FIGURE EX23.25 shows a $1.5$ g ball hanging from a string inside a parallel-plate capacitor made with 12 cm × 12 cm electrodes. The electrodes are charged to±75 nC. What is the charge on the ball in nC?

Verified step by step guidance

1. Step 1: Begin by understanding the setup of the problem. The ball is hanging inside a parallel-plate capacitor, which creates a uniform electric field between the plates. The charge on the ball will interact with this electric field, causing a force. The goal is to determine the charge on the ball.
2. Step 2: Recall the relationship between the electric field (E) and the charge density on the plates. The electric field between parallel plates is given by the formula: $E=\sigma /\epsilon$₀, where $\sigma$ is the surface charge density and $\epsilon$₀ is the permittivity of free space.
3. Step 3: Calculate the surface charge density $\sigma$ using the formula: $\sigma =Q/A$, where $Q$ is the total charge on the plates and $A$ is the area of the plates. The area of the plates can be calculated as $A=12\times 12$ cm², converted to m².
4. Step 4: Once the electric field $E$ is determined, use the force equation $F=qE$, where $F$ is the force acting on the ball, $q$ is the charge on the ball, and $E$ is the electric field. The force can be related to the tension in the string and the weight of the ball.
5. Step 5: Analyze the forces acting on the ball. The ball is in equilibrium, so the vertical component of the tension balances the weight of the ball ($\mathrm{mg}$), and the horizontal component of the tension is equal to the electric force ($\mathrm{qE}$). Use trigonometry to resolve the components of the tension and solve for $q$, the charge on the ball.

Key Concepts

Electric Field

An electric field is a region around a charged object where other charged objects experience a force. It is defined as the force per unit charge and is represented by the symbol E. In the context of a parallel-plate capacitor, the electric field is uniform between the plates and can be calculated using the voltage across the plates and the distance between them.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental in understanding how charged objects interact, including the ball in the electric field of the capacitor.

Charge Conservation

Charge conservation is a fundamental principle stating that the total electric charge in an isolated system remains constant over time. In this scenario, the ball hanging in the capacitor will acquire a charge due to the electric field created by the charged plates, but the total charge in the system (including the ball and the capacitor) will remain unchanged.