Explanation:

When a cell is connected to an external resistance, the total resistance in the circuit can be calculated using Ohm's law:

Total Resistance (Rt) = External Resistance (R) + Internal Resistance (r)

The current flowing through the circuit can be calculated using Ohm's law:

Current (I) = Voltage (V) / Total Resistance (Rt)

To find the conditions for maximum current in the external resistance (R), we need to consider the relationship between R and r.

Case (a): R = r
In this case, the total resistance in the circuit becomes:
Rt = R + r = r + r = 2r

The current flowing through the circuit becomes:
I = V / Rt = V / 2r

The current is inversely proportional to the total resistance, so when Rt is at its minimum value (2r), the current will be at its maximum value.

Case (b): R < />
If the external resistance (R) is smaller than the internal resistance (r), the total resistance in the circuit becomes:
Rt = R + r = R + r > r + r = 2r

The current flowing through the circuit becomes:
I = V / Rt = V / (R + r)

In this case, the current will be smaller than the current in case (a) because the total resistance is greater.

Case (c): R > r
If the external resistance (R) is greater than the internal resistance (r), the total resistance in the circuit becomes:
Rt = R + r

The current flowing through the circuit becomes:
I = V / Rt = V / (R + r)

In this case, the current will be smaller than the current in case (a) because the total resistance is greater.

Case (d): R = r/2
If the external resistance (R) is equal to half of the internal resistance (r), the total resistance in the circuit becomes:
Rt = R + r = r/2 + r = (3/2)r

The current flowing through the circuit becomes:
I = V / Rt = V / [(3/2)r]

In this case, the current will be smaller than the current in case (a) because the total resistance is greater.

Therefore, the correct answer is option A, R = r, which results in the maximum current in the external resistance.