Electric current density in conductors not resembling a wire

Let's say you have a square plate with a voltage applied across two opposite corners, connected by the hypotenuse.

Presumably, the electric current density distribution throughout the plate would be such that it would follow the path of minimal resistance, i.e. through the hypotenuse.

If you had a way of measuring the current density at a point in the plate (e.g. IR thermography) would the readings show that the current density has a spatial distribution throughout the plate and indeed even a change through the thickness?

What factors determine how the charge carriers are distributed in a conductive medium?

Mapes
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Let's say you have a square plate with a voltage applied across two opposite corners, connected by the hypotenuse.

Presumably, the electric current density distribution throughout the plate would be such that it would follow the path of minimal resistance, i.e. through the hypotenuse.

If you had a way of measuring the current density at a point in the plate (e.g. IR thermography) would the readings show that the current density has a spatial distribution throughout the plate and indeed even a change through the thickness?

What factors determine how the charge carriers are distributed in a conductive medium?

The current would be maximum along the hypotenuse, but would also be spatially distributed across the entire plate. Under electrostatic conditions in conductive media, the voltage $\phi$ obeys the equation $\nabla^2\phi=0$ (http://en.wikipedia.org/wiki/Laplace%27s_equation" [Broken]). The current is then $\bold{J}=\sigma\bold{E}$, where the electric field $\bold{E}=-\nabla\phi$. As it happens, this equation also describes heat flow, so it might be useful to visualize the equivalent problem of maintaining a temperature difference between the two opposite corners. Heat will flow through the entire plate, but mostly in the area near the hypotenuse. The other two corners will stabilize at half the voltage difference (equivalently, half the temperature difference).

If the plate is thin enough, there won't be much current variation through the thickness, and if both corner connections are made evenly along the corner edge, there'll be no thickness variations of current at all. Does this make sense?

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Yes it does, thanks for the clear response! :)

Is an intuitive interpretation of $\nabla^2\phi=0$ in the context of electrical voltage, that the net or sum change in voltage equals zero? IOW the vector calculus form of Kirchoff's Second Law for circuits.

Can you recommend a good online introductory source (or failing that, a book) for me to read up on this? My background is in aerospace engineering.

Mapes
Is an intuitive interpretation of $\nabla^2\phi=0$ in the context of electrical voltage, that the net or sum change in voltage equals zero? IOW the vector calculus form of Kirchoff's Second Law for circuits.