91Ó°ÊÓ

Coulomb's Law is a fundamental principle in electrostatics, describing the force between two point charges. It states that the electrostatic force (\( F_e \) ) is directly proportional to the product of the magnitudes of the charges (\( q_1 \) and \( q_2 \) ) and inversely proportional to the square of the distance (\( r \) ) between them. Mathematically, it is expressed as:\[F_e = k \frac{|q_1 q_2|}{r^2}\]where

• \( k \) is the Coulomb's constant, equal to \( \frac{1}{4\pi \varepsilon_0} \), with \( \varepsilon_0 \) being the permittivity of free space.
• The forces are along the line joining the two charges.

In our exercise, the force between the two balls arises due to their identical charges, causing a repulsion that we calculate using the above principle. The force between them can thus be expressed as \( F_e = \frac{k q^2}{x^2} \), where \( x \) is their separation.

In an electrostatic system, equilibrium occurs when all forces acting on an object sum to zero, resulting in no net movement. For our setup, each charged ball hangs in equilibrium under the influence of three forces:

• Gravitational force pulling down, \( mg \).
• Electrical repulsion force pushing sideways, \( F_e \).
• Tension in the thread, balancing both the vertical and horizontal components of other forces.

To achieve equilibrium, particularly in a static system, the forces must balance such that the tension's vertical component balances the gravitational force, and its horizontal component matches the electric force:\[T \cos \theta = mg\]\[T \sin \theta = F_e\]In this scenario, understanding these balancing points helps determine the equilibrium separation \( x \) of the balls in the equation \( x = \left( \frac{q^2 L}{2 \pi \varepsilon_0 m g} \right)^{1 / 3} \). By setting these components equal, students can comfortably solve for unknown quantities.

Trigonometric approximations are incredibly useful when dealing with small angles, as it simplifies complex equations. When the angle \( \theta \) is small, we assume \( \sin \theta \approx \tan \theta \). In terms of our exercise, this approximation helps us because:- \( \sin \theta \) represents the ratio of the horizontal separation distance to the total length of the string, given by \( \frac{x}{2L} \).- Simplifying \( \tan \theta \) and \( \sin \theta \) allows us to work with simpler linear approximations, which are easier to handle mathematically.This approximation gives us more manageable equations to balance the forces, particularly when substituting values into the force balance equations derived earlier in the equilibrium section. It transforms the complex trigonometric components into relatively straightforward linear relationships.

Analyzing the electric forces in this context involves considering how these forces interact with other forces present in the system. The electric force analysis starts by recognizing that the electric repulsion between similarly charged objects is what causes them to push apart. The formula for electric force derived from Coulomb's Law in this exercise is:\[F_e = \frac{kq^2}{x^2}\]where \( k \) is a constant and \( x \) is the distance separating the charges. By understanding these interactions, we can rearrange our equations based on the system's equilibrium conditions and solve for unknown variables. For instance, by balancing \( F_e \) with the tension's horizontal component, students derive the relationship between distance \( x \) and charge \( q \):\[\tan \theta = \frac{F_e}{mg} \]Ultimately, this step guides us to determine the unknown charge \( q \) through substitution and simplification, a fundamental aspect of the broader electric force analysis process.