Derive the formula for the resistance of two resistors in parallel. Include schematic

Question

Derive the formula for the resistance of two resistors in series. Include a schematic

Answer 1

if two resistors are in parallel the voltage across them is same

therefore

  V=I*R   (R=Equivalent resistance)

V=I1*R1

I1=V/R1

V=I2*R2

I2=V/R2

I=I1+I2

V/R=V/R1+V/R2

=> 1/R=1/R1+1/R2

If two resistors in series then current is same in both resistors.

V1=I*R1

V2=I*R2

V=V1+V2

I*R=I*R1+I*R2

R=R1+R2

Answer 2

Do it yourself. Geez.  

Answer 3

Assume...

../---WW-----\

o.......V.......o

..\----W-------/

Where WW is resistance R > r the W resistance.  V is the voltage across both r and R in parallel.  Then V = I'R' where V is the voltage across an equivalent circuit with equivalents R' resistance and I' current.

From the circuit V = iR and V = Ir because the voltage is the same across each resistance; such is the nature of parallel circuits.

Now I' = i + I, the two branch currents i and I feed into the one equivalent current I'.  Then I' = V/R + V/r = V(1/R + 1/r) = V(r + R)/rR; so that I'(rR/(r + R)) = V = I'R" and, ta da, R' = rR/(r + R)) which is the equivalent resistance you are looking for.

Key to working this is to recognize that the parallel resistors see the same voltage across them; and that the currents in each branch add up to the total current of the circuit.

Answer 4

Let resistors be Ra and Rb

1/R = 1/Ra + 1 / Rb

1/R = (Rb + Ra) / Ra Rb

R = Ra Rb / (Ra + Rb)

Cannot draw on this site.