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In a meter bridge, the wire of the length 1m has a non-uniform cross- section such that, the variation dRdl of its resistance R with length l is dRdl1l . Two equal resistances are connected as shown in figure. The galvanometer has zero deflection when the jockey is at the point P. What the length AP is:
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A.0.2mB.0.3mC.0.25mD.0.35m

Answer
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Hint: According to wheat stone’s bridge when the galvanometer shows zero deflection, we can use balanced condition. Therefore use balanced conditions. Then take resistances of point AD and DB as R1 and R2, since wire is not uniform. Apply expression, given in question. Replace R by R1 and R2, we get two equations. Integrate both the equation and equate it to get length of AD wire.

Complete answer:
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Question stated that dRdl is varying like,
 dRdl1l
Total length of wire AB is 1m which has non- uniform cross-section. Thus, for the given wire:
dR=Cdll........(1)
Where, C = constant
It is mentioned that the resistance attached is the same. Let R1 be the resistance of part AD and R2 be the resistance of part DB. According to balanced condition of wheat stone’s network,
RR=R1R2R1=R2......(2)
Now use equation (1)
Put R=R1 in equation (1), we get
dR1=Cdll
Integrate, we get
dR1=C0ldllR1=C0ll12dl=C.2l
Similarly for R=R2
Therefore, dR2=C1ll12dl=C(22l)
According to condition given in equation (2), we get,
C2l=C(22l)2l=22l4l=216l=4l=416l=14l=0.25m
Hence the length of AD is 0.25m.

Therefore option (C) is the correct option.

Note:
Meter Bridge is the modification of the wheat stone’s network. Wheatstone’s network is used to determine the value of the unknown resistance. Since the length of the wire used is one meter therefore it is called a meter Bridge. The balancing condition for Wheatstone’s network can also be determined by using ohm’s law.