A battery of e.m.f

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Calculus of change and accumulation

Mathematical Methods for Physics, Ch The derivative as a physical instrument

Problem

A battery of e.m.f. V0V_0 and internal resistance rr drives a current through a load resistor RR. The current is I=V0/(R+r)I = V_0/(R+r), so the power delivered to the load is P(R)=V02R(R+r)2.P(R) = \frac{V_0^2 R}{(R+r)^2}. ((a)) Use the quotient rule to show that dP/dR=V02(r−R)/(R+r)3\mathrm{d}P/\mathrm{d}R = V_0^2(r-R)/(R+r)^3, and check that this has the units of watts per ohm. ((b)) Find the load RR at which dP/dR=0\mathrm{d}P/\mathrm{d}R = 0, and use the sign of dP/dR\mathrm{d}P/\mathrm{d}R on either side of it to say whether PP is largest or smallest there. Give that value of PP.

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