Re: [Phys-L] Electric Current

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding:

> I have a question about electric current.  I generally think of
> the derivative of something with respect to time as the rate of
> change of that something.  In the case of a simple circuit such
> as a resistor connected across a battery, the current is
> typically written as I=dq/dt.  The question is, what is the q
> that is changing?  In the case of a different circuit, where
> charge is piling up on the plate of a capacitor, it's just the
> charge on that capacitor plate; no problem there.  But in the
> case of a resistor connected across a battery, nothing has the
> charge q that is changing with time.  We write the expression
> as if q were a state variable, but nothing has the
> corresponding state.  It seems that we would be more consistent
> if we used a script delta in front of the q the way we do for
> infinitesimal amounts of work and heat in thermodynamics.
>
> The best I can come up with is that q represents the total
> amount of charge that has crossed a boundary in one of the two
> possible directions, where the boundary is the point in the
> circuit to which the current pertains, and for charge that has
> made it all the way around the loop, double counting is
> allowed; each time that charge passes through boundary it
> contributes to the q.  What are your thoughts on this?

That's pretty much it.  The scalar I = dq/dt is actually the surface integral of the component of the current density vector normal to a surface integrated over a whole surface of interest, across which current is flowing and across which charge is being transferred.  In the case of any insulated wire, whether it is connected to a resistor or anything else, the relevant surface is any surface that cuts the wire somewhere along the wire.  The all cases the value of I = dq/dt is the net total rate at which charge is *transferred* across the monitoring surface.  In general it is both a property of the current distribution *and* the defined surface across which the charge transfer is being monitored.

The time integral of I, i.e. q (or Q), is the net amount of charge transferred across the surface during the time interval over which the transfer took place.  This q value may or may not have anything to do with a net charge on an object, and won't if the object has a second (or third) terminal connected to it with any other current(s) flowing across another conducting surface as well as the one monitored by the original surface.  In order for the time integral of I to certainly measure net change in accumulated charge on an object that object needs to be effectively surrounded by the surface in question so all kinds of charge transfers into and out of the object are being monitored and counted by the surface integral over the monitoring surface.

Dave Bowman