Gauss' law in differential form

In summary, the diff. form of Gauss' law is $4\pi\rho$ not $\rho/\epsilon_0$. Conventionally, theoretical physics, especially particle physicists, tend to go with cgs. However, today there seems to be a strong trend to go rationalized mks, aka SI. One thing I don't like about cgs is that it has no separate unit for charge Q.
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Leo Liu
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Homework Statement
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Relevant Equations
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My book claims that the diff. form of Gauss' law is
$$\nabla\cdot\mathbf E=4\pi\rho$$
Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?
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I'm sure @TSny would go into more detail but trust me you do not wish to do that and he has chosen not to. Just get used to different factors of 4pi and epsilon and mu in equations and understand there is no problem. The pictures in your head should not depend upon these details.
 
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hutchphd said:
I'm sure @TSny would go into more detail but trust me you do not wish to do that and he has chosen not to. Just get used to different factors of 4pi and epsilon and mu in equations and understand there is no problem. The pictures in your head should not depend upon these details.
Thanks. But why do we need two sets of units for EM? Doesn't SI suffice all of our needs?
 
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History and politics
 
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Leo Liu said:
Thanks. But why do we need two sets of units for EM? Doesn't SI suffice all of our needs?
You could use SI for everything. It is not always handy though.
 
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Both systems have their pluses and minuses.
Conventionally, theoretical physics, especially particle physicists, tend to go with cgs. But today there seems to be a strong trend to go rationalized mks, aka SI. Personally I won't deal with cgs though that is what I had to deal with in my own introductory physics course. A long time ago, thank goodness.

One thing I don't like about cgs is that it has no separate unit for charge Q. For an EE like myself that is unacceptable. I'll let the particle physicists defend cgs.

Of course, the presence/absence of Q is a tradeoff of sorts. In general, increasing the number of characters in a vocabulary shortens the text but at the expense of extra characters in the "alphabet". Cf. English vs. Chinese.

Another example is avoidance of extra parameters even within a given system. Some teachers prefer a minimum of parameters, others like abbreviated text. E.g. you can avoid ## \bf D ## and ## \bf H ## since ## \bf D = \epsilon \bf E ## and ## \bf B = \mu \bf H ## but personally I find that awkward. Clutters the Maxwell equations, for example. Richard Feynman even avoids using ## \mu ##, sticking to ## c ## and ## \epsilon ##. If you're SI it makes his cgs-based Lectures hard to follow at times.

Etc.
 
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1. What is Gauss' law in differential form?

Gauss' law in differential form is a mathematical equation that describes the relationship between electric charge and electric fields. It states that the divergence of the electric field is equal to the charge density divided by the permittivity of free space.

2. How is Gauss' law in differential form different from the integral form?

Gauss' law in differential form is a local form of the law, meaning it applies to a specific point in space. The integral form is a global form, meaning it applies to a larger region or volume. The differential form is used to calculate the electric field at a specific point, while the integral form is used to calculate the total electric flux through a closed surface.

3. What is the significance of Gauss' law in differential form in electromagnetism?

Gauss' law in differential form is one of the four Maxwell's equations that describe the behavior of electric and magnetic fields. It is a fundamental law in electromagnetism and is used to understand and predict the behavior of electric fields in various situations, such as in the presence of electric charges or conductors.

4. How is Gauss' law in differential form applied in practical situations?

Gauss' law in differential form is used in practical situations to calculate the electric field at a specific point due to a known charge distribution. It is also used to analyze the behavior of electric fields in various devices, such as capacitors, and in the design of electrical systems.

5. What are the limitations of Gauss' law in differential form?

Gauss' law in differential form is based on the assumption of a continuous charge distribution, which may not always be the case in real-world situations. It also does not take into account the effects of magnetic fields, which are described by a separate equation. Additionally, it may not be applicable in situations where the electric field is changing over time.

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