Question

A 24 volt battery of internal resistance 4 ohm is connected to a variable resistor at what value of the current from the battery is the rate of heat produced in the resistor maximum ?

Answer 1

Hint: To answer this question we will first find current using voltage, resistance and internal resistance. Then we will find power In the circuit, and then we will find the heat in the circuit using power and time. Then we will apply a condition to get the current required to get maximum heat in the circuit.

Formula used:
\[I = \dfrac{E}{{R + r}}\]
where \[I\] is current, \[E\] is the voltage, \[R\] is resistance and $r$ is internal resistance.
\[P = {I^2}R\]
Where \[P\]= power.
\[Q = {I^2}Rt\]
Where \[Q\] is heat and \[t\] is time.

Complete step by step answer:
Look at the following diagram:

Using the formula for the current we get:
\[I = \dfrac{E}{{R + r}}\]
\[ \Rightarrow I = \dfrac{{24}}{{R + 4}}\]
Now we will find power: we know that electric power is the rate per unit time at which electrical energy is transferred by an electric circuit. Hence it is given mathematically as a product of square of current and resistance through the circuit,
\[P = {I^2}R\]
\[ \Rightarrow P = {\left( {\dfrac{{24}}{{R + 4}}} \right)^2}R\]
\[ \Rightarrow P = \dfrac{{{{24}^2}}}{{{{\left( {R + 4} \right)}^2}}}R\]

Now we will find heat: we know that heat is the form of energy that is transferred between two substances at different temperatures. But in terms of power heat is mathematically expressed as a product of power and time.
\[Q = {I^2}Rt\]
\[ \Rightarrow Q = P \times t\]
\[ \Rightarrow Q = \dfrac{{{{24}^2}}}{{{{\left( {R + 4} \right)}^2}}}R \times t\]
Now we want this heat to be maximum. Hence the condition will be that \[{\left( {R + 4} \right)^2}\] should be minimum.
\[ \Rightarrow Q = \dfrac{{{{24}^2}}}{{\left( {R + \dfrac{{16}}{R} + 8} \right)}}R \times t\]
Hence \[\left( {R + \dfrac{{16}}{R} + 8} \right)\] should be minimum. And for this to be minimum \[R = 4\Omega \]. Now we can easily find the current in the circuit. Current will be:
\[I = \dfrac{E}{{R + r}}\]
\[\Rightarrow I = \dfrac{{24}}{{4 + 4}}\]
\[\Rightarrow I = \dfrac{{24}}{8}\]
\[\therefore I = 3A\]

Hence a 24V battery will produce maximum heat when current through it is 3 amperes.

Note: We usually find current in a circuit by dividing potential by resistance, but here the cell itself has an internal resistance. Hence it should also be considered. hence Do not forget to consider internal resistance while finding current.