Problem 1 - Fundamentals of Physics

Question

Of the charge \(Q\) initially on a tiny sphere, a portion \(q\) is to be transferred to a second, nearby sphere. Both spheres can be treated as particles and are fixed with a certain separation. For what value of \(q / Q\) will the electrostatic force between the two spheres be maximized?

Step-by-Step Solution

Answer

The electrostatic force is maximized when \( \frac{q}{Q} = \frac{1}{2} \).

Step 1: Understanding the Problem

We have two tiny spheres separated by a distance and charged with amounts of electric charge, where one sphere initially has a total charge of \(Q\). We want to find out how to divide the charge \(Q\) such that the electrostatic force between them is at its maximum. The charge on the first sphere is \(Q - q\) and on the second sphere is \(q\). We can use Coulomb's Law to determine the force, which we aim to maximize.

Step 2: Applying Coulomb's Law

The electrostatic force \(F\) between the two spheres is given by Coulomb's Law: \[ F = k \frac{(Q-q)q}{r^2} \]where \(k\) is the Coulomb's constant and \(r\) is the distance between the charges. We need to find the value of \(q\) that maximizes \(F\).

Step 3: Setting Up the Optimization Problem

To maximize the force \(F\), we rewrite it as a function of \(q\):\[ F(q) = k \frac{(Q-q)q}{r^2} = \frac{k}{r^2} (Qq - q^2) \]The constants \(k\) and \(r^2\) do not depend on \(q\), so we focus on maximizing the quadratic expression \((Qq - q^2)\).

Step 4: Finding the Critical Point

To find the maximum of \(F(q) = Qq - q^2\), we take the derivative with respect to \(q\):\[ F'(q) = Q - 2q \]Setting the derivative equal to zero gives:\[ Q - 2q = 0 \]Solving for \(q\) gives:\[ q = \frac{Q}{2} \]

Step 5: Verifying Maximum Condition

To verify we have a maximum, check the second derivative:\[ F''(q) = -2 \]Since \( F''(q) < 0 \), the function \(F(q) = Qq - q^2\) is concave down, confirming a maximum at \(q = \frac{Q}{2}\).

Step 6: Converting to Ratio

The problem asks for the ratio \( \frac{q}{Q} \), so we compute:\[ \frac{q}{Q} = \frac{\frac{Q}{2}}{Q} = \frac{1}{2} \]

Key Concepts

Electrostatic Force

When studying the interaction between charged particles or objects, understanding electrostatic force is fundamental. It is the force that charged objects exert on each other due to their electric charges, either attracting or repelling them. The direction of this force is along the line connecting the points of the charges.
This force is central to Coulomb's Law, which describes the magnitude of the electrostatic force between two point charges.

• Coulomb's Law formula: \[ F = k \frac{q_1q_2}{r^2} \]where
  • \(F\) is the force between the charges,
  • \(k\) is the Coulomb's constant, approximately \(8.9875 \times 10^9\, \text{Nm}^2/\text{C}^2\),
  • \(q_1\) and \(q_2\) are the amounts of charge, and
  • \(r\) is the separation distance between the charges.

Understanding electrostatic force not only helps us calculate the push or pull between two charges but also lays the groundwork for analyzing more complex electrical systems.

Optimization Problem

In the sphere charge exercise, our challenge involves determining how to distribute the charge \(Q\) on two spheres to maximize the electrostatic force between them. This is a classic example of an optimization problem, where one seeks to find the conditions that result in the maximum or minimum value of a particular function.
Optimization problems are common in calculus and physics, where variables need to be adjusted for optimal results.

• Solution strategy:
  • First, identify what needs to be optimized—in this case, the electrostatic force \(F\).
  • Express \(F\) as a function of one of the decision variables—in our step-by-step solution, this is \(q\).

The function is then analyzed using calculus techniques like finding the derivative to locate the maximum point.
The exercise is essential because it teaches the principles of optimization which have wide applications in engineering and science.

Quadratic Expression

Quadratic expressions often appear in problems requiring maximization or minimization. In the sphere charge problem, the expression to optimize, \(F(q) = Qq - q^2\), is quadratic.
A quadratic expression is in the form:\[ ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants.

• Key steps in finding optimum values include:
  • Taking its derivative.
  • Setting the derivative to zero to find critical points.
  • Using the second derivative test to confirm if these points are maxima or minima.
• In the sphere problem:
  • Derivative: \(F'(q) = Q - 2q\)
  • Solution for \(q\): setting \(F'(q) = 0\) gives \( q = \frac{Q}{2} \)
  • Second derivative: \(F''(q) = -2\), indicating a maximum due to negative value.

Quadratic expressions allow us to examine the changing nature of relationships in both theoretical and practical scenarios.

Charge Distribution

Charge distribution in the problem refers to how the total charge \(Q\) initially on one sphere is allocated between two spheres to achieve a particular electrostatic effect. Correct understanding of charge distribution is crucial in maximizing the electrostatic force.

• Significance of charge distribution:
  • Each sphere's charge affects the overall force since force magnitude is directly related to the product \((Q-q)q\).
  • By setting the charges such that \(q = \frac{Q}{2}\), the distribution maximizes the force as this offers a perfect balance between the charge difference and the amount on each sphere.
• The lesson from this distribution:
  • Understanding how changes in distribution affect outcomes is applicable in designing electrical components.
  • The concept helps in analyzing not only electrostatic problems but also extends to fields like electrochemistry and electronics.

Optimizing charge distribution plays a crucial role in the efficient design and functioning of many electrical systems and devices.