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.
We are given two infinite straight wires with charge densities:
- Wire 1: Charge density \( \lambda \),
- Wire 2: Charge density \( -\frac{\lambda}{2} \).
We need to find the position of the point from wire 1 where the net electric field is zero.
Step 1: Electric Fields Due to Infinite Wires
The electric field at a distance \(r\) from an infinitely long charged wire with charge density \(\lambda\) is given by:
\[
E = \frac{2k_e |\lambda|}{r}
\]
Where:
- \(k_e\) is Coulomb's constant,
- \(\lambda\) is the charge density of the wire,
- \(r\) is the distance from the wire.
For wire 1 (charge density \( \lambda \)), the electric field at a point at a distance \(x\) from wire 1 is:
\[
E_1 = \frac{2k_e \lambda}{x}
\]
For wire 2 (charge density \( -\frac{\lambda}{2} \)), the electric field at a distance \( (d - x) \) from wire 2 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
For the electric field to be zero at some point, the magnitudes of the electric fields due to the two wires must be equal and opposite in direction. Thus, we set the magnitudes of \(E_1\) and \(E_2\) equal:
\[
\frac{2k_e \lambda}{x} = \frac{k_e \lambda}{d - x}
\]
Simplifying:
\[
\frac{2}{x} = \frac{1}{d - x}
\]
Cross -multiply:
\[
2(d - x) = x
\]
\[
2d - 2x = x
\]
\[
2d = 3x
\]
\[
x = \frac{2d}{3}
\]
Thus, the point where the electric field is zero is at a distance of \( \frac{2d}{3} \) from wire 1.
Step 3: Identifying the Region
- The distance \(x = \frac{2d}{3}\) lies in Region B, which is the region between the two wires.
- Therefore, the point where the electric field is zero lies in Region B.
