Question

The total inductance${\mathbf{L}}_{\mathbf{p}}$of a solid cylindrical conductor (of radius$\mathbf{r}$) due to both internal and external flux linkages out of distance$\mathbf{D}$is given by (in$\mathbf{H}\mathbf{/}\mathbf{M}$)

(a) $\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$

(b)$\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ln}\left(\frac{D}{r}\right)$

(c) $\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ln}\left(\frac{D}{r}\right)$

Where ${\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{=}{\mathbf{e}}^{\frac{\mathbf{-}\mathbf{1}}{\mathbf{4}}}\mathbf{r}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{778}\mathbf{r}$

Short answer

Therefore, the correct option is (c) ${\mathrm{L}}_{\mathrm{p}}=2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{{\mathrm{r}}^{\text{'}}}\right)\frac{\mathrm{H}}{\mathrm{m}}$.

Step-by-step solution

Step 1: Determine the formula of internal and external inductance inside the conductor.

Write the formula of internal inductance inside the conductor.

${\mathbf{L}}_{\mathbf{int}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\frac{\mathbf{H}}{\mathbf{m}}$

Write the formula of external inductance outside the conductor.

${\mathbf{L}}_{\mathbf{ext}}\mathbf{=}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ln}\left(\frac{D}{r}\right)\frac{\mathbf{H}}{\mathbf{m}}$

Determine the total inductance.

Write the formula of total inductance of the conductor.

${\mathrm{L}}_{\mathrm{p}}={\mathrm{L}}_{\mathrm{int}}+{\mathrm{L}}_{\mathrm{ext}}$

Substitute the value and solve as,

$$\begin{gathered} {\mathrm{L}}_{\mathrm{p}}=\frac{1}{2}\times {10}^{-7}+2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{\mathrm{r}}\right) \\ =2\times {10}^{-7}\left(\frac{1}{4}+\ln \left(\frac{\mathrm{D}}{\mathrm{r}}\right)\right) \\ =2\times {10}^{-7}\left({\mathrm{lne}}^{\frac{1}{4}}+\ln \left(\frac{\mathrm{D}}{\mathrm{r}}\right)\right) \end{gathered}$$

Further solve as,

$$\begin{gathered} {\mathrm{L}}_{\mathrm{p}}=2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{{\mathrm{re}}^{{\displaystyle \frac{1}{4}}}}\right) \\ =2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{0.778\mathrm{r}}\right) \\ =2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{{\mathrm{r}}^{\text{'}}}\right)\frac{\mathrm{H}}{\mathrm{m}} \end{gathered}$$

Hence, correct option is (c) ${\mathrm{L}}_{\mathrm{p}}=2\times {10}^{-7}\ln \left(\frac{\mathrm{D}}{{\mathrm{r}}^{\text{'}}}\right)\frac{\mathrm{H}}{\mathrm{m}}$.