Problem 70 - University Physics with Modern Physics

Question

CP Two identical spheres are each attached to silk threads of length \(L=0.500 \mathrm{m}\) and hung from a common point (Fig. P21.68). Each sphere has mass \(m=8.00 \mathrm{g} .\) The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. One sphere is given positive charge \(q_{1},\) and the other a different positive charge \(q_{2} ;\) this causes the spheres to separate so that when the spheres are in equilibrium, each thread makes an angle \(\theta=20.0^{\circ}\) with the vertical. (a) Draw a free-body diagram for each sphere when in equilibrium, and label all the forces that act on each sphere. (b) Determine the magnitude of the electrostatic force that acts on each sphere, and determine the tension in each thread. (c) Based on the information you have been given, what can you say about the magnitudes of \(q_{1}\) and \(q_{2} ?\) Explain your answers. (d) A small wire is now connected between the spheres, allowing charge to be transferred from one sphere to the other until the two spheres have equal charges; the wire is then removed. Each thread now makes an angle of \(30.0^{\circ}\) with the vertical. Determine the original charges. (Hint: The total charge on the pair of spheres is conserved.)

Step-by-Step Solution

Answer

Electrostatic force: \(Fe = mg\tan\theta\), solve for charge using angles.

Step 1: Draw the Free-Body Diagram

For each sphere, two forces act vertically and one force acts horizontally. The gravitational force, \(F_g = mg\), acts downwards. The tension \(T\) in the thread acts along the thread at an angle \(\theta\) from the vertical. The electrostatic force \(F_e\) acts horizontally, repelling the two spheres from each other.

Step 2: Establish Force Equilibrium Conditions

In equilibrium, each sphere's forces balance out. Vertically, the component of tension counteracts the weight: \(T \cos \theta = mg\). Horizontally, the tension's component equals the electrostatic force: \(T \sin \theta = F_e\).

Step 3: Calculate Electrostatic Force

Using \(T \cos \theta = mg\), find \(T = \frac{mg}{\cos \theta}\). Substitute in the horizontal force equation: \(T \sin \theta = F_e\), giving \(F_e = mg \tan \theta\). Calculate \(F_e\) with \(m = 0.008\,\text{kg}\) and \(\theta = 20^\circ\).

Step 4: Calculate Tension in the Thread

Substitute values and calculate \(T\) from \(T = \frac{mg}{\cos \theta}\). This gives the tension in the threads while they are at \(20^\circ\).

Step 5: Analyzing the Charges

The assumption that each thread initially makes the same angle implies \(q_1 + q_2\) has a specific value, but the individual charge values are not determined by angle alone.

Step 6: Re-equalizing the Charge

If a wire equates charges and the angle becomes \(30^\circ\), the symmetry suggests \(q_1 = q_2 = q'\). Use modified equations from previous equilibrium steps with \(30^\circ\) to find charges when they are equal. The conservation of charge implies \(q_1 + q_2 = 2q'\). Calculate \(q'\) using \(30^\circ\).

Step 7: Determining the Original Charges

Using conservation, since each resulting charge is \(q'\), find \(q_1\) and \(q_2\) by solving \(q_1 + q_2 = 2q'\) and \(q_1 \times q_2 = (q')^2\) ensuring calculations adheres to physical constraints.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of electrostatics that describes the force between two charged objects. It states that the electrostatic force (\( F_e \)) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is given by &thenbsp;\( F_e = \frac{k |q_1 q_2|}{r^2} \), where \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)), \( q_1 \)&thenbsp; and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

This law applies well to our exercise, as the repulsion between the two charged spheres can be accurately modeled with it. Since the spheres repel each other, both have the same type of charge (either both positive or both negative). The challenge in the problem is finding how these forces and charge relationships lead to the positioning of the spheres."

Equilibrium of Forces

In physics, equilibrium refers to the condition where the sum of all forces acting on an object is zero, resulting in no motion. For the spheres hanging from threads, equilibrium occurs when the tension in the thread, gravitational force, and electrostatic force balance each other.

To fully understand this, let's look closer: - Vertically, the weight of each sphere (\( mg \)) is balanced by a vertical tension component, which can be expressed as \( T \cos \theta = mg \).- Horizontally, the electrostatic repulsion is counteracted by the horizontal tension component, i.e., \( T \sin \theta = F_e \).

When you setup these equilibrium conditions, you can solve for unknown quantities such as the tension in the string and the electrostatic force.
A deep grasp on how forces interact and reach equilibrium is crucial in understanding this scenario."

Charge Conservation

The law of charge conservation indicates that the total charge in an isolated system remains constant. In our problem, when two differently charged spheres (\( q_1 \) and \( q_2 \)) are connected by a wire, charge flows between them until they reach equilibrium, achieving equal charges.

After disconnecting the wire, the system still holds the same total charge it had originally. Therefore, once they reach equilibrium after charge transfer, each sphere will have equal charge \( q' \), resulting in \( q_1 + q_2 = 2q' \). This is a reflection of the underlying principle of charge conservation.

In such problems, understanding that electrostatic systems maintain a constant total charge helps in calculating initial individual charges before equilibrium."

Trigonometry in Physics

Trigonometry is essential in physics for analyzing forces and motions involving angles. In this exercise, trigonometric functions such as sine and cosine help resolve the tension in the thread into horizontal and vertical components.

- The relation \( \cos \theta \) helps find the vertical component of tension, supporting the weight of the spheres (\( T \cos \theta = mg \)).
- \( \sin \theta \) is vital to resolve the horizontal tension component that balances the electrostatic force (\( T \sin \theta = F_e \)).

These trigonometric identities allow us to translate physical conditions like equilibrium into solvable mathematical equations. Without trigonometry, it would be difficult to resolve complex force interactions, especially when dealing with forces at angles."