Problem 31 - Master Resource Book in JEE Main Physics

Question

In comparison of emf's of two cells using potentiometer, the balanced length for batteries having emf \(E_{1}\) and \(E_{2}\) are \(60 \mathrm{~cm}\) and \(20 \mathrm{~cm}\), respectively. Then (a) \(\frac{E_{1}}{E_{2}}=3\) (b) \(\frac{E_{1}}{E_{2}}=\frac{1}{3}\) (c) \(\frac{E_{1}}{E_{2}}=60\) (d) \(\frac{E_{1}}{E_{2}}=20\)

Step-by-Step Solution

Answer

The answer is (a) \(\frac{E_1}{E_2} = 3\).

Step 1: Understanding Potentiometer Principle

A potentiometer compares the electromotive force (emf) of two cells by comparing the length of wire to which each cell is balanced. The formula used is \(\frac{E_1}{E_2} = \frac{l_1}{l_2}\), where \(l_1\) and \(l_2\) are the balanced lengths for \(E_1\) and \(E_2\) respectively.

Step 2: Assign Values to Variables

From the problem statement, \(l_1 = 60\) cm and \(l_2 = 20\) cm. These values correspond to the balanced lengths for \(E_1\) and \(E_2\).

Step 3: Set Up the Ratio Formula

Using the formula \(\frac{E_1}{E_2} = \frac{l_1}{l_2}\), substitute the values. This becomes \(\frac{E_1}{E_2} = \frac{60}{20}\).

Step 4: Simplify the Ratio

Simplify the fraction \(\frac{60}{20}\) to its simplest form. \(\frac{60}{20} = 3\).

Step 5: Final Answer Verification

Thus, the ratio \(\frac{E_1}{E_2} = 3\). Check the given options to find the correct answer that matches this result, which is option (a).

Key Concepts

EMF Comparison

The potentiometer is a fundamental tool used to compare the electromotive force (EMF) of two cells without drawing any current from them. This non-invasive characteristic makes it highly accurate, as it avoids fluctuations and variations in potential. Here’s how it works: - A potential gradient is set up along a uniform wire. This means the voltage per unit length of the wire is constant across its span. - When you connect a cell with unknown EMF to the circuit, you adjust the length of the wire until no current flows, achieving a 'null point'. This null measurement indicates the potential difference across the wire equals the EMF of the cell. By using this method, students can effectively compare the EMFs of two different cells. Potentiometers serve as a powerful instructional apparatus because they provide an intuitive understanding of voltage and its distribution across a circuit.

Balanced Length

Balanced length refers to the specific length of wire over which the potential difference precisely matches the EMF of the cell being tested. It is central to the working of a potentiometer and helps to understand how EMFs can be compared. To find the balanced length, you adjust a sliding contact along the potentiometer wire until the galvanometer—an instrument that measures small currents—shows zero deflection. At zero deflection, the potential across this segment of the wire equals the EMF of the cell. Thus, balanced length is crucial: - It ensures that the potential difference measured is ideally equal to the EMF of the cell, ensuring accuracy. - Long balanced lengths typically correspond to stronger EMFs, while shorter lengths suggest weaker EMFs, assuming a constant potential gradient. Mastering the concept of balanced length allows students to solve exercises involving EMF comparisons with confidence.

Ratio of EMF

The ratio of EMF is determined by comparing the balanced lengths obtained for two cells under the same conditions. This is expressed mathematically using the formula:\[ \frac{E_1}{E_2} = \frac{l_1}{l_2} \]In simpler terms, the EMF of cell 1 over the EMF of cell 2 is equal to the length of wire where cell 1 balances over the length where cell 2 balances.Let’s break down what this means:- If cell 1 balances at 60 cm and cell 2 at 20 cm, the ratio is 3:1, meaning cell 1’s EMF is 3 times greater than cell 2’s.- This relationship is direct and relies heavily on maintaining uniform wire conditions. Any changes in wire resistance or uniformity can lead to inaccurate results.Understanding this ratio helps in verification of given options during exercises and ensures that theoretical knowledge can be applied practically.