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 \(-\frac{\lambda}{2}\) 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.
Show Hint
The electric field due to an infinite charged wire decreases with distance from the wire. For two wires with opposite charges, the electric fields will cancel each other out at a point where the magnitudes are equal but opposite in direction.
Given two infinite straight wires with charge densities:
- Wire 1: Charge density \( \lambda \)
- Wire 2: Charge density \( -\frac{\lambda}{2} \)
Determine the position from wire 1 where the net electric field is zero.
Step 1: Electric Fields from Infinite Wires
The electric field \(E\) at a distance \(r\) from an infinite charged wire with charge density \( \lambda \) is \( E = \frac{2k_e |\lambda|}{r} \), where \(k_e\) is Coulomb's constant.
For wire 1 (\( \lambda \)), the electric field at distance \(x\) from it is:
\[
E_1 = \frac{2k_e \lambda}{x}
\]
For wire 2 (\( -\frac{\lambda}{2} \)), the electric field at distance \( d - x \) from it is:
\[
E_2 = \frac{2k_e \left| -\frac{\lambda}{2} \right|}{d - x} = \frac{k_e \lambda}{d - x}
\]
Step 2: Condition for Zero Electric Field
The net electric field is zero when the magnitudes of the electric fields from the two wires are equal and opposite. Setting the magnitudes equal:
\[
\frac{2k_e \lambda}{x} = \frac{k_e \lambda}{d - x}
\]
Simplifying:
\[
\frac{2}{x} = \frac{1}{d - x}
\]
Cross-multiplying:
\[
2(d - x) = x
\]
\[
2d - 2x = x
\]
\[
2d = 3x
\]
\[
x = \frac{2d}{3}
\]
The point where the electric field is zero is \( \frac{2d}{3} \) from wire 1.
Step 3: Identifying the Region
The distance \(x = \frac{2d}{3}\) falls within Region B, the area between the two wires.
Therefore, the point of zero electric field is in Region B.
