Vector Calculus: Understanding Flux

Once you understand flux intuitively, you don’t need to memorize equations. The formulas become “obvious” dare I say. However, it took a lot of effort to truly understand that:

Flux is the amount of “something” (electric field, bananas, whatever you want) passing through a surface.
Thetotal fluxdepends on strength of the field, the size of the surface it passes through, and their orientation.

一旦你理解2022世界杯南美了通量，你的矢量微积分数学生活就会好很多。谁不想这样呢?

Physical Intuition

Think of flux as the amount ofsomethingcrossing a surface. This “something” can be water, wind, electric field, bananas, pretty much anything you can imagine. Math books will use abstract concepts like electric fields, which is pretty hard to visualize. I find bananas more memorable, so we’ll be using those.

To measure the flux (i.e. bananas) passing through a surface, we need to know

The surface you are considering (shape, size and orientation)
The source of the flux (strength of the field, and which way it is spitting out~~bananas~~flux)

The strength of the field is important – would you rather have a handful of \$5 or \$20 bills “flux” into your bank account? Would you rather have a big or little banana come your way? No need to answer that one.

Background Ideas

Keep a few ideas in mind when considering flux:

Vector Field: This is the source of the flux: the thing shooting out bananas, or exerting some force (like gravity or electromagnetism). Flux doesn’t have to be a physical object — you can measure the “pulling force” exerted by a field.
Surface: This is the boundary the flux is crossing through or acting on. The boundary could be a sphere, a plane, even the top of a bucket. Notice that the boundary may not exist — the top of a bucket traces out a circle, but the hole isn’t actually there. We’re considering the flux passing through the region the circle defines.
Timing: We measure flux at a single point in time. Freeze time and ask “Right now, at this moment, how much stuff is passing through my surface?”. If your field doesn’t change over time, then all is well. If your fielddoeschange, then you need to pick a point in time to measure the flux.
Measurement: Flux is a total, and is not “per unit area” or “per unit volume”. Flux is the total force you feel, the total number of bananas you see flying by your surface. Think of flux like weight. (There is a separate idea of "flux density" (flux/volume) calleddivergence，但这是另一篇文章了。)

Flux Factors

The source of flux has a huge impact on the total flux. Doubling the source (doubling the “banana-ness” of each banana), will double the flux passing through a surface.

总通量也取决于磁场和表面的方向。当我们的表面完全面向磁场时，它就会捕获最大的通量，就像直接面对风的帆一样。当表面倾斜远离磁场时，通量随着穿过表面的通量越来越少而减少。

最终，当源和边界平行时，我们得到零通量——通量穿过边界，但没有穿过它。It would be like holding a bucketsideways在一个瀑布。你不会捕捉到很多水(忽略溅起的水花)，可能会得到一些有趣的表情。

Total flux also depends on the size of our surface. In the same field, a bigger bucket will capture more flux than a smaller one. When we figure out our total flux, we need to see how much field is passing through our entire surface.

This is simple stuff so far, right? If you forget, just think about capturing water from a waterfall. What matters? The strength of the waterfall, the size of the bucket and the orientation of the bucket.

Positive and Negative Flux

One last detail – we need to decide on a positive and negative direction for flux. This decision is arbitrary, but by convention (aka your math teacher will penalize you if you don’t agree),positive flux leaves a closed surface, andnegative flux enters a closed surface.

可以把通量想象成喷水的软管。正通量表示通量离开软管;软管是通量的来源。负通量就像水进入水槽;它是一个不断变化的水槽。所以正通量=离开，负通量=进入。明白了吗?(顺便说一下，术语“源”和“汇聚”有时用于描述字段)。

Quick Summary

Quick checkpoint: Flux depends on

The size of the surface
Magnitude of the source field
The angle between them

A fire hose shooting at a tiny bucket (small surface, large magnitude) could have the same flux as a garden hose aimed at a large bucket (large surface, small magnitude). And in case you forgot, flux reminds us to hold the bucket so it is facing the source. This should be obvious – but don’t you want ideas (especially in math!) to be obvious?

Math Intuition

Now that we have a physical intuition, let’s try to derive the math. In most cases, the source of flux will be described as a vector field: Given a point (x,y,z), there's a formula giving the flux vector at that point.

We want to know how much of that vector field is acting/passing through our surface, taking the magnitude, orientation, and size into account. From our intuition, it should look something like this:

Total flux = Field Strength * Surface Size * Surface Orientation

However, this formula only works if the vector field is the same at every point. Usually, it’s not, so we’ll take the standard calculus approach to solving problems:

Divide the surface into pieces
Find the flux at each piece
将通量的小单位相加，得到总通量(积分)。

我们大胆地称这一小块曲面为dS。Total flux is:

每一个dS的总通量=(场强* dS *取向)。

Total flux = Integral (Field Strength * Orientation * dS)

到目前为止明白了吗?现在，我们需要算出方向到底有多大关系。就像我们之前说的，如果磁场和表面平行，那么通量为零。如果它们是垂直的，就有完整的通量。

(In this diagram, the flux is parallel with the top surface, and nothing enters from that direction. Mathematically, we represent surfaces by theirnormal向量，从表面伸出来。不要让这些记帐细节打乱你的设想。)

