Question

1. A plece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is R. then the ratio R/R is - (a) 1/25 (b) 175 c) 5 (d) 25​

Answer 1

Answer:

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Hint: Resistance is proportional to the length of wire and connecting n resistors parallel reduces the resistance 1n times.

Formula used: R=ρlA ; 1Reff=1R1+1R2+...+1Rn

Complete step by step answer:

We know the resistance of a wire is proportional to the length of the wire (∵R=ρlA)

Let’s say the total resistance of the wire before cutting wasR.

When we cut the wire into 5 parts, each one has a length of l5 and all other parameters remain unchanged.

So the resistance of each new segment isR5.

Connecting them parallel gives a network as shown in fig1.

Now since the question asks for The effective resistance of this combination, lets recollect that the effective resistance of a parallel combination of n resistances - R1,R2,R3, . . . Rn is given by: 1Reff=1R1+1R2+...+1Rn

So if R′ is the resistance of this new combination, then:

1R′=1R5+1R5+1R5+1R5+1R5

Simplifying this expression a bit gives us:

1R′=5R5

R′=R25

So the ratio RR′=25 , which is the required answer.

Additional Information:

Here the wire was cut and hence the area remained constant. In cases where wire is stretched to a new length, resistance is not proportional to length.

In cases of stretching, since its volume V that is conserved, The Resistance R can be related with length l as R=ρlA=ρl×lA×l=ρl2V.

Thus in case of stretching, the resistance varies as R∝l2

Note: We can answer this question quicker by remembering some special cases in parallel. When we have n equal resistances each of resistance R connected parallel, the effective resistance becomes Rn.

Here since we have 5 resistances, each of Resistance R5

R′=R55=R25

So RR′=25

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Answer 2

Answer

Correct option is

D

25

We know,

1.When a resistance is divided into n identical resistances, each of resistance (say R') become  

n

R

​

.

2.When n identical resistors each of resistance R', are connected in parallel, the equivalent resistance is  

n

R  

′

 

​

 

The resistor R ,when divided into five parts become resistors of resistance  

5

R

​

each.

The recombination of five resistors, each of resistance    

5

R

​

, gives a resistor of equivalent resistance R' of value  

5

5

R

​

 

​

=  

25

R

​

 

Hence,

R  

′

 

R

​

=  

1

25

​