Find the flux of a vector field

In summary, the person is asking for verification on their answer which equates a vector quantity with a scalar quantity. The person providing the feedback points out that there is a missing vector in the expression and that multiplying by ##\rho## is not solely for obtaining correct units. They suggest finding a better motivation for multiplying by ##\rho## and also note that using ##\times## in an expression for vector analysis is incorrect. The person's proposed result can also be simplified and evaluated exactly without using an approximate result.
  • #1
falyusuf
35
3
Homework Statement
Attached below.
Relevant Equations
Attached below.
Question:
1646436275383.png

Equation:
1646436289968.png


Attempt:
1646439292931.png

Can someone verify my answer?
 
Last edited:
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  • #2
First of all, your assertion
1647165130602.png

cannot be correct because it is equating a vector quantity with a scalar quantity (i.e., you are missing a vector in the expression). The reason to multiply by ##\rho## is also not "to get the correct units" although having the correct units is a reasonable check that you did things correctly. You could also multiply by ##z## and get the correct units but the result would be wrong. You therefore need a better motivation for multiplying by ##\rho##.

In your last expression, you should not use ##\times## like this in an expression in vector analysis but reserve it for the vector cross product. Your result may also be simplified and evaluated exactly (i.e., there is no need to use an approximate result - the result is one of the proposed answers).
 

1. What is a vector field?

A vector field is a mathematical concept that assigns a vector to each point in a given space. This can be represented visually as a collection of arrows, with each arrow pointing in a certain direction and having a certain magnitude at a specific point.

2. What is flux?

Flux is a measure of the flow of a vector field through a surface. It is calculated by taking the dot product of the vector field and the surface's normal vector at each point on the surface, and then integrating over the surface.

3. How do you find the flux of a vector field?

To find the flux of a vector field, you first need to determine the surface through which the flux is being calculated. Then, you need to find the normal vector to the surface at each point and take the dot product with the vector field. Finally, you integrate this dot product over the surface to get the total flux.

4. What is the significance of finding the flux of a vector field?

The flux of a vector field can be used to calculate the amount of fluid or energy passing through a given surface. It is also an important concept in many areas of physics and engineering, such as electromagnetism and fluid dynamics.

5. Are there any practical applications of finding the flux of a vector field?

Yes, there are many practical applications of finding the flux of a vector field. For example, in fluid dynamics, the flux of a fluid through a surface can be used to calculate the rate of flow or the amount of fluid passing through a certain area. In electromagnetism, the flux of an electric or magnetic field through a surface can be used to calculate the amount of charge or magnetic field passing through a given area.

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