- #1
kirito
- 77
- 9
- Homework Statement
- trying to understand how to derive it ,
- Relevant Equations
- gauss's theorem
I am currently studying a section from \textit{Electricity and Magnetism} by Purcell, pages 81 and 82, and need some clarification on the following concept. Here’s what I understand so far:
1. The integral of a function $ \mathbf{F} $ over a surface \( S \) is equal to the sum of the integrals of $ \mathbf{F} $ over smaller surfaces \( S_i \):
$$
\int_S \mathbf{F} \cdot d\mathbf{A} = \sum_i \int_{S_i} \mathbf{F} \cdot d\mathbf{A}_i
$$
2. This can be rewritten as:
$$
\sum_i \int_{S_i} \mathbf{F} \cdot d\mathbf{A}_i = \sum_i \int_{S_i} \frac{V_i}{V} \mathbf{F} \cdot d\mathbf{A}
$$
3. This is equal to the integral of the divergence of $ \mathbf{F} $ over a volume \( V \):
$$
\int_V \nabla \cdot \mathbf{F} \, dV
$$
Now, I want to find the divergence of $ \mathbf{F} $ in a book example, specifically the flux through the upper and lower plates in the \( z \)-direction.
In the example, I know that the function $ \mathbf{F} $ changes only in the \( z \)-direction and the area of each surface is \( dx \, dy \). The direction is \( \hat{z} \). Using the second expression above, I have:
$$
\mathbf{F}_z(x,y,z+\Delta z) \, dx \, dy - \mathbf{F}_z(x,y,z) \, dx \, dy = \left( \frac{\partial \mathbf{F}_z}{\partial z} \right) \Delta z \, dx \, dy
$$
However, in the derivation in the book, they look at the average of $ \mathbf{F}_z $ on the top and bottom plates and take the net contribution by considering the difference between them.
Why are they looking at the value of $ \mathbf{F}_z $ at the center of each plate \( \left(x + \frac{dx}{2}, y + \frac{dy}{2}, z \right) \) and at \( \left(x + \frac{dx}{2}, y + \frac{dy}{2}, z + dz \right) \)? I was only following the definition $$ \mathbf{F} \cdot d\mathbf{A}_1 + \mathbf{F} \cdot d\mathbf{A}_2 $$ and so on.
1. The integral of a function $ \mathbf{F} $ over a surface \( S \) is equal to the sum of the integrals of $ \mathbf{F} $ over smaller surfaces \( S_i \):
$$
\int_S \mathbf{F} \cdot d\mathbf{A} = \sum_i \int_{S_i} \mathbf{F} \cdot d\mathbf{A}_i
$$
2. This can be rewritten as:
$$
\sum_i \int_{S_i} \mathbf{F} \cdot d\mathbf{A}_i = \sum_i \int_{S_i} \frac{V_i}{V} \mathbf{F} \cdot d\mathbf{A}
$$
3. This is equal to the integral of the divergence of $ \mathbf{F} $ over a volume \( V \):
$$
\int_V \nabla \cdot \mathbf{F} \, dV
$$
Now, I want to find the divergence of $ \mathbf{F} $ in a book example, specifically the flux through the upper and lower plates in the \( z \)-direction.
In the example, I know that the function $ \mathbf{F} $ changes only in the \( z \)-direction and the area of each surface is \( dx \, dy \). The direction is \( \hat{z} \). Using the second expression above, I have:
$$
\mathbf{F}_z(x,y,z+\Delta z) \, dx \, dy - \mathbf{F}_z(x,y,z) \, dx \, dy = \left( \frac{\partial \mathbf{F}_z}{\partial z} \right) \Delta z \, dx \, dy
$$
However, in the derivation in the book, they look at the average of $ \mathbf{F}_z $ on the top and bottom plates and take the net contribution by considering the difference between them.
Why are they looking at the value of $ \mathbf{F}_z $ at the center of each plate \( \left(x + \frac{dx}{2}, y + \frac{dy}{2}, z \right) \) and at \( \left(x + \frac{dx}{2}, y + \frac{dy}{2}, z + dz \right) \)? I was only following the definition $$ \mathbf{F} \cdot d\mathbf{A}_1 + \mathbf{F} \cdot d\mathbf{A}_2 $$ and so on.
