6 votes
Accepted

Is it possible to introduce magnetic monopoles without breaking $∇ · B = 0$?

No, it is not possible. If you have a magnetic monopole then, by definition, there is some closed Gaussian surface $S$ over which $$\oint_S \vec B \cdot d\vec S \ne 0$$ which by implies that $$\int_V \...
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3 votes
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Gauss' law in the presence of surface charges

There are two ways of explaining this. First is that if you're considering a charge density which is concentrated on a surface, then the electric field $\mathbf{E}$ does not satisfy the hypotheses of ...
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  • 2,269
2 votes

Why can't electric field by a single charge be at an angle?

Op has been getting several answers which may be correct but appear to be above the level of what OP was looking for. So I am taking a simpler approach. Consider the following diagram where we observe ...
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  • 2,175
2 votes

Why can't electric field by a single charge be at an angle?

At an angle to what? If you are imagining a point charge, it’s a coordinate singularity. If you stand at the North Pole, every direction is “south.” If you have an object which really has zero size,...
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2 votes

Is it possible to introduce magnetic monopoles without breaking $∇ · B = 0$?

"Net magnetic charge is zero everywhere" "Magnetic current is non zero" Can be done using $$\rho_{m} = -\vec{P} \cdot \nabla \delta^3(r)$$ Where $$\frac{d\vec{P}}{dt} ≠ 0$$ Here, $\...
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  • 3,697
2 votes

Flux through faces of cube if charge is placed at an edge-center

Hint : In general the flux through an oriented open or closed surface $\:\mathrm{S}\:$ due to a point charge $\:Q\:$ is \begin{equation} \Phi_{\mathrm{S}}=\dfrac{\Theta}{4\pi}\dfrac{Q}{\epsilon_{0}} \...
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2 votes

Why is the gravitational potential inside a hollow sphere same as that of the gravitational potential on the surface of the hollow sphere?

Suppose one point on the shell is of potential V0. Due to spherical symmetry(with constant mass density throughtout the shell), every other point on the shell will be of that same potential. Now, the ...
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2 votes
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Why is the gravitational potential inside a hollow sphere same as that of the gravitational potential on the surface of the hollow sphere?

The 'Shell theorem' states that inside a hollow sphere there is no net gravitational pull. This is because the pull of all the parts of the surface cancel each other out perfectly. This is not the ...
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2 votes
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Electric displacement at the boundary of a dielectric and vaccum

Now since the electric displacement vector does not depend on the dielectric Careful -- ${\bf D}$ is affected by the presence of a dielectric, just not in an immediately obvious way. You need to use ...
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  • 2,051
1 vote

Gauss' Law in differential form applied to charged sphere

Since you said sphere, and not shell, you'll have a total charge $Q$ uniformly distributed over the sphere of radius $R$: $$\rho=\frac Q{\frac 4 3 \pi R^3}$$ Based on symmetry: $$ \vec E(r,\theta, \...
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  • 23k
1 vote

Gauss' Law in differential form applied to charged sphere

I think this problem is ill posed. The divergence operator is not generally invertible. There are many functions with the same divergence. We need further information of electric field on a region of ...
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1 vote

2D Gauss law vs residue theorem

There is a very good discussion of this in Tristan Needham's Visual Complex Analysis final chapter, but I'd like to share one other view point of it. When we have some charge configuration or so, we ...
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1 vote
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Integrate continuity equation in QM

The integral $$\int \psi^*\psi\ r^2\ dr\ d\Omega$$ is just the same as integral $$\int P(\mathbf{r},t) d^3\mathbf{r}$$ representing the total probability of finding the particle anywhere. Remember ...
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1 vote

Why can't electric field by a single charge be at an angle?

The symmetries of the charge distribution say something about the symmetries of the resulting electric field. For example $\rho$ has rotational symmetry $\implies$ $\vec E$ only depends on $r$ $\rho$ ...
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1 vote

Why can't electric field by a single charge be at an angle?

The experimental study of electricity led to modeling the data with formulas that not only modeled the data, but were also predictive of new measurements. The attraction between two point charges . ...
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1 vote

No net charge in conductor or no charge at all? Electrostatics

Since conductors are made of atoms, which are made of particles with charge, there are still charges in conductors. There is no net charge within a conductor; any net charge resides on the surface(s).
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1 vote

Charge kept inside a conducting sphere not at its center

This is a conductor and in static equilibrium all charges are on the surfaces of the conductor, and the $\vec E$ inside the conductor is $0$. This follows because, if there’s an $\vec E$-field inside, ...
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