be more suitable for different types of problems. It is important to understand all three methods and when to use each one.

## How do I perform a surface integral to find flux across a portion of a sphere?

To perform a surface integral, you first need to determine the surface area of the portion of the sphere you are interested in. Then, you need to define the vector field and use the dot product between the vector field and the surface normal vector to calculate the flux. Finally, you integrate the dot product over the surface area to find the total flux across the portion of the sphere.

## What are some real-world applications of finding flux across a portion of a sphere?

Finding flux across a portion of a sphere has many real-world applications, including predicting fluid flow in pipes, calculating the energy output of a solar panel, and understanding the behavior of electromagnetic fields in electronic devices. It is also used in weather forecasting, aerodynamics, and environmental science to model the movement of air and water.

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