Question

With suitable circuit diagram, show how emfs of 2 cells can be compared using a potentiometer?

Answer 1

Principle of a Potentiometer

A potentiometer works on the principle that the potential drop across any portion of a wire of uniform cross-section is directly proportional to the length of that portion, provided a constant current is flowing through the wire.
Mathematically, V ∝ l or V = kl, where 'k' is the potential gradient (potential drop per unit length).

Circuit Diagram and Components

The circuit is set up as shown in the image. It consists of two main parts:

1. Primary Circuit: This is the upper loop that provides a constant potential gradient across the potentiometer wire.
   Battery: A driver cell or main battery with a stable and known EMF (which must be greater than the EMFs of the cells being compared).
   Ammeter (A): To monitor the current in the primary circuit.
   Key (K): To switch the primary circuit on or off.
   Rheostat (Rh): A variable resistor used to adjust the current in the primary circuit and thus change the potential gradient.
   * Potentiometer Wire (AB): A long wire (often several meters) made of a material like Manganin or Constantan, having a uniform cross-sectional area and high resistivity.

2. Secondary Circuit: This is the lower loop used for the measurement.
   Cells E₁ and E₂: The two cells whose EMFs are to be compared.
   Two-way Key (K₁ and K₂): This key allows us to connect either cell E₁ or cell E₂ into the circuit at a time.
   Galvanometer (G): A sensitive instrument to detect the presence of current. The goal is to find a point where it shows zero deflection (the null point).
   Jockey (J): A sliding contact that can be moved along the potentiometer wire AB to find the null point.

Important Connection: The positive terminals of the driver Battery, cell E₁, and cell E₂ are all connected to the same point A (the high potential end of the potentiometer wire). The negative terminals of E₁ and E₂ are connected to the two-way key.

Procedure

1. Setup: Assemble the circuit as shown in the diagram. Ensure all connections are tight.
2. Establish Potential Gradient: Close the key K in the primary circuit. Adjust the rheostat Rh to allow a constant current to flow through the potentiometer wire AB.

3. Find the Balancing Length for Cell E₁:
   Insert the plug into the gap for key K₁, which brings cell E₁ into the secondary circuit.
   Gently tap the jockey at different points along the wire AB. Find the point J₁ on the wire where the galvanometer (G) shows zero deflection. This is the null point or balancing point for cell E₁.
   * Measure the length from point A to the null point J₁. Let this balancing length be AJ₁ = l₁.

4. Find the Balancing Length for Cell E₂:
   Without changing the setting of the rheostat (Rh), remove the plug from key K₁ and insert it into the gap for key K₂. This brings cell E₂ into the circuit.
   Again, slide the jockey along the wire AB to find the new null point J₂ where the galvanometer shows zero deflection.
   * Measure the length from point A to this new null point J₂. Let this balancing length be AJ₂ = l₂.

Theory and Calculation

Let 'k' be the potential gradient (potential drop per unit length) along the potentiometer wire AB.

• When cell E₁ is connected and the null point is at J₁, no current flows through the galvanometer. This means the potential difference across the length l₁ of the wire (from A to J₁) is exactly equal to and opposite the EMF of cell E₁.
  Therefore, *E₁ = Potential drop across AJ₁ = k l₁** --- (Equation 1)

• Similarly, when cell E₂ is connected and the null point is at J₂, the potential difference across the length l₂ of the wire (from A to J₂) is equal to the EMF of cell E₂.
  Therefore, *E₂ = Potential drop across AJ₂ = k l₂** --- (Equation 2)

• To compare the EMFs, we divide Equation 1 by Equation 2:
  E₁ / E₂ = (k l₁) / (k l₂)

  Since the current from the driver battery was kept constant, the potential gradient 'k' is the same in both cases and cancels out.

  E₁ / E₂ = l₁ / l₂

Thus, by measuring the respective balancing lengths l₁ and l₂, the ratio of the EMFs of the two cells can be determined.

Important Precautions

1. The EMF of the driver battery in the primary circuit must be greater than the EMFs of both cells E₁ and E₂.
2. The positive terminals of all cells (driver Battery, E₁, and E₂) must be connected to the same end A of the potentiometer wire.
3. The jockey should be tapped gently on the wire and not dragged, to avoid scraping the wire and changing its uniform cross-section.
4. The current in the primary circuit should be kept constant throughout the experiment.