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Electrostatic force is a fundamental concept in physics that describes the force of attraction or repulsion between two electrically charged particles. This force is calculated using Coulomb's Law, which is named after Charles-Augustin de Coulomb, who first formulated it in the 18th century. Coulomb's Law states that the electrostatic force (\( F \)) between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance (\( r \)) between them. This is expressed mathematically as:

• \[ F = k \frac{q_1 q_2}{r^2} \]

Here, \( k \) is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \). Since electrostatic forces act along the line joining the centers of the charges, they can either pull the charges together (attractive force) or push them apart (repulsive force). The forces are attractive if the charges have opposite signs and repulsive if they have the same sign. In our exercise, we consider two small spheres that act like particles, and we're interested in maximizing this force by correctly choosing the amount of charge \( q \) transferred between them.
Understanding this force helps you explain various phenomena, from adhesion of particles to the behavior of capacitors.

A quadratic function is a type of polynomial function specifically of degree two, characterized by the general equation:

• \[ f(x) = ax^2 + bx + c \]

This function is graphically represented as a parabola, which can open upward or downward. In the context of our problem, the quadratic function stems from rearranging the electrostatic force expression into:

• \[ F \, \propto \, qQ - q^2 \]

The parabola opens downward, as indicated by the negative coefficient of \( q^2 \), signifying that the function has a maximum point. Finding this maximum is crucial because it tells us the optimal amount of charge to transfer between the spheres to maximize electrostatic force. The vertex form and axis of symmetry form core methods to analyze such functions. This is valuable for students as it shares a crossover with other areas of physics and engineering, such as projectile motion.

The derivative of a function is a mathematical tool that measures how a function changes as its input changes. It provides the rate at which the function's value is changing at any given point, often regarded as the 'slope' of the function. For a quadratic function like \( f(q) \), the derivative helps identify where the function reaches its maximum or minimum value by finding critical points.

• The derivative of \( f(q) = qQ - q^2 \) is \( f'(q) = Q - 2q \)

To find the maximum point, we set this derivative to zero and solve for \( q \):

• \[ Q - 2q = 0 \]
• \[ q = \frac{Q}{2} \]

This result implies that when the charge transferred is half of the initial charge, the electrostatic force is maximized. The derivative not only helps identify that maximum position but also confirms the maximizing nature of a vertex for a parabola opening downward. Therefore, understanding derivatives is vital in optimizing functions, an essential skill across many fields of study and practical applications.