A resistance of is connected across one gap of a metre-bridge (the length of the wire is 100 cm) and an unknown resistance, greater than , is connected across the other gap. When these resistances are interchanged, the balance point shifts by 20 cm. Negl

Question

A resistance of $2 Ω$ is connected across one gap of a metre-bridge (the length of the wire is 100 cm) and an unknown resistance, greater than $2 Ω$ , is connected across the other gap. When these resistances are interchanged, the balance point shifts by 20 cm. Neglecting any corrections, the unknown resistance is [2007]

Smart Achievers · Accepted Answer

(1)

When the bridge is balanced, $x > 2 Ω$

Applying balanced Wheatstone bridge condition, $\frac{P}{Q} = \frac{R}{S}$

$\frac{R}{ℓ} = \frac{X}{100 - ℓ}$

$or, 100 R - R ℓ = ℓ X$ $or,$ $200 - 2 ℓ = ℓ X$

$∴$ $ℓ = \frac{200}{X + 2}$

When the resistances are interchanged, the jockey shifts 20 cm.

$∴$ $\frac{X}{ℓ + 20} = \frac{2}{80 - ℓ}$ $\Rightarrow 80 X - ℓ X = 2 ℓ + 40$

$or, 80 X = ℓ (X + 2) + 40$ $= (\frac{200}{X + 2}) (X + 2) + 40$ $= 200 + 40$

$∴$ $X = \frac{240}{80} = 3 Ω$

Solution

1 $3 Ω$

2 $4 Ω$

3 $5 Ω$

4 $6 Ω$

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Solution

1 3Ω

2 4Ω

3 5Ω

4 6Ω

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1 $3 Ω$

2 $4 Ω$

3 $5 Ω$

4 $6 Ω$