Answer

Verified

381.3k+ views

**Hint:**When we connect two identical batteries in series the effective resistance in series combination will be the sum of those two internal resistances and in while the batteries are connected in parallel the reciprocal of those two internal resistances are added together to find the effective internal resistances. Then by calculating ${{I}_{1}}$and ${{I}_{2}}$ , we can find the heat in those two cases. Hence by substituting in the given equation we will get the value of R.

**Complete step by step answer:**

Given that the internal resistance $1\Omega $ is connected in series with a resistor R.

Then the total resistance in series combination becomes 2r+R.

Hence the current flow through ${{I}_{1}}$ is given by,

Current ${{I}_{1}}=\left[ \frac{2E}{2r+R} \right]$

Heat in the first case can be calculated by using the equation,

${{J}_{1}}=I_{1}^{2}Rt$

$\Rightarrow {{J}_{1}}={{\left( \frac{2E}{2r+R} \right)}^{2}}Rt$

Similarly we can calculate the current ${{I}_{2}}$. Here the same battery is connected in parallel across R. Thus,

${{I}_{2}}=\left[ \frac{E}{\frac{r}{2}+R} \right]$

Hence heat produced in the second case,

${{J}_{2}}=I_{2}^{2}Rt$

$\Rightarrow {{J}_{2}}={{\left( \frac{E}{\frac{r}{2}+R} \right)}^{2}}Rt$

Given that,

${{J}_{1}}=2.25{{J}_{2}}$

Substituting the values of ${{J}_{1}}$and ${{J}_{2}}$ in the above equation we get, ${{\left( \frac{2E}{2r+R} \right)}^{2}}Rt=2.25{{\left( \frac{E}{\frac{r}{2}+R} \right)}^{2}}Rt$

${{\left( \frac{2E}{2r+R} \right)}^{2}}Rt=2.25{{\left( \frac{2E}{r+2R} \right)}^{2}}Rt$

Rearranging the equation and cancelling the common terms we get,

${{\left( r+2R \right)}^{2}}=2.25{{\left( 2r+R \right)}^{2}}$

${{\left( r+2R \right)}^{2}}=\frac{9}{4}\times {{\left( 2r+R \right)}^{2}}$

$4{{\left( r+2R \right)}^{2}}=9{{\left( 2r+R \right)}^{2}}$

Taking the square root the above equation becomes,

$2\left( r+2R \right)=3\left( 2r+R \right)$

$R=4r$

Given that,

$r=1\Omega $

Then the value of R in $\Omega $ is,

$R=4\Omega $

**Note:**In series the effective resistance in series combination will be the sum of those two resistances and parallel the reciprocal of those two internal resistances are added together to find the effective internal resistances.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE