Conceptual understanding/question of guass and Coulomb’s law

[Tryingtolearn123]

Homework Statement

This is more of a conceptual question for those already experienced with Gauss law and Coulomb’s law. As far as being able to look at a problem and knowing exactly how to solve it from the very beginning. Often looking at solutions and doing the math of a single problem can be very time consuming. In calculus courses using the technique of doing the math in your head from just looking at a problem can greatly reduce the time it takes to study for it. With the understanding of a derivative how a slope of a line changes and an integral being the area under a curve. Or in the case of Gauss law a 3 dimensional surface integral. What are the best questions to ask when looking at a Gauss law problem or coulomb problem? For example a ring of charge or a plane of charge or 4 point charges along an axis. What is the simplest reasoning one can use in the shortest explanation? My strongpoints in concepts are comparing flux to fluid flow and how the angle changes the flow. The idea of a surface integral is not challenging. Weakpoints come from a lack of practice currently.

Relevant Equations

General equations for Coulomb’s law and Gauss law.

Homework Statement: This is more of a conceptual question for those already experienced with Gauss law and Coulomb’s law. As far as being able to look at a problem and knowing exactly how to solve it from the very beginning. Often looking at solutions and doing the math of a single problem can be very time consuming. In calculus courses using the technique of doing the math in your head from just looking at a problem can greatly reduce the time it takes to study for it. With the understanding of a derivative how a slope of a line changes and an integral being the area under a curve. Or in the case of Gauss law a 3 dimensional surface integral. What are the best questions to ask when looking at a Gauss law problem or coulomb problem? For example a ring of charge or a plane of charge or 4 point charges along an axis. What is the simplest reasoning one can use in the shortest explanation? My strongpoints in concepts are comparing flux to fluid flow and how the angle changes the flow. The idea of a surface integral is not challenging. Weakpoints come from a lack of practice currently.
Homework Equations: General equations for Coulomb’s law and Gauss law.

Give me about three weeks and I could teach this to anybody. But I got a week and a half to learn all of it so I’m trying to shorten three weeks to a week and a half.

 

[Delta2]

When applying Gauss's law or Coulomb's law (which are usually equivalent and lead to the same results, however there are some extreme cases that they give different results) the concept of symmetry is quite often very important, cause it allows us to do various simplifications.

So when you facing an electrostatics problem you must look for the symmetries that this problem has, symmetries which arise by the specific geometric setup of the problem and tell us various things about the magnitude and direction of the field in various locations. These symmetries allows us to simplify the surface integral in Gauss's law or simplify the calculations we do by integrating Coulomb's law over the volume of the setup.

 

[Tryingtolearn123]

When applying Gauss's law or Coulomb's law (which are usually equivalent and lead to the same results, however there are some extreme cases that they give different results) the concept of symmetry is quite often very important, cause it allows us to do various simplifications.

So when you facing an electrostatics problem you must look for the symmetries that this problem has, symmetries which arise by the specific geometric setup of the problem and tell us various things about the magnitude and direction of the field in various locations. These symmetries allows us to simplify the surface integral in Gauss's law or simplify the calculations we do by integrating Coulomb's law over the volume of the setup.

Thanks this helps a lot. I’m sure if I keep this in mind I’ll pick up the pattern much more quickly

 

[Tryingtolearn123]

When applying Gauss's law or Coulomb's law (which are usually equivalent and lead to the same results, however there are some extreme cases that they give different results) the concept of symmetry is quite often very important, cause it allows us to do various simplifications.

So when you facing an electrostatics problem you must look for the symmetries that this problem has, symmetries which arise by the specific geometric setup of the problem and tell us various things about the magnitude and direction of the field in various locations. These symmetries allows us to simplify the surface integral in Gauss's law or simplify the calculations we do by integrating Coulomb's law over the volume of the setup.

Another question, if Gauss law uses the concept of surface area and Coulomb’s law is based off of 1/r^2 and with multiple charges at a far enough distance it becomes 1/r^3. Then it seems that the two laws being equivalent seems to be logical. Gauss law relies on surface area and Coulomb’s law relies I guess area and volume. Perhaps deriving why they are equal will give me a better understanding

 

[Delta2]

Coulomb's law and Gauss's law give always (when they are equivalent) results that approximately obey the inverse square law, when we examining the field in a position $r$ that is far away from all the given sources then we ll find that the field there is approximately proportional to $\frac{1}{r^2}$.

The equivalence of Gauss's law and Coulomb's law is as follows:
1) Gauss's Law $\Rightarrow$ Coulomb's law
This is fairly easy, if we consider a charge $q_1$ and consider a spherical surface centered at the charge and with radius $r$, then the surface integral of Gauss's law simplifies (due to the spherical symmetry) to $E(r)4\pi r^2=\frac{q_1}{\epsilon_0}$ from which we can infer that $E(r)=\frac{1}{4\pi\epsilon_0}\frac{q_1}{r^2}$ and since by definition of the force that this e-field applies to a point charge $q_2$ that is located at distance $r$ from the point charge $q_1$, we have $F=E(r)q_2=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}$ which is coulomb's law

2)Coulomb's law $\Rightarrow$ Gauss's law
This is a bit more complicated, it uses the divergence theorem and the form of known solution to Poisson's equation. I ll write this a bit later when I have more time.

From the theory of Partial Differential Equations, we know a well known result that the field we get by integrating coulomb's law over a volume charge density $\rho$, that is the field
$$\vec{E(r)}=\int\frac{\rho(\vec{r'})}{4\pi\epsilon_0|\vec{r}-\vec{r'}|^3}(\vec{r}-\vec{r'})d^3\vec{r'}\text{(1)}$$
satisfies the partial differential equation
$$\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_0}$$

By using divergence theorem and IF the E-field of (1) is continuously differentiable we can conclude that $$\oint_{\partial V} \vec{E}\cdot d\vec{S}=\int_V \nabla\cdot \vec{E}d^3\vec{r}=\int_V \frac{\rho(\vec{r})}{\epsilon_0}d^3\vec{r}=\frac{q_{enclosed}}{\epsilon_0}$$

 

[Tryingtolearn123]

Coulomb's law and Gauss's law give always (when they are equivalent) results that approximately obey the inverse square law, when we examining the field in a position $r$ that is far away from all the given sources then we ll find that the field there is approximately proportional to $\frac{1}{r^2}$.

The equivalence of Gauss's law and Coulomb's law is as follows:
1) Gauss's Law $\Rightarrow$ Coulomb's law
This is fairly easy, if we consider a charge $q_1$ and consider a spherical surface centered at the charge and with radius $r$, then the surface integral of Gauss's law simplifies (due to the spherical symmetry) to $E(r)4\pi r^2=\frac{q_1}{\epsilon_0}$ from which we can infer that $E(r)=\frac{1}{4\pi\epsilon_0}\frac{q_1}{r^2}$ and since by definition of the force that this e-field applies to a point charge $q_2$ that is located at distance $r$ from the point charge $q_1$, we have $F=E(r)q_2=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}$ which is coulomb's law

2)Coulomb's law $\Rightarrow$ Gauss's law
This is a bit more complicated, it uses the divergence theorem and the form of known solution to Poisson's equation. I ll write this a bit later when I have more time.

Dude your amazing thanks so much no worries about writing coulombs to gauss law. My class isn’t going to go that advanced I’m only an engineer after all but thanks a lot for the help. If I didn’t do engineering then I’d certainly do physics