Problem 96 - University Physics with Modern Physics

Question

CP A small sphere with mass \(m\) carries a positive charge \(q\) and is attached to one end of a silk fiber of length \(L .\) The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density \(\sigma\) . Show that when the sphere is in equilibrium, the fiber makes an angle equal to arctan \(\left(q \sigma / 2 m g \epsilon_{0}\right)\) with the vertical sheet.

Step-by-Step Solution

Answer

The angle is \( \arctan\left(\frac{q \sigma}{2mg\epsilon_0}\right) \).

Step 1: Identify the Forces

First, identify the forces acting on the charged sphere when it is in equilibrium. The forces are: the gravitational force downward, \( F_g = mg \); the electric force due to the charged sheet, \( F_e \); and the tension in the fiber.

Step 2: Determine the Electric Force

The electric force \( F_e \) exerted on the charge \( q \) by the charged sheet is given by the formula \( F_e = qE \), where \( E \) is the electric field due to a charged infinite plane. The electric field \( E \) is given by \( \sigma / (2\epsilon_0) \). Thus, \( F_e = q \sigma / (2\epsilon_0) \).

Step 3: Establish Equilibrium Conditions

For the sphere to be in equilibrium, the sum of the horizontal and vertical forces must be zero. Thus, horizontally, \( T \sin(\theta) = F_e = q \sigma / (2 \epsilon_0) \), and vertically, \( T \cos(\theta) = mg \).

Step 4: Express Angle with Trigonometric Identity

By dividing the horizontal and vertical equilibrium equations, we obtain \( \tan(\theta) = \frac{q \sigma / (2 \epsilon_0)}{mg} \). This simplifies to \( \theta = \arctan\left(\frac{q \sigma}{2mg\epsilon_0}\right) \).

Step 5: Conclusion

The angle \( \theta \) the fiber makes with the vertical sheet when the sphere is in equilibrium is given by \( \theta = \arctan\left(\frac{q \sigma}{2mg\epsilon_0}\right) \), matching the problem's requirement.

Key Concepts

Electric Force

Electric force is one of the fundamental interactions in nature between charged objects. It is described by Coulomb's law, which states that like charges repel each other and opposite charges attract. This force is responsible for numerous phenomena in our daily lives and forms the basis for understanding electrostatic interactions.

In the problem we are considering, the electric force (\( F_e \)) acts on a small charged sphere due to a charged sheet. The force can be calculated using the electric field \( E \) and the charge \( q \) of the sphere. The formula for electric force is \( F_e = qE \). The electric field \( E \) for an infinite charged plane is given as \( \sigma / (2\epsilon_0) \), where \( \sigma \) is the surface charge density.

• The electric force influences how the sphere will move or remain in position.
• The balance of this force with others, like gravity, determines the equilibrium position.

Equilibrium of Forces

In physics, equilibrium refers to a state where all forces acting on an object are balanced. In simple terms, it means that the object is at rest or moving with constant velocity because the net force is zero. For the charged sphere, equilibrium occurs when the gravitational force, electric force, and tension in the fiber all cancel each other out.

Here, the equilibrium condition is established through two main equations derived from balancing horizontal and vertical forces. Horizontally, we have: \( T \sin(\theta) = F_e \), and vertically: \( T \cos(\theta) = mg \). These equilibrium conditions allow us to solve for the angle \( \theta \), showing the sphere's fiber inclination.

• Equilibrium is essential to understand static structures or constant velocity conditions.
• It helps predict how objects will react to external forces.

Trigonometric Identity

Trigonometric identities play a critical role in solving problems involving angles and forces. In our problem, they are used to relate the different components of forces acting on the sphere.

To find the angle \( \theta \) that the fiber makes with the vertical, we use the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). By dividing the horizontal equilibrium equation by the vertical, we derive:\[ \tan(\theta) = \frac{T \sin(\theta)}{T \cos(\theta)} = \frac{q \sigma / (2 \epsilon_0)}{mg} \]This simplifies to finding \( \theta \) as: \( \theta = \arctan\left(\frac{q \sigma}{2mg\epsilon_0}\right) \).

• Trigonometric identities help in transforming complex force relations into simple angle equations.
• They are a powerful tool for simplifying and solving physics problems.

Electric Field

An electric field represents the force per unit charge exerted at any point in space around a charged object or within a charge distribution. In the context of this exercise, it is generated by the charged sheet, which affects the charged sphere and determines the direction and magnitude of electric forces acting on it.

The field due to an infinite plane charge is given by \( E = \sigma / (2\epsilon_0) \). This expression shows that the electric field is only dependent on the surface charge density \( \sigma \) and the permittivity of free space \( \epsilon_0 \).

• An understanding of the electric field allows for the prediction of the force on a charge placed in the field.
• The electric field describes how strongly another charge will be influenced by the source charge.

Electrostatic Equilibrium

Electrostatic equilibrium is the state where charges within a system are at rest, and the electric field within the conductor is zero. In this scenario, the charged sphere attached to the fiber reaches an equilibrium position where all acting forces balance out.

This concept applies when the resultant electric force, gravitational force, and tension in the fiber reach a zero net force condition. Thus, the charged sphere is neither moving nor rotating, remaining in a stable position. The problem demonstrates this equilibrium by solving for \( \theta \), ensuring the forces' balance.

• Electrostatic equilibrium governs how charges distribute themselves on conductors or within systems.
• Understanding this concept helps in analyzing and predicting the behavior of static charge systems.