Question

A battery of e.m.f. E and internal resistance is connected to a variable resistor R as shown. Which one of the following is true?
(A) Potential difference across the terminals of the battery is where $R = r$.
(B) The power delivered to the resistor is maximum when $R = r$.
(C) Current in the circuit is maximum when $R = r$.
(D) Current in the circuit is maximum when $R > > r$.

Answer 1

Hint
The maximum power transfer theorem explains that “the maximum amount of power will be disposed of in the load resistance if it is the same in value to the Thevenin or Norton source resistance of the network supplying the power “.
The current that flows in the circuit is given by the formula;
$\Rightarrow I = \dfrac{E}{{R + r}}$
Where, $I$ denotes the current flowing through the circuit, $E$ denotes the power source in the circuit, $R$ denotes the resistance of a variable resistor in the circuit, $r$ denotes the resistance of the constant resistor.

Complete step by step answer
The data given in the circuit are as follows;
The power source of the circuit is, $E$.
Resistance of a variable resistor is, $R$.
Resistance of a constant resistor is, $r$.
The current that flows in the circuit is given by the formula;
$\Rightarrow I = \dfrac{E}{{R + r}}$
$\Rightarrow I = \dfrac{E}{{{{\left( {\sqrt R - \sqrt r } \right)}^2} + 2\sqrt {Rr} }}$
That is the current $I$ that flows through the circuit is maximum, if the resistance of a variable resistor is equal to resistance of the constant resistor, that is, $R = r$.
Since the power of the circuit is given as;
$\Rightarrow P = {I^2}R$
Where, $P$ denotes the power consumed by the circuit.
So, if the current $I$ of the circuit is maximum then the power in the circuit is also maximum;
$\Rightarrow {P_{\max }} = I_{\max }^2R$
So, this is called the maximum power theorem.
Therefore, the power delivered to the resistor is maximum when $R = r$ and current in the circuit is maximum when $R = r$.
Hence, the option (B) and (C) the power delivered to the resistor is maximum when $R = r$ and current in the circuit is maximum when $R = r$ are the correct answer.

Note
Ohm's law explains that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship.