Question: The electric power P in watts in a direct-current circuit with two resistors R1 and R2 connected in series is P=(vR1R2)/(R1+R2)^2 where v is the voltage. If v and R1 are held constant, what resistance R2 produces maximum power?
v and R1 are held constant, therefore P is a function of R2 only. Thus we have,
P(R2) = C1.R2/(C2+R2)^2
where C1 and C2 are constants and C1 = vR1 and C2 = R1.
Differentiate P wrt R2 and set to zero.
dP/dR2 = C1/(C2 + R2)^2 - 2C1.R2/(C2 + R2)^3 = 0
{C1.(C2 + R2) - 2C1.R2}/(C1 + R2)^3 = 0
C1.(C2 + R2) - 2C1.R2 = 0
C2 + R2 - 2R2 = 0
C2 = R2
i.e. R1 = R2 (since C2 = R1)
Power is at a maximum when R2 = R1