Question 343: Electrostats & Capacitance

Question

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

Accepted Answer

Electric field due to infinitely long wire
$$\overrightarrow E  =  - {{2k\lambda } \over r}\widehat r$$
We know, $$V =  - \int {\overrightarrow E \,.\,\overrightarrow {dr} } $$
$$ =  - \int {{{ - 2k\lambda } \over r}\widehat r\,.\,\overrightarrow {dr} } $$
$$v = \int {{{2k\lambda } \over r}dr = 2k\lambda \,\ln r + c} $$
Net potential due to all wires,
$$V = {v_1} + {v_2} + {v_3}$$
$$ = 2k\lambda \ln \left( {\sqrt {{x^2} + {y^2}} } \right) + 2k\lambda \ln \left( {\sqrt {{y^2} + {z^2}} } \right) + 2k\lambda \ln \left( {\sqrt {{z^2} + {x^2}} } \right) + c$$
for equipotential surface, $$v = c$$
$$ \Rightarrow 2k\lambda \left[ {\ln \left( {\sqrt {{x^2} + {y^2}} \,.\,\sqrt {{y^2} + {z^2}} \,.\,\sqrt {{z^2} + {x^2}} } \right)} \right] = c$$
$$ \Rightarrow \sqrt {({x^2} + {y^2})\,.\,({y^2} + {z^2})\,.\,({z^2} + {x^2})}  = c$$
$$ \Rightarrow ({x^2} + {y^2})({y^2} + {z^2})({z^2} + {x^2}) = c$$
where, c = constant.

Answer

$\left(x^2+y^2\right)\left(y^2+z^2\right)\left(z^2+x^2\right)=$ constant

Answer

$x y z=$ constant

Answer

$x y+y z+z x=$ constant

Answer

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

Electrostats & Capacitance

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

Choose the correct answer: