Two identical cells whether connected in prallel or in series gives the same current when connected to an external resistance 1.5 ohm.Find the value of the internal resistance of the cell

To solve the problem of finding the internal resistance of identical cells connected in both series and parallel configurations, we can apply Ohm's Law and the principles of electrical circuits. Let's break it down step by step.

Understanding the Circuit Configurations

When two identical cells are connected to an external resistance, the way they are connected—either in series or parallel—affects the total voltage and current in the circuit. However, in this case, it's stated that both configurations yield the same current through the external resistance of 1.5 ohms.

Defining Variables

Let's define some variables for clarity:

• V: The voltage of each cell
• r: The internal resistance of each cell
• R: The external resistance (1.5 ohms)
• I: The current flowing through the circuit

Analyzing the Series Connection

In a series connection, the total voltage is the sum of the voltages of the two cells, while the total internal resistance is the sum of their internal resistances. Therefore, the total voltage and resistance can be expressed as:

Total Voltage (V_total) = 2V

Total Internal Resistance (r_total) = 2r

Using Ohm's Law (V = IR), the current in the series configuration can be calculated as:

I_series = V_total / (R + r_total) = 2V / (1.5 + 2r)

Examining the Parallel Connection

For the parallel connection, the total voltage remains the same as that of one cell, while the total internal resistance is halved. Thus, we have:

Total Voltage (V_total) = V

Total Internal Resistance (r_total) = r/2

The current in the parallel configuration can be expressed as:

I_parallel = V_total / (R + r_total) = V / (1.5 + r/2)

Setting the Currents Equal

Since both configurations yield the same current, we can set the two equations for current equal to each other:

2V / (1.5 + 2r) = V / (1.5 + r/2)

Solving the Equation

To eliminate V from the equation, we can multiply both sides by (1.5 + 2r)(1.5 + r/2):

2V(1.5 + r/2) = V(1.5 + 2r)

Dividing both sides by V (assuming V is not zero), we get:

2(1.5 + r/2) = 1.5 + 2r

Expanding both sides leads to:

3 + r = 1.5 + 2r

Rearranging gives:

3 - 1.5 = 2r - r

1.5 = r

Final Result

The internal resistance of each cell is therefore 1.5 ohms. This result shows that the cells have equal internal resistance, which is consistent with the problem's conditions. By understanding how series and parallel circuits operate, we can effectively analyze and solve for unknowns in electrical systems.