We have to show that in the circuit shown in the figure the power dissipated in the resistance R is maximum when , in which r is the internal resistance of the AC generator.

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Let the emf of the AC-generator be

.

As the total resistance in the resistance in the circuit is , the current in the circuit will be

The average power dissipated in the circuit will be

The function will have an extremum at R for which

This condition gives

By calculating , we will show that it is <0, and therefore is a maximum.

The power dissipation is maximum when the external load is equal to the load of the generator.