Why Does the Divergence Theorem Not Apply in This Flux Calculation?

My guess is that the flux is 0, but I don't know how to determine if the flux is 0 or a value when using the divergence theorem. Additionally, it is difficult to determine if the volume encloses a charge when only given a vector function.
  • #1
jesuslovesu
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Homework Statement


A volume in cylindrical coords is defined as [1,2]x[pi/6,pi/3]x[1,2]
Calculate the flux of the vector field A(rowe,phi,z) = 4z (rowehat)

Well I used the divergence theorem
Volume integral( div(A) dot A dV) = pi
I got pi as an answer, but apparently that is not correct.

Does anyone know why the divergence theorem in this case doesn't work? I tried converting to Cartesian coords, the surface integrals are really nasty so I haven't evaluated them.

My guess is that the flux is 0, however, how does one tell when the flux through a surface is 0 as opposed to a value? How do I know if the volume encloses a 'charge' in the case of gauss's law if I am just given a vector function?


Homework Equations





The Attempt at a Solution

 
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  • #2
I used the divergence theorem and got pi as an answer, but it turns out that that is not correct. I have not evaluated the surface integrals.
 
  • #3


I would like to offer some explanations and possible solutions to the questions posed.

Firstly, the divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the enclosed volume. In this case, the volume is not closed as it is defined as [1,2]x[pi/6,pi/3]x[1,2]. Therefore, the divergence theorem cannot be directly applied to this problem.

One possible solution is to convert the cylindrical coordinates to Cartesian coordinates and evaluate the surface integrals. However, as mentioned, this may be a difficult and time-consuming process.

Another approach is to use the concept of flux through a surface. Flux is defined as the flow of a physical quantity through a given surface. In this case, the physical quantity is the vector field A, and the surface is the volume defined in cylindrical coordinates. To calculate the flux, we can use the formula:

Flux = surface integral (A dot n dS)

where n is the unit normal vector to the surface and dS is the differential surface area element. This formula takes into account the direction of the vector field and the orientation of the surface.

In this case, the surface is a rectangular prism with sides parallel to the x, y, and z axes. The unit normal vector to each face can be easily calculated, and the dot product with the vector field A can be evaluated. The resulting flux will be a scalar value.

To determine if the flux is 0 or a non-zero value, we need to consider the physical meaning of the problem. In this case, the vector field A represents a flow of a physical quantity (4z) in the direction of the z-axis. The surface is a closed volume, which means that the flow of this physical quantity is confined within the volume. Therefore, the flux through the surface will be non-zero, as there is a flow of the physical quantity through the surface.

In conclusion, the divergence theorem may not be applicable in this case, but we can still calculate the flux through the surface using the formula above. The resulting flux will be a non-zero scalar value, representing the flow of the physical quantity through the given volume.
 

Related to Why Does the Divergence Theorem Not Apply in This Flux Calculation?

1. What is flux through a surface?

Flux through a surface is a measure of the flow of a physical quantity, such as energy, mass, or electric charge, through a surface. It is a vector quantity and is defined as the dot product of the surface area and the component of the quantity's flow that is perpendicular to the surface.

2. How is flux through a surface calculated?

Flux through a surface is calculated using the formula: Flux = surface area * component of flow perpendicular to surface. The surface area is typically given in square meters, while the flow component is given in units specific to the physical quantity being measured.

3. What is the significance of flux through a surface in physics?

Flux through a surface is an important concept in physics as it allows us to quantify the flow of physical quantities through a specified surface. It is used in various fields such as electromagnetism, fluid mechanics, and thermodynamics to understand and analyze the behavior of physical systems.

4. What factors affect the flux through a surface?

The flux through a surface can be affected by several factors, including the magnitude and direction of the physical quantity being measured, the orientation of the surface, and the size and shape of the surface. Other factors such as the presence of barriers or obstacles can also impact the flux through a surface.

5. Can the flux through a surface be negative?

Yes, the flux through a surface can be negative. This occurs when the direction of the flow component is opposite to the direction of the surface normal vector. It is important to consider the sign of the flux when interpreting the results of a calculation, as it can give insight into the direction and magnitude of the flow of the physical quantity through the surface.

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