State the working principle of potentiometer. Explain with the help of circuit diagram. How the potentiometer is used, to determine the internal resistance of the primary cell?

Principle of Potentiometer

The potentiometer class 12 operates on the principle that the potential difference across any part of a uniform wire is directly proportional to the length of the wire, provided a constant current flows through it. Mathematically, this relationship can be expressed as:

V ∝ l

Where:

• V = Potential difference across the wire
• l = Length of the wire segment

This means that if the current remains constant, the potential difference across the wire depends on the length of the wire. The constant factors include the resistivity (ρ) of the wire and its cross-sectional area (A). The equation can be derived from Ohm’s law:

V = I × R

Where R is the resistance of the wire, given by:

R = ρ × l / A

Therefore, the potential difference V can be written as:

V = I × ρ × l / A

For simplicity, we can express this as:

V = K × l

Where K is a constant that depends on the material and the dimensions of the wire. This relationship shows that potential difference (V) is directly proportional to the length (l) of the wire, assuming other factors (I, ρ, A) remain constant. This is known as the principle of potentiometer class 12.

Determination of Internal Resistance of a Primary Cell Using Potentiometer

To determine the internal resistance of a primary cell, we use a potentiometer class 12 setup, which involves two circuits: a primary circuit (for the cell under test) and a secondary circuit (with the potentiometer wire).

Procedure:

1. Primary Circuit: A primary cell with an EMF 'E' and internal resistance 'r' is connected in series with a key 'K' and a rheostat (Rh).
2. Secondary Circuit: A cell of known EMF 'E1' and internal resistance 'r' is connected in series with a galvanometer (G) and a jockey (J). A resistance box (R) and a plug key (K1) are also included.
3. Initial Setup: Close the plug key 'K' and open key 'K1'. Adjust the jockey on the potentiometer wire until the balance point is obtained at length 'l1'. The potential difference for this condition is given by:

E1 = I × ρ × l1 → (1)

1. After Closing Both Keys: Now close both the keys, 'K' and 'K1'. Introduce a known resistance 'R' from the resistance box. Adjust the jockey to find the new balance point 'l2'. The terminal potential difference for this case is:

V = I × ρ × l2 → (2)

1. Divide Equations (1) and (2): By dividing the two equations, we get:

E1 / V = l1 / l2 → (3)

1. Substitute the Value of V: Using Ohm’s law, V = I × R, and for the EMF of the cell, E1 = I × (r + R), we get:

E1 / V = (r + R) / R → (4)

1. Relating Equations (3) and (4): From equations (3) and (4), we can equate:

(l1 / l2) = (r + R) / R

1. Solve for Internal Resistance (r): Rearranging the terms, we get:

r = R × (l1 / l2 - 1)

This gives the internal resistance of the primary cell, which can be calculated from the known values of resistance (R) and lengths (l1 and l2).

Conclusion

The potentiometer is an accurate device for determining the internal resistance of a primary cell. By comparing the balance points in the potentiometer circuit, we can calculate the internal resistance with high precision. This method is widely used in potentiometer class 12 experiments and provides a clear understanding of the behavior of electrical circuits.