Calculating the Resistance for Maximum Power
To determine the value of the resistance (R) that will result in the maximum power being delivered to the circuit, we need to follow a specific calculation method.

Formula for Power
The power (P) delivered to a circuit can be calculated using the formula:
P = (EMF^2 * R) / (4 * (R + r)^2), where
- EMF is the electromotive force (10 volts in this case),
- R is the resistance of the variable resistor,
- r is the internal resistance of the battery (0.5 ohms).

Calculating the Derivative
To find the value of R that maximizes power, we need to take the derivative of the power equation with respect to R and set it equal to zero. By doing this, we can find the critical points where the power is maximized.

Maximum Power Condition
The condition for maximum power transfer is when the resistance of the load (R) is equal to the internal resistance of the source (r). Therefore, the value of R that will deliver maximum power is equal to the internal resistance of the battery, which in this case is 0.5 ohms.

Conclusion
In conclusion, to achieve maximum power transfer in a circuit with an EMF of 10 volts and an internal resistance of 0.5 ohms, the resistance of the load should be equal to the internal resistance of the source. This ensures that the maximum amount of power is delivered to the circuit.