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0answers
50 views

Debye screening in $\mathbb{R^d}$

Consider the Poisson-Boltzmann equation $$ \nabla^2 V(r) = -\frac{1}{\epsilon_0}en\left(1 - e^{e V(r)/k_BT}\right) $$ which models the electrostatic potential in a spherically symmetric ideal gas of ...
2
votes
1answer
75 views

Is there a higher-level reason why $\nabla\cdot(\hat{\bf r}/r^2) = 0$ in three dimensions but not two? [duplicate]

I am working through Griffiths, Introduction to Electrodynamics, and finding the divergence of the electric field generated by a single charge sitting at the origin. $$\mathbf{E}(\mathbf{r}) = \frac{\...
0
votes
0answers
20 views

Scepticism regarding the exponent of -2 in Coulombs law? [duplicate]

So lately I was speculating why nature choose the number 2 in Coulomb's law like why not 2.$10^{100}$ trailing zeros and then 1 or anything else. I find 2 a bit arbitrary the given explanation being ...
0
votes
2answers
226 views

Coulomb Potential in 1+1 and 1+2 dimensions?

It is know that in 1+1 dimensions, the Coulomb potential is of linear form: $$ V(x) = Cx$$ and in 1+2 dimensions, of the form: $$V(x,y) = - \ln\left(\frac{L}{\sqrt{x^2 + y^2}}\right).$$ And I am ...
0
votes
2answers
129 views

Dimension of an equipotential surface

How many dimensions has an equipotential surface? My book says it has $3$ dimensions but I think it has $2$ dimensions because a plane is an object of dimension $2$ as well. From linear algebra, I ...
2
votes
1answer
76 views

Zero net charge in two space dimensions?

I recently came across the statement (without further explanation) that the net charge in a two dimensional system has to be zero. Obviously in two dimensions, the electric field $\vec{E}$ due to a ...
2
votes
1answer
3k views

Why is Coulomb's law is an inverse square law? [duplicate]

I read about Coulomb's law it says the force of attraction or repulsion between two charges is inversely proportional to the square of the distance between them. So generally if I say then Coulomb'...
0
votes
1answer
134 views

How come divergence of $\vec E$ is zero in this case where $\vec E=\xi\frac{\left[x,y\right]}{\sqrt{(x^2+y^2)^3}}\,?$

I hope you could help me clearing some doubts about Gauss' law of the electric field that states $\epsilon_0\nabla\cdot\vec E=\rho$. Take for instance the case of a point charge in the origin in empty ...
6
votes
2answers
154 views

Significance of electrical fields of infinite objects

I've noticed a pattern with the electric fields of charged objects of infinite dimensions. A point charge, which can be considered a charge of 0 dimensions, has an electric field that goes as $r^{-2}$...
3
votes
6answers
6k views

2 dimensional Coulomb's law equation

We can notice that in the Coulomb's law equation, $$\begin{equation}\tag{1}F=\frac{1}{4\pi\epsilon}\cdot\frac{q_1q_2}{r^2}\end{equation} $$ $4\pi r^2$ factor in the denominator expresses directly ...
5
votes
4answers
11k views

Electric field and electric potential of a point charge in 2D and 1D

in 3D, electric field of a piont charge is inversely proportional to the square of distance while the potential is inversely proportional to distance. We can derive it from Coulomb's law. however, I ...
2
votes
2answers
4k views

Electric potential due to a point charge in Gaussian/CGS units

I learned electrostatics in SI units. In SI, the electrostatic potential due to a point charge $q$ located at $\textbf{r}$ is given by $\Phi(\textbf{r}) = \frac{q}{4 \pi \epsilon_0 |\textbf{r}|}$. ...
3
votes
1answer
193 views

Scaling of Static Electric Field

The electric field of a point charge goes like $\displaystyle\frac{1}{r^2}$ The electric field of an infinite line goes like $\displaystyle\frac{1}{s}$ The electric field of an infinite plane is ...
71
votes
6answers
6k views

Does Coulomb's Law, with Gauss's Law, imply the existence of only three spatial dimensions?

Coulomb's Law states that the fall-off of the strength of the electrostatic force is inversely proportional to the distance squared of the charges. Gauss's law implies that a the total flux through a ...