2) Since the terminal Voltage of a battery is the current in the battery and r t

Question

2) Since the terminal Voltage of a battery is the current in the battery and r the internal resistance, the power output of the battery is given by AV-e-Ir where is the emf, I given by P= IAV 1-12r . P is then the power transferred to the external circuit a) Since the power output is a quadratic function of the current, there is a value of I for which the power reaches a maximum. Use calculus and determine both the value of the current for which there is maximum power and the maximum amount of power at this current in terms of and r (no numerical substitution please). You have to clearly show your work b) At what rate is thermal energy developed in the battery when it transfers maximum power to the circuit. At what rate is chemical energy consumed in the battery at maximum power? (Answer both in terms of and r.) c) When the maximum amount of curreni that can be drawn from the battery is drawn, the terminal voltage will be zero. In terms of and r how much current is this? How much power is transferred to the circuit under these conditions? he circuit below contains a battery (+ and - represent its terminals) with internal resistance, several external resistors, two ammeters and a voltmeter. Assume the meters are ideal, the voltmeter has very high resistance R 00 and the ammeters very low

Explanation / Answer

P= E I - I^2 r ------ (1)

a) power reaches maximum when

dP/dI = 0 and d^2P/dI^2 <0 ( from Calculus-Maxima and minima )

Or E- 2I r =0 Here were see second derivative = -2r and hence dP/dI gives a mximum

I = E/2r is value I for which power delivered is maimum.

P max = E(E/2r) -(E/2r)^2 r = E^2/2r - E^2/4r = E^2/4r  

(b) Maximum Thermal Energy tranferred

= I^2 r = (E/2r)^2 (r)= E^2/2r

(c) Maximum Chemical energy is consumed at rate = maximum thermal energy rate + max Power delivered to external circuit

= E^2/2r + E^2/4r = 3E^2/4r

(c) The max current = E/r,in this case entire emf is consumed internally and no power can be transferred to extenal circuit.

Because from (1) P = EI - I^2r = E(E/r) - (E/r)^2 r = 0

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