Question

Two infinitely long straight wires having linear charge densities $-\lambda$ and $3\lambda$ are held vertically parallel to each other, distance r apart in free space. Find the nature and magnitude of the force/length exerted by one wire on the other.

Hint:

Do not mistakenly confuse this pure electrostatic geometrical setup with the widely known Ampere's magnetic force existing between two parallel current-carrying wires; linear charge density $\lambda$ refers exclusively to static, non-moving charges.
Opposite charges attract, identical charges repel—a universal fundamental rule spanning all physics.

Answer 1

Concept:
• An infinitely long, perfectly straight charged wire inherently creates a radially outward or inward electric field in its completely surrounding three-dimensional space.

• A second charged wire placed gracefully within this already established electric field will consequently and inevitably experience a continuous electrostatic force.

• The total experienced force per unit length is elegantly and systematically derived using the mathematical definition of linear charge density tightly coupled with Gauss's law for electrostatics.

Step 1: Determine the Electric Field of the First Wire
Let the primary infinite wire possess a strictly negative linear charge density mathematically defined as $\lambda_1 = -\lambda$.
Using the standard application of Gauss's law for a cylindrical geometry, the magnitude of the electric field perfectly generated by this infinite wire at a perpendicular radial distance $r$ is:
\[ E_1 = \frac{|\lambda_1|}{2\pi\epsilon_0 r} = \frac{\lambda}{2\pi\epsilon_0 r} \]
Because the primary wire's charge density is firmly negative, this resulting electric field vector is directed radially strictly inwards, pointing directly toward the primary wire.

Step 2: Calculate the Force Exerted on the Second Wire
The secondary parallel wire possesses a distinctly positive linear charge density logically given as $\lambda_2 = 3\lambda$.
We carefully and meticulously consider a microscopically small arbitrary length segment $dl$ of this secondary wire.
The accumulated static electric charge meticulously contained entirely within this tiny segment is strictly $dq = \lambda_2 dl = 3\lambda dl$.
The infinitesimal electrostatic force $dF$ experienced physically by this charged segment residing strictly in the presence of the external field $E_1$ is:
\[ dF = dq \cdot E_1 \]
Substituting our securely established mathematical expressions into the foundational force equation forcefully gives:
\[ dF = (3\lambda dl) \left( \frac{\lambda}{2\pi\epsilon_0 r} \right) \]

Step 3: Determine Force Per Unit Length and its Fundamental Nature
To firmly and absolutely find the exact force per unit length acting continuously along the wires, we systematically divide the derived infinitesimal force by the segment length $dl$:
\[ \frac{F}{l} = \frac{dF}{dl} = \frac{3\lambda^2}{2\pi\epsilon_0 r} \]
This mathematically represents the exact, uncompromising magnitude of the continuous electrostatic force per unit length strictly acting mutually on the wires.
To decisively determine the physical nature of this specific force, we must deeply examine the opposing algebraic signs of the respective charge densities.
Since one wire is heavily negatively charged ($-\lambda$) and the other is heavily positively charged ($+3\lambda$), they actively and constantly experience mutual electrostatic attraction.
Thus, the resulting continuous force is decisively attractive in its fundamental physical nature.