Judge the equivalent resistance when Die following are connected in parallel— (a) 1 Ω , 106 Ω , (b) 1 Ω , 103 Ω and 106 Ω.
Answers
(a) 1 ohm and 106 times ohm R1 = 1 ohm R2 = 106 times = 1000000 ohm Total resistance in parallel 1/R = 1/R1 + 1/R2 = 1/1 + 1/1000000 = 1000000+1/1000000 = 1000001/1000000 = 1/R = 1/1 ohm (approx) Resistance = 1 ohm (approx)
(b) R1 = 1 ohm R2 = 103 ohm R3 = 106 ohm Total resistance in parallel
Answer:
Part 'a':
R = 0.99 ohms
Part 'b':
R = 0.914 ohms
Explanation:
For parallel combination, equivalent can be calculated by:
1/R = 1/R1 + 1/R2 + 1/R3 + ...
Where,
'R' is the equivalent or net resistance,
'R1' is the resistance of first resistor,
'R2' is the resistance of second resistor,
'R3' is the resistance of third resistor,
And so on,
Now to compute and compare the equivalent resistance in each case:
Part 'a':
We are given:
R1 = 1ohm
R2 = 106ohms
Thus, for equivalent resistance:
1/R = 1/R1 + 1/R2
1/R = 1/1 + 1/106
Thus;
1/R = 1+1/106
1/R = (106+1)/106
Or, reciprocating the equation;
R = 106/107
Or,
R = 0.99 ohms.
Part 'b':
In this case,
R1 = 1 ohm
R2 = 103 ohm
R3 = 106 ohm
Now, again, for equivalent resistance;
1/R = 1/R1 + 1/R2 + 1/R3
1/R = 1/1 + 1/103 + 1/106
1/R = 1.109
Or,
R = 0.914 ohms
Thus, resistance in part 'b' is lesser as compared to the resistance in part 'a', as for parallel combination, greater resistances reduces the final resistances.