Question

Two sources of equal emf are connected to an external resistance R. The internal resistance of the two sources are \[R1\] and \[R2{\rm{ }}\left( {R2 > R1} \right)\]. If the potential difference across the source having internal resistance R2 is zero, then
A. \[R = R2 - R1\]
B. \[R = R2 \times \left( {R1 + R2} \right)/\left( {R2 - R1} \right)\]
C. \[R = R1R2/\left( {R2 - R1} \right)\]
D. \[R = R1R2/\left( {R1 - R2} \right)\]

Answer 1

Hint: The ratio of the cell's potential variations to the resultant resistance across each cell must be zero for the flow of current through the battery to be zero. Rearrange the equation to find R's value. This data will assist you in answering this question.

Formula used:
Total emf can be calculated using the formula:
\[{\rm{e = IR + Ir}}\]
Where ‘e’ is electromotive force, ‘I’ is current, ‘R’ is resistance and ‘r’ is internal resistance.


Complete answer:

The internal resistance of the battery plus the external resistance will add up to the overall resistance in the circuit.

Let us assume that two sources are connected in series. So, the total net emf \[{E_n} = E + E = 2E\]
Then, the total net internal resistance \[ = {R_1} + {R_2}\]
So, the Total equivalent resistance \[{R_{eq}} = R + {R_1} + {R_2}\]
The circuit’s current is calculated as,
 \[I = \frac{{{E_n}}}{{{{\mathop{\rm R}\nolimits} _{eq}}}} = \frac{{2E}}{{R + {R_1} + {R_2}}}\]
So, the 2nd source’s across the potential is,
\[E - I{R_2} = 0\] or
\[E - \frac{{2E}}{{R + {R_1} + {R_2}}} \times {R_2} = 0\] or
\[R = {R_2} - {R_1}\]
Therefore, the correct option for this problem is A.



Therefore, the correct option is A.


Note:Students most likely make mistakes in these types of problems because these types of problems include many formulas such as electromotive formulas and resistance formulas. Electromotive force is commonly known as EMF. And one should be very careful while substituting the values from the previous calculations to get the desired result.