Problem 2

An ideal source of emf

\mathcal E

and internal resistance

r>0

is connected to a variable resistive load

R>0

. In steady state the circuit current is

I=\mathcal E/(R+r)

, and the load power is the electrical energy delivered to

R

per unit time. Ignore temperature changes and any other circuit elements. (a) Express the load power in terms of

R

. (b) Determine the load resistance that maximizes that power. (c) Find the maximum power and compare it with the source emf and internal resistance.

problems.proof.stepOf

\boxed{P(R)=\frac{\mathcal E^2R}{(R+r)^2},\quad R_{max}=r,\quad P_{max}=\frac{\mathcal E^2}{4r}}

problems.proof.analysis

Collect the results for all parts in one place. This makes clear that they use the same data and a consistent sign convention.