Mathematics

Problem

Consider a source of emf

\mathcal{E}

having internal resistance

r

that supplies current to an external load resistor

R

. The current in the circuit is

I=\dfrac{\mathcal{E}}{R+r}

. Answer the following. a. (i) Derive an expression for the power delivered to the load

P_R

in terms of

\mathcal{E}, r

and

R

. b. (ii) Show that the efficiency

\eta

of power transfer, defined as

\eta = \dfrac{P_R}{\mathcal{E} I}

, can be written as

\eta = \dfrac{R}{R+r}

. c. (iii) Explain qualitatively, without further calculation, how the efficiency changes when (a)

R \gg r

and (b)

R \ll r

. d. (iv) A student claims that choosing

R

equal to zero would yield

100\%

efficiency. Identify and explain the flaw in this reasoning.