Electric field in an arbitrary number of dimensions

[spaghetti3451]

I am looking to use Gauss's law to find the electric field in $1+1$ dimensional spacetime:

$\int \vec{E}\cdot d\vec{A}=\frac{Q}{\epsilon_{0}}$

Now, for a point charge in $1+1$ dimensional spacetime, the Gaussian surface is the two endpoints (a distance $r$ away from the point charge) along which the electric field points outwards. How do I account for $d\vec{A}$ of the two endpoints?

 

[mfb]

The integral gets a sum over the two endpoints.

 

[spaghetti3451]

I know that, but I am finding it difficult to form a quantitative value for the infinitesimal area of the two points.

 

[mfb]

Just add the two electric field strengths? Your field will have different dimensions anyway, so it fits.

 

[spaghetti3451]

I don't exactly understand what you mean when you say that the field will have different dimensions.

 

[mfb]

Sorry, I didn't mean the field itself, I meant the equation. To work, $\epsilon_0$ needs different units, which means the whole equation has different units.

 

[spaghetti3451]

Thanks!

 

[robphy]

The more interesting question is the magnetic field in an arbitrary number of dimensions...

 

[spaghetti3451]

Well, $\nabla\cdot{\vec{B}}=0$ generalises to $\partial_{\mu}B^{\mu}=0$, I suppose, which means that $B$ has inverse dimensions of the length in any spacetime.

Am I right?

 

[mfb]

But how is it generated? ;)

 

[spaghetti3451]

It's generated by a changing electric field, which comes from Faraday's Law.

$\epsilon_{ijk}\partial_{i}E_{i}=-\partial_{t}B_{i}$ is Faraday's law in 3 dimensions.

I guess the epsilon symbol ought to have more (or less) spatial indices as the number of spatial dimensions increases (or decreases)?

 

[mfb]

There is no obvious generalization of this law to N dimensions.

 

[spaghetti3451]

Is this why the magnetic field cannot be generalised to more than 3 dimensions?

 

[robphy]

Crudely...

The electric field is "time-space part" of the [antisymmetric] field tensor, orthogonal to the observer's 4-velocity. In (n+1)-spacetime, it has n independent components. In 4+1, Ex, Ey, Ez, Ew.

The magnetic field is the remaining part... also orthogonal to the observer's 4-velocity. In (n+1)-spacetime, it has n(n-1)/2 independent components... Counting the number of strictly lower triangular space-space components. In 4+1, Bxy, Byz, Bzx, Bxw, Byw, Bzw (up to signs). The epsilon symbol or its equivalent will appear.

For n=3, the electric and magnetic field each have 3 components.