Electrical circuits and charge - caution, lengthy

I am a reading of the math group sci.math where a person asked the
following question.


Op's original post:


I am pretty familiar with first year calculus (including FTOC and
convergence of improper integrals).

I am now reading an EE circuit theory book where there is a common
calculus step in several derivations that I can not follow 100%.

For example:

i(t) = dq(t)/dt (i=current is time rate of change of q=charge)

Then book would say, therefore....

q(t) = int[from -inf to t] i(t)dt


This happens a lot in the book, where the "reverse of a derivative
formula" results in an improper integral. Now, it makes sense when I
think of things like this as "the charge that passed the point from
beginning of time to current time (where negative infinity is
beginning of time)," but what exactly is the calculus step taken? I
mean, how do we get an improper integral from the original derivative
formula (what manipulation or theory or reasoning do we use to do that
step)? What about issues of convergence?


Basically, what allows me to do such a step?

z(t) = dy(t)/dt ==> y(t) = int[from -inf to t] z(t)dt

(assume only that y(t) is differentiable on the interval considered)


My reply


The integral undoes the derivative.

Start with

i(t) = dq(t)/dt

and integrate both sides with respect to t

Int( i(t) dt ) = Int( dq(t)/dt dt)

From here, insert any limits of integration that are needed.

Int[-inf to t] i(t) dt = Int[-inf to t] dq(t)/dt dt

Int[-inf to t] i(t) dt = q(t) - lim(t --> -inf) q(t)

I am guessing that it must be the case that lim(t --> -inf) q(t) = 0,
which makes sense and the charge must be decaying.

Hope this helps,

Brian


Math professor reply


I am pretty familiar with first year calculus (including FTOC and
convergence of improper integrals).

I am now reading an EE circuit theory book where there is a common
calculus step in several derivations that I can not follow 100%.

For example:

i(t) = dq(t)/dt (i=current is time rate of change of q=charge)

Then book would say, therefore....

q(t) = int[from -inf to t] i(t)dt

I hope it's assuming somewhere that the limit of q(t) as t ->
-infinity is 0.

This happens a lot in the book, where the "reverse of a derivative
formula" results in an improper integral. Now, it makes sense when I
think of things like this as "the charge that passed the point from
beginning of time to current time (where negative infinity is
beginning of time)," but what exactly is the calculus step taken? I
mean, how do we get an improper integral from the original derivative
formula (what manipulation or theory or reasoning do we use to do that
step)? What about issues of convergence?


Basically, what allows me to do such a step?

z(t) = dy(t)/dt ==> y(t) = int[from -inf to t] z(t)dt

(assume only that y(t) is differentiable on the interval considered)

It's not true then...

The Fundamental Theorem of Calculus tells you, if z(t) is continuous
then
y(b) - y(a) = int_a^b z(t) dt. Assuming the limit of y(a) as a ->
-infinity
is L, that implies y(b) - L = int_{-infinity}^b z(t) dt (that improper
integral being defined as the limit of integrals from a to b as
a -> -infinity). So your formula is right if L happens to be 0, but
not
otherwise. If the limit of y(a) as a -> -infinity exist, then the
improper
integral doesn't exist either.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


I replied

Dr. Israel:

The limit of q(t) is most likely zero as t goes to -inf because it is
a charge, and those are known to decay to nothingness. So I believe,
in this particular context, it is right to do what the authors did.

Thanks,

Brian


Dr. Isreal replied


Electrical charges decay? That's news to me, not to mention
Maxwell...

Any real electrical circuit is unlikely to have existed from "time
immemorial".
It was set up at some finite time in the past, and anything that your
mathematical model says about it before that time is a fictional
extrapolation. In that context, there's no reason to assume a limit
of 0.


Is this really how things work? I know that in a circuit electrical
charge is constant, but is that how it works in real life? I'm
looking for the answer as I am not a physicist, so you might have to
use little words for me to understand.

Thanks,

Brian

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