### Related Formula mathrmV = -int vecmathrmEcdotmathrmdvecmathrmr = 2mathrmklambda ln mathrmr + mathrmc$\mathrm{V} = -\int \vec{\mathrm{E}}\cdot\mathrm{d}\vec{\mathrm{r}} = 2\mathrm{k}\lambda \ln \mathrm{r} + \mathrm{c}$ ### Core Logic The potential at a point due to a line charge on an axis is proportional to the logarithm of its perpendicular distance. For the wire along the z-axis: mathrmV_z = -mathrmklambda ln(x^2 + y^2)$\mathrm{V}_z = -\mathrm{k}\lambda \ln(x^2 + y^2)$ For the wire along the x-axis: mathrmV_x = -mathrmklambda ln(y^2 + z^2)$\mathrm{V}_x = -\mathrm{k}\lambda \ln(y^2 + z^2)$ For the wire along the y-axis: mathrmV_y = -mathrmklambda ln(z^2 + x^2)$\mathrm{V}_y = -\mathrm{k}\lambda \ln(z^2 + x^2)$ Summing the individual potentials to find the net configuration potential: mathrmV_textnet = -mathrmklambda left[ ln(x^2+y^2) + ln(y^2+z^2) + ln(z^2+x^2) right] + mathrmC'$\mathrm{V}_{\text{net}} = -\mathrm{k}\lambda \left[ \ln(x^2+y^2) + \ln(y^2+z^2) + \ln(z^2+x^2) \right] + \mathrm{C}'$ mathrmV_textnet = -mathrmklambda ln left[ (x^2+y^2)(y^2+z^2)(z^2+x^2) right] + mathrmC'$\mathrm{V}_{\text{net}} = -\mathrm{k}\lambda \ln \left[ (x^2+y^2)(y^2+z^2)(z^2+x^2) \right] + \mathrm{C}'$ For an equipotential surface, set mathrmV_textnet = textconstant$\mathrm{V}_{\text{net}} = \text{constant}$: ### Step 1: Final Expression (x^2 + y^2)(y^2 + z^2)(z^2 + x^2) = textconstant$(x^2 + y^2)(y^2 + z^2)(z^2 + x^2) = \text{constant}$ This maps perfectly to option (3). ### Pattern Recognition Logarithmic combination rules transform scalar potential additions into products inside the functional argument: sum ln(mathrmr_i^2) = ln(prod mathrmr_i^2)$\sum \ln(\mathrm{r}_i^2) = \ln(\prod \mathrm{r}_i^2)$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Electrostatic Potential and Capacitance