Final answer:
The maximum power transfer theorem indicates that the power dissipated by a load resistor is maximum when the load resistance matches the internal resistance of the power source, a condition known as impedance matching.
Step-by-step explanation:
Maximum Power Transfer
To determine the maximum power that can be extracted by a load resistor from a power source such as a battery, we apply the maximum power transfer theorem. This theorem states that to extract maximum power, the load resistance (Rload) should be equal to the internal resistance (Rint) of the power source.
Proof of Maximum Power Transfer
The power dissipated by the load resistor can be expressed as P = V2/Rload, where V is the voltage across the load. The voltage V can also be written as the current I times the load resistance (V = IRload). Substituting into our power expression gives P = I2Rload. Considering Ohm's law (V = IR), the current can be expressed as V/(Rint + Rload), leading to the formula for power as a function of load resistance: P = V2Rload/(Rint + Rload)2. To find the value of Rload that maximizes P, we take the derivative of P concerning Rload and set it equal to zero. This results in Rload being equal to Rint.
When the resistances are equal, the circuit reaches a condition of impedance matching, which leads to the maximum power being transferred from the source to the load.