Problem 69 - University Physics with Modern Physics

Question

CP Two small spheres with mass \(m=15.0\) g are hung by silk threads of length \(L=1.20 \mathrm{m}\) from a common point (Fig. P21.68). When the spheres are given equal quantities of negative charge, so that \(q_{1}=q_{2}=q,\) each thread hangs at \(\theta=25.0^{\circ}\) from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of \(q .\) (c) Both threads are now shortened to length \(L=0.600 \mathrm{m},\) while the charges \(q_{1}\) and \(q_{2}\) remain unchanged. What new angle will each thread make with the vertical? (Hint: This part of the problem can be solved numerically by using trial values for \(\theta\) and adjusting the values of \(\theta\) until a self- consistent answer is obtained.)

Step-by-Step Solution

Answer

(a) Forces: gravity, tension, electric repulsion. (b) Magnitude of charge \(q = 4.68 \times 10^{-8} \, C\). (c) New angle \(\theta \approx 18.0^{\circ}\).

Step 1: Diagram of Forces

When the charged spheres are hanging, there are three main forces acting on each sphere: the gravitational force \(F_g\) pointing downward, the tension \(T\) in the thread pointing along the thread, and the electric force \(F_e\) pointing horizontally due to the repulsion between the charges. Draw these forces for one of the spheres on a diagram: the electric force \(F_e\) is horizontal, the tension \(T\) is along the thread, and the gravitational force \(F_g = mg\) points vertically downward.

Step 2: Set Up Equilibrium Equations

At the equilibrium position, the sum of forces in both vertical and horizontal directions must be zero. For vertical direction: \(T \cos \theta = mg\). For horizontal direction: \(T \sin \theta = F_e\). Thus, from these, the equilibrium conditions are \(mg = T \cos \theta\) and \(F_e = T \sin \theta\).

Step 3: Express Electric Force

The electric force between the two spheres can be expressed by Coulomb's Law: \(F_e = k \frac{q^2}{r^2}\), where \(k = 8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2\) is Coulomb's constant, and \(r\) is the distance between the charges, which can be found using \(r = 2L \sin \theta\).

Step 4: Solve for Charge Magnitude

From the equilibrium equations, substitute \(T = \frac{mg}{\cos \theta}\) into \(T \sin \theta = F_e\), simplifying to \(mg \tan \theta = F_e = k \frac{q^2}{(2L \sin \theta)^2}\). Plugging in the given values and solving for \(q\), we find: \(q = \sqrt{\frac{(mg \tan \theta)(2L \sin \theta)^2}{k}} = 4.68 \times 10^{-8} \, \text{C}\).

Step 5: Recalculate with Shorter Threads

With shorter threads \(L = 0.600 \, m\), we need to solve the equation numerically for the new \(\theta\). Using the relationship from equilibrium, \(mg \tan \theta = k \frac{q^2}{(2L' \sin \theta)^2}\). Try different \(\theta\) values that satisfy this equation and adjust until consistent. Now, \(\theta \approx 18.0^{\circ}\) when threads are shortened.

Key Concepts

Coulomb's Law

Coulomb's Law explains how electric charges interact. When two small spheres are given the same quantity of charge, they exert forces on each other. The force is proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:

• \( F_e = k \frac{q_1 q_2}{r^2} \)

Here, \( k \) is Coulomb's constant, equal to \( 8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2 \). In the exercise, the two spheres each have a charge \( q \) and repel each other, creating a horizontal force on each sphere. The distance \( r \) between the spheres determines how strong this force is.
Understanding Coulomb's Law helps us predict the electric force that small spheres exert on one another when suspended and charged.

Equilibrium of Forces

When the spheres hang at an angle, they are in equilibrium. This means that the forces acting on them are balanced. Three forces interact on each sphere: the gravitational force \( F_g \), the electrical force \( F_e \), and the tension in the thread \( T \). These forces can be broken down into components:

• Vertical Equilibrium: The tension's vertical component opposes gravity.
• Horizontal Equilibrium: The tension's horizontal component equals the electric force.

The conditions for equilibrium are:

• Vertical: \( T \cos \theta = mg \)
  Horizontal: \( T \sin \theta = F_e \)

Solving these equations allows us to find relationships between forces and solve for unknown quantities like charge. It's key to resolving how these forces interact to maintain the stability of the charged spheres.

Angle of Thread Suspension

The angle at which the threads suspend the charged spheres is crucial. This angle changes based on the forces and length of the thread. Initially, when the threads are longer, the angle \( \theta \) is 25 degrees. This angle is derived from balancing the electric and gravitational forces.
Even when the thread length changes, the same equilibrium conditions maintain:

• For shorter threads, angles must adapt so forces stay balanced.
• We recompute using the equation \( mg \tan \theta = k \frac{q^2}{(2L \sin \theta)^2} \).

By checking the math or using trial and error, we find the angle where forces re-balance. New angles form at around 18 degrees, showing how pendulum mechanics play a role in the physical setup. Understanding this process helps explain how equilibrium shifts with changing physical conditions.