Three infinitely long wires with linear charge density are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface? [2025]

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Q.
Three infinitely long wires with linear charge density $λ$ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface [2025]

1 xy + yz + zx = constant

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

3 $(x^{2} + y^{2}) (y^{2} + z^{2}) (z^{2} + x^{2})$ = constant

4 xyz = constant

Ans.
(3)

$V = - \int \vec{E} \cdot d \vec{r} = \int \frac{2 k λ}{r} d r = 2 k λ \ln r + c$

Net potential due to all wires

$V = 2 k λ \ln \sqrt{x^{2} + y^{2}} + 2 k λ \ln \sqrt{y^{2} + z^{2}} + 2 k λ \ln \sqrt{z^{2} + x^{2}} + c$

for V to be constant

$\sqrt{(x^{2} + y^{2}) (y^{2} + z^{2}) (z^{2} + x^{2})} = c'$

$∴ (x^{2} + y^{2}) (y^{2} + z^{2}) (z^{2} + x^{2}) = c$

where c = constant

Solution

Q.
Three infinitely long wires with linear charge density $λ$ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface [2025]

1 xy + yz + zx = constant

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

3 $(x^{2} + y^{2}) (y^{2} + z^{2}) (z^{2} + x^{2})$ = constant

4 xyz = constant

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Solution

Q. Three infinitely long wires with linear charge density λ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface [2025]

1 xy + yz + zx = constant

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

3 (x2+y2)(y2+z2)(z2+x2) = constant

4 xyz = constant

Ans. (3) V=–∫E→·dr→=∫2kλrdr=2kλ ln r+c Net potential due to all wires V=2kλ ln x2+y2+2kλ ln y2+z2+2kλ ln z2+x2+c for V to be constant (x2+y2)(y2+z2)(z2+x2)=c' ∴ (x2+y2)(y2+z2)(z2+x2)=c where c = constant

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Q.
Three infinitely long wires with linear charge density $λ$ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface [2025]

3 $(x^{2} + y^{2}) (y^{2} + z^{2}) (z^{2} + x^{2})$ = constant