Partial differentiation problem! help please?

Question

P=(ε^2)R/ (R+r)^2

differentiate P with respect to R to show that R is maximum when it is equal to r

Help much appreciated

Answer 1

P=ε²R/(R+r)²

dp/dR = (R+r)²(ε²)-(ε²R)2(R+r)(1) / (R+r)^4
dp/dR = ε²(R²+2Rr+r²)-(ε²R)(2R+2r) / (R+r)^4
dp/dR = ε²R²+2Rrε²+r²ε²-2ε²R²-2Rrε²/(R+r)^4

For Power Max dp/dR = 0 .
0=ε²R²+2Rrε²+r²ε²-2ε²R²-2Rrε²/(R+r)^4
0=-ε²R²+r²ε²/(R+r)^4
0=-ε²R²+r²ε²
ε²R²=r²ε²
R²=r²
R=r ( There Proved )

For maximum power , External resistance(R) = internal resistance(r) .

Answer 2

Master is right on.

Thought I'd add two things. That R = r for max power is a prime example of something called impedance matching. Impedance Z = f(R, L, C); where R is resistance, L is inductance, and C is capacitance. So if you have two systems connected, power transfer will be max when Z = z, the impedance Z of one system matches (equals) the impedance z of the other system.

In your case impedance is all resistor, but the concept is the same. That is, power transfer between two systems always maxes out when the impedances of the two systems are the same where they are connected. [This is why you buy R = 75 ohm coax cable for your TV, the TV is r = 75 ohms at the connector; so max power from the signal transfers into the TV.]

The second point is that your problem is not a "partial" differential problem. There is but one independent variable on the RHS...that is R. There would be more than one if the problem were a partial differentiation problem.