Step 1: Understanding the Concept:
A potentiometer measures the unknown electromotive force (e.m.f.) of a cell by finding a point on the potentiometer wire where the potential drop equals the cell's e.m.f.
The principle is that the potential drop across any length of a uniform wire is directly proportional to that length (\( E \propto l \)).
Step 2: Key Formula or Approach:
The basic potentiometer equation is:
\[ E = k \cdot l \]
where \( E \) is the e.m.f., \( l \) is the balancing length, and \( k \) is the potential gradient (voltage per unit length) of the wire.
When cells are connected in opposition, their net e.m.f. is the difference of their individual e.m.f.s.
Step 3: Detailed Explanation:
In the first case, only the cell with e.m.f. \( E_1 \) is connected. The balancing length is given as \( l_1 \).
Using the potentiometer principle:
\[ E_1 = k l_1 \quad \dots \text{(Equation 1)} \]
In the second case, another cell with e.m.f. \( E_2 \) is connected in opposition to \( E_1 \). Since \( E_1>E_2 \), the effective net e.m.f. is \( (E_1 - E_2) \).
The balancing length for this combination is given as \( l_2 \).
Applying the potentiometer principle again:
\[ E_1 - E_2 = k l_2 \quad \dots \text{(Equation 2)} \]
We need to find the ratio \( \frac{E_1}{E_2} \).
From Equation 2, we can isolate \( E_2 \):
\[ E_2 = E_1 - k l_2 \]
Substitute \( E_1 \) from Equation 1 into this expression:
\[ E_2 = k l_1 - k l_2 \]
\[ E_2 = k (l_1 - l_2) \]
Now, calculate the required ratio:
\[ \frac{E_1}{E_2} = \frac{k l_1}{k (l_1 - l_2)} \]
The potential gradient \( k \) cancels out:
\[ \frac{E_1}{E_2} = \frac{l_1}{l_1 - l_2} \]
Step 4: Final Answer:
The ratio \( E_1 : E_2 \) is \( \frac{l_1}{l_1-l_2} \).