7 votes

Wrong solution for Green function of one-dimensional Poisson equation

The general solution of $$G^{\prime \prime}(r)=-\delta(r) \tag{1} \label{1}$$ for the Green function $G(r)$ can indeed be obtained from your ansatz $$G(r)=(ar+b) \theta(r)+(cr+d) \theta(-r). \label{...
Hyperon's user avatar
  • 5,998
7 votes
Accepted

Wrong solution for Green function of one-dimensional Poisson equation

Integration over the singularity not only yields $$ \Big[\frac{d}{dr} G \Big]_{-\epsilon}^{\epsilon} = -1 $$ but also: $$ \big[G \big]_{-\epsilon}^{\epsilon} = 0. $$ That will fix $b$ and $d$ to be ...
Jos Bergervoet's user avatar
2 votes
Accepted

Uniform electric field. Find the flux of this field through a square side of 20 cm, parallel to the $y$-$z$ plane?

If I understand correctly, you're interested in the flux through the square not the flux through some cube. You are actually correct in that the flux of this specific electric field through a cube ...
WillHallas's user avatar
2 votes

Why is the differential form of Gauss's Law equivalent to the integral form?

An easy way to understand this equivalence is by using the intuitive definition of the divergence as the surface integral per volume (in the limit of small volume): $$\nabla\cdot\textbf{E} = \lim_{V\...
Thomas Fritsch's user avatar
1 vote
Accepted

Doubt regarding proof of Earnshaw's Theorem using Gauss's theorem

there is a net flux in the sphere precisely due to the presence of the test charge No, the flux arises from the electric field of the other charges. From basic electrostatics, the force on a test ...
Vincent Thacker's user avatar
1 vote
Accepted

Why charges reside only on the surface on conductor?

Let's take mole of free electrons in a copper plate, say $d=1$ mm thick. We can estimate the electric field energy by considering the a mole of $+1$ charges separated by $d$ from a mole of electron (...
JEB's user avatar
  • 33.7k
1 vote

Why is the differential form of Gauss's Law equivalent to the integral form?

Basically you can think of the divergence of a vector field as its tendency to converge (or diverge) to a point in space. For example, think of a fluid: the velocity of every element of fluid is a ...
Tato's user avatar
  • 11
1 vote

Uniform electric field. Find the flux of this field through a square side of 20 cm, parallel to the $y$-$z$ plane?

The concept of flux describes how much of something (in this case, electric field lines) passes through an area. The equation for electric flux through an area is $$\Phi=E.A=EAcos\theta$$ It can also ...
wonderingwhy's user avatar
1 vote

What exactly does charge density mean in Gauss's law?

You can imagine that all charge distributions are composed of infinitesimal charges. $\rho$ describes how these infinitesimal charges are distributed. It does not need to be uniform (in this case, one ...
Níckolas Alves's user avatar
1 vote
Accepted

What exactly does charge density mean in Gauss's law?

The charge density in Gauss' law is simply a scalar function that specifies the distribution of charge in a region of space. The region may be 1, 2, or 3-dimensional as the case demands. Essentially, $...
Albertus Magnus's user avatar
1 vote

Volume Distribution of Bound Charge in an Isotropic Dielectric: when does $ \nabla \cdot \vec{E }=0$?

Your working is correct for free charges, but what about the bound ones? Generally (without making any assumptions about the dielectric's properties), we can write: $$\vec{D}=\varepsilon_0\vec{E}+\vec{...
AlanFox86's user avatar
  • 744
1 vote

Using Gauss' law to determine field

As a purely theoretical endeavor, it is possible to envision a universe where symmetric charge distributions simply do not give rise to symmetric fields. There is no fundamental reason that the ...
Albertus Magnus's user avatar
1 vote

Using Gauss' law to determine field

The spherical symmetry of the object (it could be a point charge, or a ball of uniform charge density) means that, for any two points in space ($A$ and $B$) that are the same distance $d$ from the ...
Ben H's user avatar
  • 1,290

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