Divergence Theorem-Electromagnetism

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In summary, the conversation is about the application of the divergence theorem in solving a question on a problem sheet. The poster has attempted to use the continuity equation and has come up with an equation that may be the solution, but is unsure if it is correct. Another poster suggests that the equation is indeed the solution and no further calculation is necessary.
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Homework Statement


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Homework Equations


The divergence theorem is quoted on the problem sheet.


The Attempt at a Solution


I am struggling with the last question (2)c)). I have tried to put the continuity equation into the divergence theorem and have got:

∫S J.ds=-d/dt∫V ρdV

But I'm not sure if I can just sub it in in this way. Do I then need to set up an intergral to get the final solution? I am not really sure what form the answer will be in.

Thanks.
 
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sclatters said:
I am struggling with the last question (2)c)). I have tried to put the continuity equation into the divergence theorem and have got:

∫S J.ds=-d/dt∫V ρdV

But I'm not sure if I can just sub it in in this way. Do I then need to set up an intergral to get the final solution? I am not really sure what form the answer will be in.

Hello, sclatters. I'm not sure what you mean when you say "I'm not sure if I can just sub it in in this way."

But, it looks to me that you have the result! No further calculation should be necessary. You just need to interpret the meaning of the left and right sides of the equation.
 
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FAQ: Divergence Theorem-Electromagnetism

1. What is the Divergence Theorem in electromagnetism?

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental principle in electromagnetism that relates the flow of electric and magnetic fields through a closed surface to the charge and current enclosed within that surface. It states that the net flux of a vector field through a closed surface is equal to the volume integral of the divergence of that field over the region enclosed by the surface.

2. How is the Divergence Theorem used in electromagnetic theory?

The Divergence Theorem is used in electromagnetic theory to relate the behavior of electric and magnetic fields to the sources of those fields, such as charges and currents. It allows us to calculate the total amount of electric and magnetic flux through a closed surface, which is important in many applications, such as determining the strength of an electric or magnetic field at a particular point.

3. What is the mathematical formula for the Divergence Theorem in electromagnetism?

The mathematical formula for the Divergence Theorem in electromagnetism is written as: ∫∫∫V ∇ ⋅ E dV = ∫∫ E ⋅ dA, where ∇ ⋅ E represents the divergence of the electric field, V is the volume enclosed by the closed surface, and dA is an infinitesimal surface element.

4. What are some real-life applications of the Divergence Theorem in electromagnetism?

The Divergence Theorem has many practical applications in electromagnetism, including calculating the electric and magnetic fields of charged particles or objects, analyzing the behavior of electromagnetic waves, and designing electrical circuits and devices. It is also used in areas such as electromagnetic shielding, antenna design, and electromagnetic compatibility testing.

5. How does the Divergence Theorem relate to other fundamental equations in electromagnetism?

The Divergence Theorem is closely related to other fundamental equations in electromagnetism, such as Gauss's Law, Ampere's Law, and Faraday's Law. It provides a mathematical framework for understanding the behavior of electric and magnetic fields and their sources, and is an essential tool in solving problems and analyzing systems in electromagnetism.

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