A copper wire of radius r and length ell is c | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The resistance of a wire is given by $R = \rho \frac{\ell}{A}$.The resistance of the copper core is $R_{cu} = \rho_c \frac{\ell}{\pi r^2}$.The cro

Vedclass · Accepted Answer

$\left( \frac{\rho_c \rho_n}{3\rho_c + \rho_n} \right) \frac{\ell}{\pi r^2}$

Vedclass · Answer

(C) The resistance of a wire is given by $R = \rho \frac{\ell}{A}$.The resistance of the copper core is $R_{cu} = \rho_c \frac{\ell}{\pi r^2}$.The cross-sectional area of the nickel layer is $A_{ni} = \pi(2r)^2 - \pi r^2 = 3\pi r^2$.Thus,the resistance of the nickel layer is $R_{ni} = \rho_n \frac{\ell}{3\pi r^2}$.Since the copper core and the nickel layer are in parallel,the equivalent resistance $R_{eq}$ is given by $\frac{1}{R_{eq}} = \frac{1}{R_{cu}} + \frac{1}{R_{ni}}$.$R_{eq} = \frac{R_{cu} R_{ni}}{R_{cu} + R_{ni}} = \frac{(\rho_c \frac{\ell}{\pi r^2}) (\rho_n \frac{\ell}{3\pi r^2})}{\rho_c \frac{\ell}{\pi r^2} + \rho_n \frac{\ell}{3\pi r^2}}$.Simplifying the expression: $R_{eq} = \frac{\frac{\rho_c \rho_n \ell^2}{3\pi^2 r^4}}{\frac{\ell}{\pi r^2} (\rho_c + \frac{\rho_n}{3})} = \frac{\rho_c \rho_n \ell}{3\pi r^2 (\rho_c + \frac{\rho_n}{3})} = \frac{\rho_c \rho_n \ell}{\pi r^2 (3\rho_c + \rho_n)}$.Therefore,$R_{eq} = \left( \frac{\rho_c \rho_n}{3\rho_c + \rho_n} \right) \frac{\ell}{\pi r^2}$.

$A$ copper wire of radius $r$ and length $\ell$ is coated with a nickel layer until its radius becomes $2r$. If the resistivities of copper and nickel are $\rho_c$ and $\rho_n$ respectively,find the equivalent resistance of the wire.

Similar Questions

Three unequal resistors in parallel are equivalent to a resistance $1 \ \Omega$. If two of them are in the ratio $1:2$ and if no resistance value is fractional,then the largest of the three resistances in ohms is

For the shown circuit,find the effective resistance between the points $A$ and $B$.

The equivalent resistance between $A$ and $B$ is $:-$

Two heater wires of equal length are first connected in series and then in parallel. The ratio of heat produced in the two cases is

The total power dissipated in watts in the circuit shown is ............. $W$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module