CBSE 2025, Set 1

Question

Two infinitely long straight wires '1' and '2' are placed $d$ distance apart, parallel to each other, as shown in the figure. They are uniformly charged having charge densities $\lambda$ and $-\dfrac{\lambda}{2}$ respectively. Locate the position of the point from wire '1' at which the net electric field is zero and identify the region in which it lies.

![Two parallel infinite wires with charge densities lambda and -lambda/2](https://www.edvaya.com/images/cbse/physics/2025/set1/q28b-two-infinite-wires-electric-field-zero.webp)

Edvaya · Accepted Answer

The correct answer is Subjective. Let the point P where the net electric field is zero be at a distance $x$ from wire 1. **Electric field due to an infinitely long charged wire:** $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ (Directed radially outward for a positive charge, inward for a negative charge). Wire 1 has charge density $+\lambda$ and Wire 2 has charge density $-\lambda/2$. The separation between them is $d$. **Direction analysis:** - Wire 1 ($+\lambda$) produces a field pointing **away** from itself. - Wire 2 ($-\lambda/2$) produces a field pointing **towards** itself. For the net field to be zero, $\vec{E}_1$ and $\vec{E}_2$ must point in opposite directions and have equal magnitudes. This is only possible in the regions outside the two wires (Region A to the left of wire 1, or Region C to the right of wire 2). Since wire 2 has a smaller charge magnitude ($|-\lambda/2| < |\lambda|$), the null point must lie closer to wire 2 to compensate. Thus, it lies in **Region C**. **Calculation in Region C** (Point P is at a distance $x$ from wire 1, and $x-d$ from wire 2): Equating the magnitudes of the electric fields: $$\frac{\lambda}{2\pi\varepsilon_0 x} = \frac{\lambda/2}{2\pi\varepsilon_0 (x-d)}$$ $$\frac{1}{x} = \frac{1}{2(x-d)}$$ $$2(x-d) = x$$ $$2x - 2d = x \implies x = 2d$$ The point lies at a distance $x = 2d$ from wire 1, which firmly places it in **Region C** (at a distance $d$ to the right of wire 2).

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