Question

A tapered bar of length $L$ and end diameters $D_1$ and $D_2$ is made of a material of electrical resistivity $\rho .$ The electrical resistance of the bar is

• A. $\frac{4\rho L}{\pi {\left(D_1+D_2\right)}^2}$
• B. $\frac{4\rho L}{\pi {\left(D_1-D_2\right)}^2}$
• C. $\frac{\rho \pi \sqrt{D_1{D}_2}}{4L^2}$
• D. $\frac{4\rho L}{\pi D_1{D}_2}$

Answer 1

Answer: (D)

Similarity of triangles,

$\frac{r-a}{x}=\frac{b-a}{l}$
(Here $a=D_1/2,b=D_2/2$)
$\Rightarrow r=a+\left(\frac{b-a}{l}\right)x$
Resistance of elemental disc is
$dR=\frac{\rho dx}{\pi r^2}=\frac{\rho dx}{\pi {\left[a+\left(\frac{b-a}{l}\right)x\right]}^2}$
Total resistance of tapered wire is
$R=\int dR={\int }_0^l\frac{\rho dx}{\pi [a+\left(\frac{b-a}{l}\right)x{]}^2}$
$\Rightarrow R=\frac{\rho }{\pi }{\left[-\frac{1}{[a+\left(\frac{b-a}{l}\right)x}\right]}_0^l$
$=\frac{\rho }{\pi }\times \frac{l}{b-a}\left[\frac{1}{a}-\frac{1}{b}\right]=\frac{\rho l}{\pi ab}$
$=\frac{4\rho l}{\pi D_1{D}_2}$