In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure.

S. No.   R(Ω)       l(cm)
1.        1000         60
2.        100          13
3.        10            1.5
4.        1              1.0

Which of the reading is inconsistent?

Step-by-Step Solution

1. Understanding the Meter Bridge Principle

A meter bridge works on the principle of a balanced Wheatstone bridge. When the galvanometer shows zero deflection (null point), the ratio of the resistances in opposite arms is equal:

$$\frac{R}{X} = \frac{\ell}{100 - \ell}$$

Where:

• $$R$$ = Known resistance from the resistance box
• $$X$$ = Unknown resistance
• $$\ell$$ = Balancing length from the left end (in $$\text{cm}$$)
• For Reading 1: ($$R = 1000\,\Omega$$, $$\ell = 60\text{ cm}$$)
• For Reading 2: ($$R = 100\,\Omega$$, $$\ell = 13\text{ cm}$$)
• For Reading 3: ($$R = 10\,\Omega$$, $$\ell = 1.5\text{ cm}$$)
• For Reading 4: ($$R = 1\,\Omega$$, $$\ell = 1.0\text{ cm}$$)
• $$X_1 \approx 666.7\,\Omega$$
• $$X_2 \approx 669.2\,\Omega$$
• $$X_3 \approx 656.7\,\Omega$$
• $$X_4 = 99.0\,\Omega$$

Rearranging this formula to express the unknown resistance $$X$$ gives:

$$X = R \left( \frac{100 - \ell}{\ell} \right)$$

2. Testing Each Observation Row

Since the unknown resistance $$X$$ remains constant throughout the entire experiment, calculating its value for each row will reveal which reading is incorrect.

$$X_1 = 1000 \left( \frac{100 - 60}{60} \right) = 1000 \left( \frac{40}{60} \right) = \frac{4000}{6} \approx 666.67\,\Omega$$

$$X_2 = 100 \left( \frac{100 - 13}{13} \right) = 100 \left( \frac{87}{13} \right) \approx 100 \times 6.692 \approx 669.23\,\Omega$$

$$X_3 = 10 \left( \frac{100 - 1.5}{1.5} \right) = 10 \left( \frac{98.5}{1.5} \right) = \frac{985}{1.5} \approx 656.67\,\Omega$$

$$X_4 = 1 \left( \frac{100 - 1.0}{1.0} \right) = 1 \left( \frac{99}{1} \right) = 99.0\,\Omega$$

3. Identifying the Inconsistent Reading

Comparing the calculated values of $$X$$:

The values from observations 1, 2, and 3 are close to each other ($$\sim 660\,\Omega$$), whereas the value from observation 4 ($$99\,\Omega$$) is completely out of line. Therefore, reading 4 is inconsistent.

Final Answer

The inconsistent reading is 4.

The correct option matching this is (D).