Problem on Parallel Networks

PARALLEL NETWORKS:

Problem 1: Figure shows three resistors, R1, R2 and R3 connected across each other, i.e. in parallel, across a battery source

of V volts.

In a parallel circuit:

(a) the sum of the currents I1, I2 and I3 is equal to the total circuit current, I, i.e. I =I1 +I2 +I3, and 

the source p.d., V volts, is the same across each of the

From       Ohm’s   law:

I1 = V/R1

,  I2 = V/R2 

,  I3 = V/R3 and I = V/R 

where R is the total circuit resistance. Since I =I1 + I2 + I3 

then 

V/R= V/R1+ V/R2+ V/R3 Dividing throughout by V gives:

This equation must be used when finding the total resistance R of a parallel circuit. For the special case of two resistors in parallel

Problem 2: For the circuit shown in Figure , determine (a) the reading on the ammeter, and (b) the value of resistor R2.

P.d. across R1 is the same as the supply voltage V.

Hence supply voltage, V =8 × 5=40V

(a) Reading on ammeter, I = V R3= 40/20=2A 

Current flowing through R2 =11−8−2=1A 

Hence, R2 = V/I2= 40/1=40 Ω

(a) The total circuit resistance R is given by 1/R= 1/R1+ 1/R2= 1/3+ 1/6 

1/R= 2 + 1/6= 3/6 Hence, R = 6/3= 2 Ω 

(b) Current in the 3 Ωresistance, I1 = V R1= 12/3= 4A 

Problem 3: For the circuit shown in Figure find (a) the value of the supply voltage V and (b) the value of current I.

(a) P.d. across 20 Ωresistor = I2R2 = 3× 20 = 60V, hence supply voltage V =60V since the circuit is connected in parallel. 

(b)Current I1 = V/R1= 60/10= 6A; I2 = 3A 

I3 = V/R3= 60/60= 1A

Current I =I1+I2+I3 and hence I =6+3+1=10A Alternatively,

1/R= 1/60+ 1/20+ 1/10= 1 + 3 + 6/60= 10/60 Hence total resistance R= 6010=6 Ω Current I = V/R= 60/6=10A

Problem 4: Find the equivalent resistance for the circuit shown in Figure

R3, R4 andR5 are connected in parallel and their equivalent resistance R is given by: 1/R= 1/3+ 1/6+ 1/18=6 + 3 + 1/18= 10/18

Hence R= 18/10=1.8 Ω

The circuit is now equivalent to four resistors in series and the equivalent circuit resistance =1+2.2+1.8+4=9 Ω