Problem 2

A battery is modeled by a constant emf

E

in series with internal resistance

r>0

and supplies a purely resistive load

R>0

; neglect lead resistance and temperature changes. Let

I

be the circuit current,

U

the terminal voltage (also the load voltage), and

P_R

the load power. (a) Find

I

and

U

in terms of

E,r,R

. (b) Calculate

P_R

and the power dissipated inside the source. (c) Determine the value of

R

that maximizes

P_R

, the maximum power, and the efficiency at that operating point.

problems.proof.stepOf

\frac{dP_R}{dR}=E^2\frac{r-R}{(R+r)^3}=0\Longrightarrow R=r

problems.proof.analysis

Substitution gives $P_R=E^2R/(R+r)^2$ . Its derivative changes from positive to negative where $(R+r)-2R=0$ , so this is a maximum.