If there is an angle, then it is some factor in-between:

How much, exactly? Well, this is a job for thedot product, which is theprojectionof the field onto the surface. The dot product gives us a number (from 0 to 1) that tells us what percent of the field is passing through the surface. So, the equation becomes:

Total flux = Integral( Vector Field Strength dot dS )

And finally, we convert to the stuffy equation you’ll see in your textbook, whereFis our field,Sis a unit of area andnis the normal vector of the surface:

$\displaystyle{\text{flux} = \int_{S} \vec{F} \cdot \vec{n} \ dS}$

Time for one last detail — how do we find the normal vector for our surface?

Good question. For a surface like a plane, the normal vector is the same in every direction. For a sphere, the normal vector is in the same direction as $\vec{r}$, your position on the sphere: the top of a sphere has a normal vector that goes out the top; the bottom has one going out the bottom, etc.

More complicated shapes may have a normal vector that varies quite a bit. In this case, try to break the shape into smaller regions (like spheres, cylinders and planes) and find the flux in each part. Then, add up the flux in each region to get the total flux (keeping in mind positive and negative flux).

If the shape is more complicated than that, you may need a computer model or more advanced theorems; but at least you know what is happening behind the scenes.

Flux Examples

Let’s do a few thought experiments to understand flux. Imagine a tube, that lets water pass right through it. We hold the tube under a waterfall, wait a few seconds, then ask what the flux is. I want a numeric answer – what is the flux?

You might think we need to know the speed of the waterfall, the size of the tube, the orientation, etc. But that isn’t the case.

Remember our convention for flux orientation: positive means flux is leaving, negative means flux is entering. In this example, water is falling downward, or entering the tube. This means the top surface has negative flux (it appears to be siphoning up water).

然而，盒子的底部发生了什么?The water passed through the top and is now leaving the bottom, which is positive flux:

Ah, this beautiful diagram shows what is going on. The top of the box / tube says that water is entering, and the bottom says water is leaving. Assuming the same amount of water is leaving and entering (the rate of water falling is a constant), the net flux would be zero. Think of it as X + (-X) = 0.

What if we had increased the rate of water? Decreased? What would happen?

My (possibly incorrect) answer: If we increased the rate, it means more water would enter than leaves, for a brief moment. We’d have a momentary spike in negative flux (the tube would look like a sink), until the rates equalized. Vice versa if we decreased the rate of water – we’d have a brief spike of positive flux (more water was leaving than entering), until the rate equalized.

Even though net flux is zero, this is different from having zero flux pass through each surface. If you are in an empty field, no shape will generate any flux. But if you are in a field where flux is canceling, changing your shape or orientation could create a non-zero flux. Recognize the difference between having zero flux because the field is zero, vs. having all the flux cancel.

One more point – the “tube” we are considering is a region we define, not a physical tube. Measuring flux is about drawing imaginary boundaries, not having a physical shape. So, when we define the region of a “bucket”, it would not “fill up” with flux. Flux is what is passing through the sides of a bucket at a moment in time. Clearly, if we put in a physical bucket it would fill up, but that’s not what we’re measuring. We’re seeing how much flux would be entering a region we define, from any and all sides (not just the opening). Got it?

还有一点。我们还没有讲到流量的单位。它是用什么测量的?据我所知，单位可以是任何东西，这取决于向量场的单位。所以，你的向量场可以表示香蕉，在这种情况下，你得到香蕉总数穿过一个表面。或者，你的场可以表示每秒的香蕉数，在这种情况下，你会得到每秒穿过你表面的香蕉数。通量的单位取决于向量场的单位。

Flux is relatively simple to understand, and is really helpful in vector calculus and physics. Trying to understand flux by looking at a mess of integrals is not the way to go. First get an intuitive understanding, and the details will make more sense.

Insights

Here’s a few insights that hit me after learning about flux:

You can take the time derivative of flux. If the vector field (F) changes with time (t), you can use dF/dt to see how the total flux changes over time. Even though flux is taken at a unit in time, you can measure flux at two consecutive moments to see how fast it is changing.
You can integrate flux, which means finding how much flux has crossed over a certain time. If the field F is constant over time, you can multiply the flux at one instant by your duration. But if F changes with time, then you need to measure at each moment and integrate. Each flux calculation is done at an instant of time, then they are summed together. Again, this is the standard calculus technique.

在我们的瀑布例子中，我们关注的是一个时间点，在那里水已经流动了一段时间。如果我们选择一个较早的时间点，我们会得到负通量:水进入了顶部，但还没有离开底部。如果我们关掉水，就会有一个正通量的瞬间:水停止进入，但继续离开。

通量对数学、电学和磁学都很重要，你的科学生活会因为知道它而变得更好。你的社交生活——就没那么多了。

这是一篇很长的文章。休息一下。洗澡。走到户外。看到你的家人。Or, read on aboutdivergence. It’s your call.