Problem 13 - University Physics with Modern Physics

Question

Three point charges are arranged on a line. Charge \(q_{3}=+5.00 \mathrm{nC}\) and is at the origin. Charge \(q_{2}=-3.00 \mathrm{nC}\) and is at \(x=+4.00 \mathrm{cm} .\) Charge \(q_{1}\) is at \(x=+2.00 \mathrm{cm} .\) What is \(q_{1}\) (magnitude and sign) if the net force on \(q_{3}\) is zero?

Step-by-Step Solution

Answer

The charge \( q_1 \) is \( +0.75 \text{ nC} \).

Step 1: Understanding the Setup

The problem involves three charges on a line: \( q_3 = +5.00 \text{ nC} \) at the origin, \( q_2 = -3.00 \text{ nC} \) 4 cm to the right of the origin, and \( q_1 \) at 2 cm to the right of the origin. We need to find the magnitude and sign of \( q_1 \) such that the net force on \( q_3 \) is zero.

Step 2: Identifying Forces on \( q_3 \)

Calculate the forces exerted on \( q_3 \) by \( q_1 \) and \( q_2 \). The force between two charges is given by Coulomb's Law: \[F = k \frac{|q_1 q_2|}{r^2}\]where \( k = 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \).

Step 3: Calculating Force by \( q_2 \) on \( q_3 \)

The distance between \( q_2 \) and \( q_3 \) is 4 cm. Calculate the force:\[F_{23} = k \frac{|-3.00 \times 5.00| \times 10^{-18}}{(0.04)^2}\]

Step 4: Calculating Force by \( q_1 \) on \( q_3 \)

The distance between \( q_1 \) and \( q_3 \) is 2 cm. Suppose the charge \( q_1 = x \), then the force:\[F_{13} = k \frac{|x \times 5.00| \times 10^{-18}}{(0.02)^2}\]

Step 5: Setting the Net Force to Zero

To satisfy the condition that net force on \( q_3 \) is zero, set the magnitudes of \( F_{13} \) and \( F_{23} \) equal because they act in opposite directions:\[k \frac{|x \times 5.00| \times 10^{-18}}{(0.02)^2} = k \frac{|-3.00 \times 5.00| \times 10^{-18}}{(0.04)^2}\]

Step 6: Simplifying the Equation

Divide both sides of the equation by \( k \) and solve for \( x \):\[\frac{|x \times 5.00|}{(0.02)^2} = \frac{|-3.00 \times 5.00|}{(0.04)^2}\]Solving gives:\[|x| = \frac{-3.00}{4^2/2^2} = -3/4 = -0.75 \text{ nC}\]

Step 7: Determining the Sign of \( q_1 \)

Since the force exerted by \( q_1 \) should be in the opposite direction of the attractive force due to \( q_2 \) (which is attractive for opposite charges), \( q_1 \) must be positive. Thus, \( q_1 = +0.75 \text{ nC} \).

Key Concepts

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest. It involves understanding how forces between charged particles behave when they are not in motion. This field of study explores the interactions and distributions of static electric charges and the fields they produce. Key features of electrostatic interactions include:

• The concept that like charges repel and opposite charges attract each other.
• The use of Coulomb's law to determine the force between two point charges.
• Understanding electric fields, which represent the region around a charged particle where forces can be exerted on other charges.

By analyzing these static forces, we can explain how materials and objects interact based on their electric charges, predict the behavior of charges in various configurations, and apply this knowledge in practical applications.

Point Charges

Point charges are idealized charges that are assumed to be located at a single point in space with no physical size. This simplification is useful in electrostatics for calculating electric forces and fields. By treating charges as point charges, it allows us to focus on the effects of the charges without concerns about their physical dimensions.

• Point charges are often used in theoretical physics to simplify calculations involving electric forces and fields.
• They are characterized by having a specific magnitude and sign, determining whether they are positive or negative.

In the given exercise, we deal with three point charges each located at specific positions on a line, allowing us to use Coulomb's law to find the forces between them. By focusing on their interactions, we can determine the conditions needed for a particular point charge to experience zero net force.

Net Force

The concept of net force is fundamental in physics and describes the total force acting on an object or charge after all individual forces are considered. It determines whether the object will accelerate or remain in equilibrium. In electrostatics, the net force on a charge is the vector sum of all forces exerted on that charge by other charges.

• Forces between charges can either be attractive or repulsive, and their direction plays a crucial role in finding the net force.
• To achieve equilibrium where the net force is zero, the opposing forces must balance each other out.

In the exercise, ensuring that the net force on charge \( q_3 \) is zero requires that the attractive and repulsive forces from \( q_1 \) and \( q_2 \) counterbalance each other.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: positive and negative. Charges can exert forces on each other over a distance, and the way they interact depends on their types:

• Like charges repel each other, while opposite charges attract.
• The amount of force exerted between two charges is proportional to their magnitudes and inversely proportional to the square of the distance between them, as expressed in Coulomb's law.

In the scenario of the exercise, each charge has a specific value indicated in nanocoulombs (nC), and interactions among these charges define the net forces and equilibrium conditions. Understanding the nature of electric charges is essential for solving problems involving the forces and fields they generate.