Three infinitely long wires with linear charge density are placed… - Vidyadip

MCQ

Three infinitely long wires with linear charge density $\lambda$ are placed along the x -axis, y -axis and z axis respectively. Which of the following denotes an equipotential surface?

A
$x y+y z+z x=$ constant
✓
$(x+y)(y+z)(z+x)=$ constant
C
$\left(x^{2}+y^{2}\right)\left(y^{2}+z^{2}\right)\left(z^{2}+x^{2}\right)=$ constant
D
$x y z=$ constant

✓

Answer

Correct option: B.

$(x+y)(y+z)(z+x)=$ constant

(B)
Sol. $\quad v=-\int \overrightarrow{\mathrm{E}} \cdot \mathrm{d} \overrightarrow{\mathrm{r}}=\int \frac{2 \mathrm{k} \lambda}{\mathrm{r}} \mathrm{dr}=2 \mathrm{k} \lambda \ln \mathrm{r}+\mathrm{c}$
Net potential due to all wire
$
\mathrm{v}=2 \mathrm{k} \lambda \ln \sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}+2 \mathrm{k} \lambda \ln \sqrt{\mathrm{y}^{2}+\mathrm{z}^{2}}+2 \mathrm{k} \lambda \ln \sqrt{\mathrm{z}^{2}+\mathrm{x}^{2}}+\mathrm{c}
$
for $v=c$
$\sqrt{\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\left(\mathrm{y}^{2}+\mathrm{z}^{2}\right)\left(\mathrm{z}^{2}+\mathrm{x}^{2}\right)}=\mathrm{c}$
$\therefore\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\left(\mathrm{y}^{2}+\mathrm{z}^{2}\right)\left(\mathrm{z}^{2}+\mathrm{x}^{2}\right)=\mathrm{c}$
where $\mathrm{c}=$ constant

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