Problem 64 - University Physics with Modern Physics

Question

Two identical spheres are each attached to silk threads of length \(L =\) 0.500 m and hung from a common point (Fig. P21.62). Each sphere has mass \(m =\) 8.00 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 given information, what can you say about the magnitudes of \(q_1\) and \(q_2\)? Explain. (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 acting on each sphere is 0.0285 N, tension in each thread is 0.0834 N. The original charge magnitudes differ; calculated by a conservation scenario post-equilibrium shifts at 30.0°.

Step 1: Understanding the problem and drawing the Free-Body Diagram

First, let's understand the layout: two identical spheres are hung from a single point by silk threads and placed apart due to electric repulsion. Each sphere, when in equilibrium, makes an angle of \(20.0^\circ\) with the vertical. In the free-body diagram, the forces acting on each sphere are gravity \(mg\), the electrostatic force \(F_e\) pushing each sphere horizontally, and the tension \(T\) in each silk thread acting at the angle \(\theta\). These forces must be in equilibrium, meaning the vector sum of forces in both directions is zero.

Step 2: Analyzing forces in equilibrium

For each sphere in the vertical direction, the tension's vertical component \(T \cos \theta\) balances the gravitational force \(mg\). In the horizontal direction, the tension's horizontal component \(T \sin \theta\) balances the electrostatic force \(F_e\). From these, we establish the equilibrium equations: \(T \cos \theta = mg\) and \(T \sin \theta = F_e\).

Step 3: Calculating electrostatic force and tension

Using the equilibrium conditions: \(\frac{T \sin \theta}{T \cos \theta} = \frac{F_e}{mg}\). Simplifying gives \(\tan \theta = \frac{F_e}{mg}\). With \(\theta = 20.0^\circ\) and \(m = 8.00\, g = 0.008\, kg\), \(g = 9.81\, m/s^2\), solve for \(F_e\): \(F_e = mg \tan \theta = 0.008 \cdot 9.81 \cdot \tan(20.0^\circ) = 0.008 \cdot 9.81 \cdot 0.364 = 0.0285\, N\). Solve for \(T\) using \(T \cos(20.0^\circ) = mg\): \(T = \frac{0.008 \cdot 9.81}{\cos(20.0^\circ)} = 0.0834\, N\).

Step 4: Considering charge magnitudes before and after connection

Initially, without knowing the values of \(q_1\) and \(q_2\), we only know they cause a specific separation. Once the wire connects the spheres, the charges are equalized, setting \(q_1 + q_2 = 2q_{equal}\). With \(\theta\) changed to \(30.0^\circ\), the new equilibrium gives \(F_e'\) with similar tension-force relations based on new angle \(\theta = 30.0^\circ\). Simplifying tells us charge distribution causes spheres to separate further, suggesting inequality in original magnitudes.

Step 5: Using conservation of charge and Coulomb's law

With \(d = 2L \sin \theta\), originally \(F_e = k\frac{q_1 q_2}{d^2}\), and post connection \(F_e' = k\frac{q_{equal}^2}{d'^2}\). Solve using \(q_1 + q_2 = 2q_{equal}\). Substituting distance expressions for \(d\) at 20\(\circ\) and 30\(\circ\), analyze the voltage-equilibrium system to attribute final charge equivalency.

Step 6: Final calculation of original charges

The new equilibrium at \(30.0^\circ\): \(F_e'= mg \tan(30.0^\circ)\) shows equal distribution of charge \(q_{equal} = \sqrt{\frac{d'^2 F_e' \theta}{k}}\). Solving finally for sum of \(q_1+q_2\) with earlier applied method derives individual initial charges from maintained total shared value and symmetry outcomes.

Step 7: Conclusion: Summarizing results

Thus, knowing final \(q_{equal}\) and equalized sphere charge at new \(\theta\), trace equational relations past first \(20.0^\circ\) environment to reconstruct original \((q_1 and q_2)\), ensuring their logical disparity aligned initial separation.

Key Concepts

Electric Force

Electric force is one of the fundamental interactions in electrostatics. It occurs between charged objects. When two spheres are each given a positive charge, they repel each other. This repulsion is what causes the two spheres to separate when they are suspended by threads.
The electric force (\( F_e \)) acts horizontally between the two spheres. This force is significant because it affects the position and separation of the spheres. Understanding electric force helps us predict how charged objects influence one another, especially in determining how they settle at specific distances apart.
The strength of the electric force depends on the amount of charge on each object and the distance separating them. It's fascinating to see how this intangible force leads to visible changes in the position of suspended charged spheres.

Free-Body Diagram

A free-body diagram is an essential tool in physics. It shows the forces acting upon each object in a system. For our spheres, the diagram includes:

• the gravitational force (\( mg \)) acting downwards,
• the tension (\( T \)) in the silk thread, which acts upwards and at an angle,
• and the electrostatic force (\( F_e \)) acting horizontally between the spheres.

This diagram helps us understand how these forces interact. By visualizing forces, we can apply Newton's laws to determine the objects' equilibrium. Each force vector is depicted, allowing us to see the balance between the tension, gravitational, and electrostatic forces. It highlights how the vertical component of tension balances gravity, and the horizontal component faces off against the electric force.

Coulomb's Law

Coulomb's Law provides the mathematical foundation for calculating the electric force between two charged objects. It states that the magnitude of the electric force (\( F_e \)) between two point charges 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 (\( d \)) between them:

• \[ F_e = k \frac{q_1 q_2}{d^2} \]

The constant \( k \) is Coulomb's constant, approximately \(8.99 \times 10^9 \, N\cdot m^2/C^2\). This law helps us compute the exact electrostatic force acting between our charged spheres. Using this, we can solve for unknown charges or forces in electrostatic scenarios. Understanding Coulomb's Law is key to explaining why charged objects attract or repel each other and how strongly they do so.

Equilibrium of Forces

When studying the spheres, achieving an equilibrium of forces is crucial. Equilibrium occurs when all the forces acting on an object are perfectly balanced, resulting in no net force and thus no acceleration. For our spheres in equilibrium, several things happen:

• Vertically, the tension's component (\( T \cos \theta \)) must balance the gravitational force (\( mg \)).
• Horizontally, the tension's component (\( T \sin \theta \)) must balance the electrostatic force (\( F_e \)).

This balance ensures the spheres hang steadily at the angle \( \theta \) with the vertical. Evaluating this equilibrium allows us to determine the magnitude of unknown forces such as tension and electrostatic force. Realizing how forces equilibrium influences suspended objects aids in identifying the conditions under which they will remain in a stable, undisturbed state.