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. $\frac{\rho \pi \sqrt{D_{1} D_{2}}}{4 L^{2}}$
• 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
$d R=\frac{\rho\, d x}{\pi r^{2}}=\frac{\rho \,d x}{\pi\left[a+\left(\frac{b-a}{l}\right) x\right]^{2}}$
Total resistance of tapered wire is
$R=\int d R=\displaystyle \int_{0}^{l} \frac{\rho\ d x}{\pi[a+\left(\frac{b-a}{l}\right) x]^{2}}$
$\Rightarrow R= \frac{\rho}{\pi}\left[-\frac{1}{\left[a+\left(\frac{b-a}{l}\right) x\right.}\right]_{0}^{l} $
$=\frac{\rho}{\pi} \times \frac{l}{b-a}\left[\frac{1}{a}-\frac{1}{b}\right]=\frac{\rho l}{\pi\,a b} $
$=\frac{4 \,\rho l}{\pi \,D_{1} \,D_{2}}$