A question on electric current

Question

In my physics book the definition of current is

If a charge $∆Q$ crosses an area in time $∆t$, we define the average electric current through the area during this time as
$I=frac{∆Q}{∆t}$. The current at time $t$ is $I=lim_{∆tto0} frac{∆Q}{∆t}=frac{dQ}{dt}$

And the definition of current density is

If $∆I$ is the current through the area $∆S$, the (same as the top) ...  $J=frac{dI}{dS}$

My question is what is the difference between the 1st definition area and the 2nd one?

GiorgioP · Accepted Answer

The total current $I$ through a surface is a global quantity (with respect to the surface). The current density $bf J$ is a vector and it is a function of the point, in general. The integral over the surface of $bf J cdot n$ is the total current $I$, $bf n$ being a unit vector orthogonal to each surface element. If the surface is so small that variations of $bf J$ over the surface can be neglected, and if the surface is orthogonal to the direction of $bf J$ there is a relation between the small surface area $Delta S$, $|{bf J}|$, and the total current through the same surface: $I=|{bf J}| Delta S.$ If the unit vector makes an angle $theta$ with $bf J$ the previous formula becomes:
$$
I=|{bf J}| Delta Scos theta.tag{1}
$$
The right-hand side can be interpreted as the product of the modulus of the density current by the cross-sectional area of the surface element.
Summarizing: if $I$ is the total current through a small surface element $Delta S$ and we want to connect it to an underlying current density ${bf J}$ the surface element must be the same, in general. However, formula $(1)$ tells us that many surface areas, oriented to give the same cross-sectional area, will give the same total current.

A question on electric current

One Answer

Add your own answers!

Ask a Question