Problem 84 - University Physics with Modern Physics

Question

Two tiny balls of mass \(m\) carry equal but opposite charges of magnitude \(q\) . They are tied to the same ceiling hook by light strings of length L. When a horizontal uniform electric field \(E\) is turned on, the balls hang with an angle \(\theta\) between the strings (Fig, 21.46\()\) . (a) Which ball (the right or the left) is positive, and which is negative? (b) Find the angle \(\theta\) between the strings in terms of \(E, q, m,\) and \(g\) . (c) As the electric field is gradually increased in strength, what does your result from part (b) give for the largest possible angle \(\theta ?\)

Step-by-Step Solution

Answer

(a) Left ball is positive, right is negative. (b) \( \theta = 2 \tan^{-1}(\frac{Eq}{mg}) \). (c) Maximum \(\theta\) is \(180^\circ\).

Step 1: Analyze the System with Electric Field

Let's analyze the system when the electric field is turned on. The electric field exerts a force on the charges. Given the electric field acts horizontally, the ball carrying the positive charge will experience a force in the direction of the field, while the negatively charged ball will experience a force in the opposite direction.

Step 2: Identify which ball is positive or negative

The positively charged ball experiences a force in the direction of the electric field E, moving to the right. Conversely, the negatively charged ball will move to the left, as it experiences a force opposite to E. Therefore, the left ball must be positive and the right ball negative.

Step 3: Set Up Forces on a Charged Ball

Consider one of the balls (say the right one), analyze the equilibrium of forces. The forces include tension T of the string, gravitational force mg, and the force due to electric field Eq. At equilibrium, the components must balance.

Step 4: Find Vertical and Horizontal Forces

For vertical forces, the tension's vertical component balances the gravitational force: \[ T \cos(\theta/2) = mg \]For horizontal forces, the tension's horizontal component balances the electric force: \[ T \sin(\theta/2) = Eq \]

Step 5: Solve for Angle \(\theta/2\)

From the vertical and horizontal force balance: Divide horizontal equation by vertical:\[ \tan(\theta/2) = \frac{Eq}{mg} \]Solve for \(\theta\): \[ \theta = 2 \tan^{-1}\left(\frac{Eq}{mg}\right) \]

Step 6: Evaluate Limits for Maximum \(\theta\) with Increasing E

As the electric field E increases, \(\tan(\theta/2)\) approaches infinity, implying \(\theta/2\) approaches \(90^\circ\). Thus, \(\theta\) approaches \(180^\circ\) as E increases indefinitely.

Key Concepts

Electric Field

The concept of an electric field is essential in understanding how charged objects interact. An electric field is a region around a charged particle where a force is exerted on other charges. It is a vector field, which means it has both a magnitude and direction. In our scenario, when a uniform electric field \(E\) is applied horizontally, it exerts a force on charged bodies, illustrating the influence of the field.

Key aspects of electric fields include:

• Direction: The direction of the electric field is defined by the direction of the force experienced by a positive charge.
• Magnitude: The magnitude of the electric field \(E\) indicates how strong the field is and how much force it can exert on a charge.

Understanding electric fields is crucial for predicting the motion and equilibrium of charged objects, like the tiny balls on strings in this scenario.

Coulomb's Law

Coulomb's Law provides a fundamental principle to calculate the electric force between two charged particles. According to Coulomb's Law, the force \(F\) between two charges \(q_1\) and \(q_2\), which are separated by a distance \(r\), is given by:\[F = k \frac{|q_1 q_2|}{r^2}\]where \(k\) is Coulomb's constant.

For our exercise, Coulomb's Law explains:

• Force Direction: Like charges repel, and opposite charges attract each other. Therefore, the electric force helps us determine that the negatively charged ball moves opposite the electric field, while the positive one moves with it.
• Magnitude Relation: Although we don't directly use Coulomb's constant in this problem, understanding how charge and distance affect force is pivotal.

Thus, Coulomb's Law allows us to validate the behaviors and interactions between charged particles in electric fields.

Equilibrium of Forces

Equilibrium of forces ensures that the tiny balls tied with strings remain at a certain angle, rather than swinging wildly. At equilibrium, the sum of forces acting on each ball is zero. This means the forces acting vertically and horizontally need to balance out.

There are two main components:

• Vertical Forces: The vertical component of tension \(T\) in the string must balance the gravitational force \(mg\):
  \[ T \cos(\theta/2) = mg \]
• Horizontal Forces: The horizontal component of tension must balance the electric force \(Eq\):
  \[ T \sin(\theta/2) = Eq \]

By achieving equilibrium, we can derive relationships between forces, allowing us to find the angle \( \theta \) between the hanging strings when the ball system is under the influence of the electric field.

Trigonometric Functions in Physics

Trigonometric functions often play an essential role in physics by helping us resolve forces into components. In this exercise, they are critical to determining the angle \( \theta \), using functions like sine and cosine.In the case of the balls on strings:

• Trigonometric Ratios: Using \( \sin \) and \( \cos \) ratios helps analyze the force components on the balls, with respect to the angle \( \theta/2 \).
• Tangent Function: To find the angle specifically, we used the tangent function:
  \[\tan(\theta/2) = \frac{Eq}{mg}\]Solving gives:
  \[\theta = 2 \tan^{-1}\left(\frac{Eq}{mg}\right)\]
• Limit Considerations: As the electric field increases, \( \theta \) reaches its maximum value, showing the limiting behavior of trigonometric functions in physics.

These functions allow us to connect the angles and forces, providing valuable tools for problem-solving in mechanics and electromagnetism.