In apotentiometer arrangement `E_(1)` is the cell establishing current in primary circuit `E_(2)` is the cell to be measured . AB is the potentiometer wire and G is a galvanometer . Which of the following are the essential condition for balance to be obtained.

Options

A. The emf of `E_(1)` must be greater than the emf of `E_(2)`
B. Either the positive terminals of both `E_(1)andE_(2)` or the negative terminals of both `E_(1)andE_(2)` must be joined to one end of potentiometer wire.
C. The positive terminals of `E_(1)andE_(2)` must be joined to one end of potentiometer wire.
D. The resistance of G must be less than the resistance of AB.

Answer

To determine the essential conditions for balance in a potentiometer arrangement, we need to analyze the setup and the requirements for achieving a balanced condition. Here’s the step-by-step solution:

### Step 1: Understand the Potentiometer Setup
In a potentiometer setup, we have:
- `E1`: The cell that establishes the current in the primary circuit.
- `E2`: The cell whose voltage we want to measure.
- `AB`: The potentiometer wire where the voltage drop occurs.
- `G`: The galvanometer used to detect the balance point.

### Step 2: Condition for Balance
For balance to be obtained in the potentiometer:
- The voltage across the potentiometer wire (V_AB) must be equal to the voltage of the cell being measured (E2). This means that the potential difference across the length of the wire must match the voltage of the cell.

### Step 3: Voltage Relationship
The relationship can be expressed as:
- \( V_{AB} \geq E2 \)
This means that the voltage across the potentiometer wire must be greater than or equal to the voltage of the cell being measured.

### Step 4: Current in the Circuit
The current established by the primary cell (E1) must be sufficient to create a voltage drop across the potentiometer wire that can equal or exceed E2. Therefore:
- The electromotive force (emf) of E1 should be greater than E2 for the potentiometer to function correctly.

### Step 5: Connection of Terminals
The terminals of the cells must be connected properly:
- The positive terminal of E1 and E2 must be connected to one end of the potentiometer wire (or the negative terminals can be connected to the same end). This ensures that the potentials are referenced correctly.

### Step 6: Resistance Considerations
The resistance of the galvanometer (G) should be less than the resistance of the potentiometer wire (AB):
- This is to ensure that when the galvanometer is connected, it does not draw significant current that would affect the balance condition.

### Conclusion
The essential conditions for balance in a potentiometer arrangement are:
1. \( V_{AB} \geq E2 \)
2. The emf of E1 should be greater than E2.
3. The terminals of E1 and E2 must be connected correctly.
4. The resistance of the galvanometer must be less than the resistance of the potentiometer wire.