Solving for Internal Resistance ($r$) in Series and Parallel Cell Connections
This problem requires us to find the internal resistance ($r$) of identical cells when the current through an external resistor is the same under both series and parallel connections.
Series Connection Analysis
When two identical cells, each with EMF $E$ and internal resistance $r$, are connected in series:
- Total EMF = $E + E = 2E$
- Total internal resistance = $r + r = 2r$
- The total resistance of the circuit is the sum of total internal resistance and the external resistance ($6 \Omega$). Total resistance = $2r + 6 \Omega$.
- Using Ohm's law ($I = \frac{V}{R}$), the current ($I_{series}$) flowing through the $6 \Omega$ resistor is: $I_{series} = \frac{2E}{2r + 6}$
Parallel Connection Analysis
When the same two cells are connected in parallel:
- Total EMF = $E$ (since the cells are identical)
- Total internal resistance = $\frac{r \times r}{r + r} = \frac{r}{2}$
- The total resistance of the circuit is the sum of total internal resistance and the external resistance ($6 \Omega$). Total resistance = $\frac{r}{2} + 6 \Omega$.
- The current ($I_{parallel}$) flowing through the $6 \Omega$ resistor is: $I_{parallel} = \frac{E}{\frac{r}{2} + 6} = \frac{E}{\frac{r + 12}{2}} = \frac{2E}{r + 12}$
Equating Currents and Solving for $r$
The problem states that the current is the same in both configurations:
$I_{series} = I_{parallel}$ $\frac{2E}{2r + 6} = \frac{2E}{r + 12}$Since the EMF ($E$) is non-zero, we can equate the denominators:
$2r + 6 = r + 12$Now, we solve for $r$:
- Subtract $r$ from both sides: $r + 6 = 12$
- Subtract $6$ from both sides: $r = 6$
Result
The value of the internal resistance $r$ is $6 \Omega$.
Similar Questions
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
- The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x,t) = 25 \sin(2.0 \times 10^{15} t - 10^7 x) \hat{n}$
then the refractive index of the medium is ________.
(All given measurement are in SI units) Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \text{ \mu T}$. The magnetic field at a distance $x = \sqrt{3}R$ on its axis from the centre is ________$\text{\mu T}$.
- Two point charges of $1 \text{ nC}$ and $2 \text{ nC}$ are placed at the two corners of equilateral triangle of side $3 \text{ cm}$. The work done in bringing a charge of $3 \text{ nC}$ from infinity to the third corner of the triangle is ________$\text{\mu J}$.
$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N.m}^2\text{/C}^2$ The equivalent resistance between the points $A$ and $B$ in the following circuit is $\frac{x}{5} \text{ }\Omega$. The value of $x$ is ________.
- Three identical coils $C_1, C_2$ and $C_3$ are closely placed such that they share a common axis. $C_2$ is exactly midway. $C_1$ carries current $I$ in anti-clockwise direction while $C_3$ carries current $I$ in clockwise direction. An induced current flows through $C_2$ will be in clockwise direction when
A meter bridge with two resistances $R_1$ and $R_2$ as shown in figure was balanced (null point) at 40 cm from the point $P$. The null point changed to 50 cm from the point $P$, when $16 \ \Omega$ resistance is connected in parallel to $R_2$. The values of resistances $R_1$ and $R_2$ are _________.
XPQY is a vertical smooth long loop having a total resistance $R$ where PX is parallel to QY and separation between them is $l$. A constant magnetic field $B$ perpendicular to the plane of the loop exists in the entire space. A rod CD of length $L \ (L > l)$ and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is _________ m/s. (g = acceleration due to gravity)
Important Questions from Electricity and Magnetism
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
- The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x,t) = 25 \sin(2.0 \times 10^{15} t - 10^7 x) \hat{n}$
then the refractive index of the medium is ________.
(All given measurement are in SI units) Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \text{ \mu T}$. The magnetic field at a distance $x = \sqrt{3}R$ on its axis from the centre is ________$\text{\mu T}$.
