consider a metre bridge whose length of wire is 2m. A resistance of 10 ohm is connected across one gap of the metre bridge and an unknown resistance is connected across another gap. When these resistances are interchanged, the balance point shifts by 50 cm. What is the value of unknown resistance?
Answers
Answer:
Case 1 :- Without interchanging
⇒
R
2
=
100−l
l
⇒R=
l
2(100−l)
−(1)
Case 2 :- After interchanging
2
R
=
100−(20+l)
l+20
- (2)
⇒ Equating R from (1) and (2)
we get L = 40 cm
⇒R=
40
2(100−40)
=
40
2×60
cm
⇒R=3Ω
Given: Length = 2m
Resistance R₁ = 10Ω
The shift in the balance point = 50cm
To find: The value of unknown resistance
Solution:
Let the value of the unknown resistance be R
The balance point of when the resistance R is connected to the other gap be L. According to the Wheatstone bridge,
R₁/R₂ = L/200-L
10/R = L/200-L
10(200 - L) = RL
2000 - 10L = RL
2000 = RL + 10L
2000 = L(R + 10)
L = 2000/(R + 10) ( Let this be the equation 1)
Now, when the Resistance R is at the other end, then the balance point will be at X,
R/10 = X/200-X
R(200-X) = 10X
200R - RX = 10X
200R = 10X + RX
200R = X(10 +R)
X = 200R/(10 +R) (Let this be equation 2)
The shift in the balance point is 50cm, that L-X = 50cm
{ 2000/(R + 10)} - {200R/(10 +R)} = 50
1800/10+R = 50
180/10+R = 5
180 = 50 + 5R
130 = 5R
R = 26Ω
Therefore, the value of the unknown resistance will be 26Ω