Re: Series, Parallel, and Resistivity Equations

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

You can play the same game with the parallel plates capacitor
formula:
                           C=eps_o*A/d

Substitute A1+A2 for A (cutting charged plates) and you
have C=C1+C2 (two capacitors in parallel)

Substitute d1+d2 for d (inserting a neutral plate between the
two charged plates) and you have  1/C=1/C1+1/C2 (in seies).
Ludwik Kowalski

Tim Folkerts wrote:

> I don't know why it had never occurred to me before, but the equations for
> the series an parallel resistors are contained within the equation for
> resistivity.  The texts I just checked all do the traditional derivation
> using either equal currents or equal voltage differences, but none seem to
> appeal to this simple argument.
>
> Consider a resistor of uniform composition and diameter, with length L.
> Cut it into two pieces of length L1+L2=L, which will have resistances of
> R1 = (rho/A)L1 and R2 = (rho/A)L2.
>      So R1+R2  = (rho/A)(L1+L2) = (rho/A)L =  R
>
> Consider a resistor of uniform composition and diameter, with length L.
> Slice it lengthwise into two pieces of area A1+A2=A, which will have
> resistances of  R1 = (rhoL)/A1 and R2 = (rhoL)/A2.
>      So 1/R1 +1/R2 = A1 /(rhoL) + A2 / (rhoL) = (A1+A2) / (rhoL) = 1/R
>
> But the circuit doesn't care about the geometry of individual resistors, so
> any other resistors with R1 & R2 would behave the same.
>
> Surely this isn't a new idea, and it's not surprising, but it was new to
> me.  Does this seem like a reasonable way to introduce series & parallel
> resistance?  Or perhaps just use it as a way to reinforce the traditional
> derivation?
>
> Tim Folkerts

Tim Folkerts wrote:

> I don't know why it had never occurred to me before, but the equations for
> the series an parallel resistors are contained within the equation for
> resistivity.  The texts I just checked all do the traditional derivation
> using either equal currents or equal voltage differences, but none seem to
> appeal to this simple argument.
>
> Consider a resistor of uniform composition and diameter, with length L.
> Cut it into two pieces of length L1+L2=L, which will have resistances of
> R1 = (rho/A)L1 and R2 = (rho/A)L2.
>      So R1+R2  = (rho/A)(L1+L2) = (rho/A)L =  R
>
> Consider a resistor of uniform composition and diameter, with length L.
> Slice it lengthwise into two pieces of area A1+A2=A, which will have
> resistances of  R1 = (rhoL)/A1 and R2 = (rhoL)/A2.
>      So 1/R1 +1/R2 = A1 /(rhoL) + A2 / (rhoL) = (A1+A2) / (rhoL) = 1/R
>
> But the circuit doesn't care about the geometry of individual resistors, so
> any other resistors with R1 & R2 would behave the same.
>
> Surely this isn't a new idea, and it's not surprising, but it was new to
> me.  Does this seem like a reasonable way to introduce series & parallel
> resistance?  Or perhaps just use it as a way to reinforce the traditional
> derivation?
>
> Tim Folkerts