Question

An experiment is performed for comparing EMF of two cells using a potentiometer. For 1st cell, balancing length was achieved at 200 cm and for 2nd cell it was 150 cm. If least count of measurement of length of potentiometer wire is 1cm, the percentage error in the ratio of emf of two cells is :

• A. \(\frac{8}{7}\)
• B. \(\frac{7}{6}\)
• C. \(\frac{5}{6}\)
• D. \(\frac{3}{2}\)

Hint:

When dealing with errors, remember the simple rules: for addition/subtraction, absolute errors are added; for multiplication/division, fractional (or percentage) errors are added.
A larger measurement generally has a smaller percentage error, which is why longer balancing lengths are desirable in a potentiometer experiment for better accuracy.

Answer 1

The Correct Option is B

To find the percentage error in the ratio of EMF of two cells using a potentiometer, we begin by using the principle that the EMF of a cell is directly proportional to the balancing length on the potentiometer. The formula relating the EMFs and balancing lengths is:

\(E_1/E_2 = L_1/L_2\)

Given: 

• Balancing length for the first cell, \(L_1 = 200 \, \text{cm}\)
• Balancing length for the second cell, \(L_2 = 150 \, \text{cm}\)
• Least count of length measurement = \(1 \, \text{cm}\)

First, calculate the ratio of EMFs:

\(\frac{E_1}{E_2} = \frac{200 \, \text{cm}}{150 \, \text{cm}} = \frac{4}{3}\)

The percentage error in the measurement is calculated using the formula for percentage error in ratios:

\(\Delta(R)/R \times 100 = (\Delta(L_1)/L_1 + \Delta(L_2)/L_2) \times 100\%\)

where \( \Delta(L_1) = \Delta(L_2) = 1 \, \text{cm} \) and \( R = E_1/E_2 \).

Calculate the percentage error:

\(Percentage\ Error = \left(\frac{1}{200} + \frac{1}{150}\right) \times 100\)

\(= \left(\frac{1.0}{200} + \frac{1.0}{150}\right) \times 100\)

\(= \left(\frac{150 + 200}{30000}\right) \times 100\)

\(= \frac{350}{300} = \frac{7}{6}\)

Therefore, the percentage error in the ratio of the EMFs of the two cells is \(\frac{7}{6}\%\).

Thus, the correct answer is \(\frac{7}{6}\).