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

P_R=I^2R=\frac{E^2R}{(R+r)^2},\quad P_r=I^2r

problems.proof.analysis

Resistive power is $I^2R$ in the load and $I^2r$ internally. Their sum is $I^2(R+r)=EI$ , consistent with the ideal source power.