
What is the maximum power output that can be obtained from a cell of emf $E$ and internal resistance $r$?
A. $\dfrac{{2{E^2}}}{r}$
B. $\dfrac{{{E^2}}}{{2r}}$
C. $\dfrac{{{E^2}}}{{4r}}$
D. None of these
Answer
476.1k+ views
Hint:Calculate the current in a circuit with a battery having external resistance and external resistance and then calculate the maximum power by using the maxima and differentiation property of function.
Complete step-by-step solution:
According to the question it is given that the a cell has emf of value $E$ and the internal resistance value is $r$.
Assume a battery with emf $E$ and internal resistance $r$ is connected with an external resistance.
Let the External resistance is $R$ and the value of current is $i$.
The value of current in the circuit is calculated as,
$i = \dfrac{E}{{r + R}}$
Write the formula of power across the external resistance $R$.
$P = {i^2}R$
Substitute $\dfrac{E}{{r + R}}$ for $i$ in the above formula.
$P = {\left( {\dfrac{E}{{r + R}}} \right)^2}R$ …..(1)
If we have to find the maximum power output then it will be differentiated with respect to $R$ and then make it equal to zero and substitute the value of $R$ to get the maximum value of power output.
Differentiate the equation (1) with respect to $R$.
\[\dfrac{{dP}}{{dR}} = \dfrac{d}{{dR}}\left[ {{{\left( {\dfrac{E}{{r + R}}} \right)}^2}R} \right]\] …..(2)
Use the property of differentiation as shown below.
$d\dfrac{{f\left( x \right)}}{{g\left( x \right)}} = \dfrac{{g\left( x \right)df\left( x \right) - f\left( x \right)dg\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}$
Use the property of differentiation in the equation (2).
\[
\Rightarrow \dfrac{{dP}}{{dR}} = \dfrac{d}{{dR}}\left[ {{{\left( {\dfrac{E}{{r + R}}} \right)}^2}R} \right] \\
\Rightarrow \dfrac{{dP}}{{dR}} = {E^2}\left[ {\dfrac{{{{\left( {r + R} \right)}^2} - 2\left( {r + R} \right)R}}{{{{\left( {r + R} \right)}^4}}}} \right] \\
\]
For the condition of maxima put \[\dfrac{{dP}}{{dR}}\] equals to zero.
\[
\Rightarrow {E^2}\left[ {\dfrac{{{{\left( {r + R} \right)}^2} - 2\left( {r + R} \right)R}}{{{{\left( {r + R} \right)}^4}}}} \right] = 0 \\
\Rightarrow {\left( {r + R} \right)^2} - 2\left( {r + R} \right)R = 0 \\
\]
Simplify the above expression to get the relation.
\[
\Rightarrow {\left( {r + R} \right)^2} = 2\left( {r + R} \right)R \\
\Rightarrow \left( {{r^2} + {R^2} + 2rR} \right) = 2rR + 2{R^2} \\
\Rightarrow {R^2} = {r^2} \\
\Rightarrow R = r \\
\]
So, the power will be maximum when the value of external resistance is equal to the value of internal resistance.
Substitute $r$ for $R$ in the equation (2) to get the maximum power output.
\[
P = {\left( {\dfrac{E}{{r + r}}} \right)^2}r \\
P = \dfrac{{{E^2}}}{{4{r^2}}} \times r \\
P = \dfrac{{{E^2}}}{{4r}} \\
\]
So, the maximum power output that can be obtained from a cell of emf $E$ and internal resistance $r$ is \[\dfrac{{{E^2}}}{{4r}}\].
Hence, the correct option is C.
Note:-
In the case of finding the maxima of a function always differentiate with variable and get the value of variable. Always solve the equation in the known variable and eliminate the unknown variables.
Complete step-by-step solution:
According to the question it is given that the a cell has emf of value $E$ and the internal resistance value is $r$.
Assume a battery with emf $E$ and internal resistance $r$ is connected with an external resistance.
Let the External resistance is $R$ and the value of current is $i$.
The value of current in the circuit is calculated as,
$i = \dfrac{E}{{r + R}}$
Write the formula of power across the external resistance $R$.
$P = {i^2}R$
Substitute $\dfrac{E}{{r + R}}$ for $i$ in the above formula.
$P = {\left( {\dfrac{E}{{r + R}}} \right)^2}R$ …..(1)
If we have to find the maximum power output then it will be differentiated with respect to $R$ and then make it equal to zero and substitute the value of $R$ to get the maximum value of power output.
Differentiate the equation (1) with respect to $R$.
\[\dfrac{{dP}}{{dR}} = \dfrac{d}{{dR}}\left[ {{{\left( {\dfrac{E}{{r + R}}} \right)}^2}R} \right]\] …..(2)
Use the property of differentiation as shown below.
$d\dfrac{{f\left( x \right)}}{{g\left( x \right)}} = \dfrac{{g\left( x \right)df\left( x \right) - f\left( x \right)dg\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}$
Use the property of differentiation in the equation (2).
\[
\Rightarrow \dfrac{{dP}}{{dR}} = \dfrac{d}{{dR}}\left[ {{{\left( {\dfrac{E}{{r + R}}} \right)}^2}R} \right] \\
\Rightarrow \dfrac{{dP}}{{dR}} = {E^2}\left[ {\dfrac{{{{\left( {r + R} \right)}^2} - 2\left( {r + R} \right)R}}{{{{\left( {r + R} \right)}^4}}}} \right] \\
\]
For the condition of maxima put \[\dfrac{{dP}}{{dR}}\] equals to zero.
\[
\Rightarrow {E^2}\left[ {\dfrac{{{{\left( {r + R} \right)}^2} - 2\left( {r + R} \right)R}}{{{{\left( {r + R} \right)}^4}}}} \right] = 0 \\
\Rightarrow {\left( {r + R} \right)^2} - 2\left( {r + R} \right)R = 0 \\
\]
Simplify the above expression to get the relation.
\[
\Rightarrow {\left( {r + R} \right)^2} = 2\left( {r + R} \right)R \\
\Rightarrow \left( {{r^2} + {R^2} + 2rR} \right) = 2rR + 2{R^2} \\
\Rightarrow {R^2} = {r^2} \\
\Rightarrow R = r \\
\]
So, the power will be maximum when the value of external resistance is equal to the value of internal resistance.
Substitute $r$ for $R$ in the equation (2) to get the maximum power output.
\[
P = {\left( {\dfrac{E}{{r + r}}} \right)^2}r \\
P = \dfrac{{{E^2}}}{{4{r^2}}} \times r \\
P = \dfrac{{{E^2}}}{{4r}} \\
\]
So, the maximum power output that can be obtained from a cell of emf $E$ and internal resistance $r$ is \[\dfrac{{{E^2}}}{{4r}}\].
Hence, the correct option is C.
Note:-
In the case of finding the maxima of a function always differentiate with variable and get the value of variable. Always solve the equation in the known variable and eliminate the unknown variables.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Trending doubts
Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is a transformer Explain the principle construction class 12 physics CBSE

Draw a labelled sketch of the human eye class 12 physics CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

How much time does it take to bleed after eating p class 12 biology CBSE
