Dot product vs trigonometry in Gauss' law

In summary, the conversation revolves around the use of the dot product in Maxwell's equations and its advantages over using trigonometry. The dot product is preferred due to its ability to calculate without knowing the angle between vectors, making it more useful in advanced problems. In addition, it has other mathematical properties that make it a better tool for calculations.
  • #1
Korosenai
3
1
I'm currently writing my EP on various physical equations including Maxwell's equations, and I had to justify using the dot product of the normal unit vector and the electric field in the integral version. However, I can't think of a reason for not using trigonometry as opposed to the |a||b|cos<(a,b). Any clarification or explanation is very welcome.
 
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  • #2
I am not sure I understand your question. What, exactly, do you mean by "use trigonometry"? It seems to me that if you were to "use trigonometry" (in the usual sense) you would get exactly the same result as using the dot product.
 
  • #3
What's the trigonometry you meant in Guass' Law?
 
  • #4
Sorry, I just realized that they would give the same result :P
But why is the dot product written instead of trig? Is it because it's easier to write out in an equation?
 
  • #5
Korosenai said:
Sorry, I just realized that they would give the same result :P
But why is the dot product written instead of trig? Is it because it's easier to write out in an equation?

This is getting to be rather silly.

Don't be lazy. Write down the exact equations that you are talking about, because it is obvious that the rest of us have no idea what you are talking about. This forum has the ability to use LaTex math formatting. Use that and show us exactly the type of equations you are referring to.

Otherwise, we have this rather puzzling description from you which makes very little sense!

Zz.
 
  • #6
I'm new to the forum so I apologize. However, there is no need to be quite that rude to me.
I am not being lazy, I'm being ignorant :)
Have a nice day now
 
  • #7
Korosenai said:
Sorry, I just realized that they would give the same result :P
But why is the dot product written instead of trig? Is it because it's easier to write out in an equation?
I presume you're asking why we write the integrand as the dot product of the E vector and the normal unit vector, instead of using the expression you posted (product of their magnitudes and the angle between them)?

It's because the dot product of vectors can be calculated without knowing the angle between them, and in more advanced problems it can be difficult or impossible to find this angle. This is one of many nice mathematical properties that make the dot product more generally useful than a one-off trig-based calculation.

You probably won't see how much more powerful the dot product is until you get into linear algebra and non-trivial coordinate transforms. Until then, you may have to take our word for it that's it a better tool and that you'll want to get comfortable with it.
 
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Related to Dot product vs trigonometry in Gauss' law

What is the dot product and how is it used in Gauss' law?

The dot product is a mathematical operation that takes two vectors and produces a scalar quantity. In Gauss' law, the dot product is used to calculate the electric flux through a surface by multiplying the electric field vector with the surface normal vector.

How does trigonometry relate to Gauss' law?

Trigonometry is used in Gauss' law to calculate the angle between the electric field vector and the surface normal vector. This angle is necessary for accurately calculating the dot product and determining the electric flux through the surface.

Why is the dot product a useful tool in Gauss' law?

The dot product allows for a simple and efficient calculation of the electric flux through a surface, which is a crucial component of Gauss' law. It also allows for a clear understanding of the relationship between the electric field vector and the surface normal vector.

Can Gauss' law be applied without using the dot product?

No, the dot product is an essential part of Gauss' law and cannot be omitted. It is necessary for accurately calculating the electric flux through a surface and determining the strength of the electric field.

Are there any limitations to using trigonometry and the dot product in Gauss' law?

Trigonometry and the dot product are limited to use in situations where the electric field and surface normal vectors are perpendicular to each other. If the angle between these vectors is not 90 degrees, a more complex calculation method is required.

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