Question

Apply the following questions to both the circuit shown in Figure (a) (resistors in series) and the circuit shown in Figure (b) (resistors in parallel).

(a) Use Ohm's law with Kirchhoff's current and voltage laws to derive equations for currents i1, i2 and i (through Resistor R1, resistor R2, and the voltage source , respectively) in terms of R1, R2, and v.

(b) R1 and R2 can be interchanged with an equivalent resistor rith resistance R without changing the values of v and i. Show the derivation of an equation for the equivalent resistance R in terms of R1 and R2.

 

                       

                      (a)                                                                                        (b)

Answer 1

Ans

        

               fig a                                                        fig b

 

A) let us assume that current i , i1 and i2 is flowing through voltage source v , the resistor r1 and r2 respectively in fig a and b respectively.

In fig a i1=i2=i

using kvl in fig a 

V=i1*R1+i2*R2=i(R1+R2)..........................................................1

hence, i=i1=i2=V/(R1+R2)

in fig b

i=i1+i2

v=i1*R1=i2*R2=(i-i1)*R2........................................................2

i1*R1+i1*R2=i*R2

hence,i1=i*R2/(R1+R2)

v=i*R1*R2/(R1+R2).......................................................................3

i=v*(R1+R2)/R2*R1

i1=v/R1 (from eq. 2)

i2=v/R2 (from eq. 2)

 

B) We knoe that V=I*R

from the equation 1 it is clear that

Equivalent resistor R=R1+R2  (in series)

from equation 3 it is clear that

Equivalent Resistor R=R1*R2/(R1+R2)  (in parallel)

Answer 2

 

 

 

B). 

We knoe that V=IR

from the equation 1 it is clear that

Equivalent resistor R=r1+r2  (in series)

from equation 3 it is clear that

Equivalent Resistor R=r1*r2/(r1+r2)  (in parallel)