Question

When connected to a battery, which arrangement of three equal resistors would produce the most current?1 All in Parallel2 All in Series 3All arrangements produce the same current. 4Two in parallel one in series 5Two in Series one in parallel

Answer 1

We know that the resistances in series always get added up i.e the total resistanec Rs is given by

RS =R1+R2+R3 ........eqn 1

Now according to question, R1=R2=R3=(say) R

Hence, RS = R+R+R = 3R ............eqn 2

For resistances in parallel,

The total Resistance, Rp is given by'

1/Rp = 1/R1+1/R2+1/R3 = 1/R+1/R+1/R= 3/R

1/RP = 3/R

Taking reciprocal on both sides;

Rp = R/3 .................eqn 3

If the two resistances are in parallel and the third one is in series with two,

Firstly we will calculate total resistance of the resistors in parallel; i.e

1/R' = 1/R+ 1/R

1/R' = 2/R

Taking reciprocal on both sides

R' = R/2

Now this combined resistance (of two resistances in parallel,) i.e R' is in series with R

Hence, total resistance of the combination,

Rps = R' + R = R/2+R = R(1+1/2) = 3R/2

i.e Rps = 3R/2 ...............eqn 4

Taking the fourth combination, if the two reistances are in series and the third resistance is in parallel to both of them;

Again, we will calculate the resistance for the two resistances in series; i.e R'

R' = R+R = 2R

Now, this R' is in parallel combintion with R

Hence, total Resistance Rsp of the entire combination,

1/Rsp = 1/R' + 1/R

1/Rsp = 1/2R+1/R

1/Rsp = 1/R(1/2+1)

1/Rsp = 3/2R

Taking reciprocal on both sides;

Rsp = 2R/3 .............eqn 5

We know that according to Ohm's law,current I= V/R

where V is the potential difference and R is the total resistance of the circuit

In other words we can say that I is inversely proprtional to R

i.e greater the value of R,lesser is the current.

Among eqn 2 to eqn 5, eqn 3 corresponds to the least resistance.

Hence,there would be maximum value of current through this combination

i.e when all the resistors are in parallel, they would have maximmum current through them.

Therefore, OPTION 1 IS THE CORRECT ANSWER