A battery with EMF E and internal resistance r is connected across a resistance R . The power consumption in R will be maximum when :

Step-by-step solution

Power consumed in external resistance R is P = E^2 R (R + r)^2 . To find maximum, differentiate with respect to R: dP dR = E^2 (R + r)^2 - 2R(R + r) (R + r)^4 = E^2 r - R (R + r)^3 . dP dR = 0 gives R = r . This is the maximum power transfer theorem: maximum power is delivered when the external load resistance equals the source internal resistance.