Question:
Two sources of equal emf are connected to a resistance $R$. The internal resistance of these sources are $r_1$ and $r_2 (r_1 > r_2 )$. If the potential difference across the source having internal resistance $r_2$ is zero, then
Two sources of equal emf are connected to a resistance $R$. The internal resistance of these sources are $r_1$ and $r_2 (r_1 > r_2 )$. If the potential difference across the source having internal resistance $r_2$ is zero, then
Updated On: Apr 22, 2024
- $R = \frac{r_1 r_2}{r_2 - r_1}$
- $R = r_2 \left( \frac{r_1 + r_2}{r_2 - r_1} \right)$
- $R = \left( \frac{r_1 r_2}{r_2 - r_1} \right)$
- $R = r_2 - r_1$
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The Correct Option is D
Solution and Explanation
Let E be the emf of each source. When they are connected in series, then the current in the circuit is given by
$I = \frac{E_{tot}}{R_{tot}} = \frac{E +E}{r_{1 } + r_{2} + R} $
$= \frac{2E}{r_{1} + r_{2} + R} $
So, potential drop across the cell of internal resistance
$ r_{2} , \left(\frac{2E}{r_{1} + r_{2} + R}\right) r_{2}$
Hence, $ E - \frac{2E}{ \left(r_{1} + r_{2} + R\right)} r_{2} = 0 $
$ r_{1} + r_{2} + R = 2r_{2} $
So $ R = r_{2} - r_{1} $
$I = \frac{E_{tot}}{R_{tot}} = \frac{E +E}{r_{1 } + r_{2} + R} $
$= \frac{2E}{r_{1} + r_{2} + R} $
So, potential drop across the cell of internal resistance
$ r_{2} , \left(\frac{2E}{r_{1} + r_{2} + R}\right) r_{2}$
Hence, $ E - \frac{2E}{ \left(r_{1} + r_{2} + R\right)} r_{2} = 0 $
$ r_{1} + r_{2} + R = 2r_{2} $
So $ R = r_{2} - r_{1} $
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Concepts Used:
Magnetic Field
The magnetic field is a field created by moving electric charges. It is a force field that exerts a force on materials such as iron when they are placed in its vicinity. Magnetic fields do not require a medium to propagate; they can even propagate in a vacuum. Magnetic field also referred to as a vector field, describes the magnetic influence on moving electric charges, magnetic materials, and electric currents.
A magnetic field can be presented in two ways.
- Magnetic Field Vector: The magnetic field is described mathematically as a vector field. This vector field can be plotted directly as a set of many vectors drawn on a grid. Each vector points in the direction that a compass would point and has length dependent on the strength of the magnetic force.
- Magnetic Field Lines: An alternative way to represent the information contained within a vector field is with the use of field lines. Here we dispense with the grid pattern and connect the vectors with smooth lines.
Properties of Magnetic Field Lines
- Magnetic field lines never cross each other
- The density of the field lines indicates the strength of the field
- Magnetic field lines always make closed-loops
- Magnetic field lines always emerge or start from the north pole and terminate at the south pole.