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 _________.
To find the values of resistances \( R_1 \) and \( R_2 \), we will use the principle of a meter bridge, which is a practical application of Wheatstone bridge. The principle states that at balance (null point), the ratio of resistances is equal to the ratio of the lengths of the bridge wire:
Initially, when the null point is at 40 cm from point \( P \):
\[ \frac{R_1}{R_2} = \frac{40}{60} \]When a 16 Ω resistor is connected in parallel with \( R_2 \), the effective resistance becomes:
\[ R_{2 \text{ eff}} = \frac{R_2 \cdot 16}{R_2 + 16} \]Now, the null point changes to 50 cm from \( P \):
\[ \frac{R_1}{R_{2 \text{ eff}}} = \frac{50}{50} = 1 \]Using the equation, we get:
\[ R_1 = R_{2 \text{ eff}} \] \[ R_1 = \frac{R_2 \cdot 16}{R_2 + 16} \]Substitute the value of \( R_1 \) from the first condition:
\[ \frac{R_1}{R_2} = \frac{2}{3} \implies R_1 = \frac{2}{3} R_2 \]Equate both expressions for \( R_1 \):
\[ \frac{2}{3} R_2 = \frac{R_2 \cdot 16}{R_2 + 16} \]Cross-multiplying gives:
\[ 2R_2 (R_2 + 16) = 3 \cdot 16 \cdot R_2 \]Solving it:
\[ 2R_2^2 + 32R_2 = 48R_2 \] \[ 2R_2^2 = 16R_2 \] \[ R_2^2 = 8R_2 \] \[ R_2 = 8 \, \text{or} \, R_2 = 0 \]Since a zero resistance is not practical, \( R_2 = 8 \, \Omega \). Plugging back to find \( R_1 \):
\[ R_1 = \frac{2}{3} R_2 = \frac{2}{3} \times 8 = \frac{16}{3} \, \Omega \]Therefore, the resistances are \( R_2 = 8 \, \Omega \) and \( R_1 = \frac{16}{3} \, \Omega \).
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________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
- 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
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________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
