# Is current density always constant for an isotropic conductor of any shape?

I came across this question online, and I was just thinking how to find the transverse voltage ΔV. If we divide the large semicircular strip into very thin semicircular strips of width dx, then the outermost strip will have more resistance than the innermost one, due to its longer length. But does this mean that the current in the outer strip will be less than the current in the inner one? In that case, there shouldn't be a transverse voltage ΔV developed, and the current density would be non-uniform throughout the conductor. One way I could explain the transverse voltage ΔV is to say that the current density is constant, and hence the current in each small strip would be constant (due to uniform cross sectional area). This would explain the transverse voltage ΔV. Is my second model, in which I say current density is uniform throughout an isotropic conductor, correct?

But does this mean that the current in the outer strip will be less than the current in the inner one? - yes.

The $$\Delta V$$ effect is real and has been measured. It is sometimes called the Hall effect without a magnetic field being present.

The charged particles move in a circular path which means that there must be a force acting on them to produce a centripetal acceleration, $$qE = \dfrac{mv^2}{r}$$.
$$E$$ is the radial electric field which in turn means that there must be a potential gradient and hence a transverse potential difference $$\Delta V$$ ("Hall" voltage) across the conductor.
Using the definition of resistivity $$\rho$$, $$R=\rho\dfrac{\ell}{A}$$ and the Drude model $$I = nqAv_{\rm drift}$$ for a semicircular element and then integrating across the whole conductor one can show that $$\Delta V \propto I^2$$.

• It is a very well known effect, measured way before 2019 (the reference you link). This is why a lot of care is usually taken when measuring the resistivity of a sample using the 4 probes method. The current contacts must ensure the current density is as homogeneous as possible, otherwise the value retrieved for $\rho$ gets wrong if using the formula $R=\rho \frac{l}{A}$. But very good answer, +1 from me. Commented Mar 26, 2023 at 11:40
• So the current in the outer strip is less than the current in the inner strip (because of the difference in resistance), but still, there would be a transverse voltage purely due to centripetal acceleration? Also, that would imply that the current density is not necessarily uniform in an isotropic conductor, am I right?
– R H
Commented Mar 26, 2023 at 15:04
• Does this also mean that whenever there's a bend in a thick wire, there's a transverse voltage developed across the cross section of the wire due to centripetal acceleration of the electrons?
– R H
Commented Mar 27, 2023 at 17:48
• @RH : Yes, there's a transverse voltage at every bend in a wire. Commented Mar 14 at 10:47