Problem 19 - College Physics

Question

\(\bullet$$\bullet\) 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

Charge \(q_1 = -0.75\, \mathrm{nC}\) for zero net force on \(q_3\).

Step 1: Understand the Problem

We need to find the charge \(q_1\) such that the net force on charge \(q_3\), located at the origin, is zero. The task involves three point charges: \(q_3 = +5.00\, \mathrm{nC}\), \(q_2 = -3.00\, \mathrm{nC}\) at \(x = +4.00\, \mathrm{cm}\), and \(q_1\) at \(x = +2.00\, \mathrm{cm}\).

Step 2: Forces Exerted on \(q_3\)

The force on \(q_3\) due to \(q_2\) is \[ F_{32} = k \cdot \frac{|q_3 \, q_2|}{r_{32}^2} \]where \(r_{32} = 4.00\, \mathrm{cm} = 0.04\, \mathrm{m}\). The force direction is attractive as \(q_2\) is negative and \(q_3\) is positive, so it acts towards \(x=+4.00\, \mathrm{cm}\).

Step 3: Force Balance Condition

For \(q_3\) to have zero net force, the force due to \(q_1\) must cancel out the force due to \(q_2\). Therefore, the force exerted by \(q_1\) on \(q_3\) is:\[ F_{31} = k \cdot \frac{|q_3 \, q_1|}{r_{31}^2} \]with \(r_{31} = 2.00\, \mathrm{cm} = 0.02\, \mathrm{m}\). Since \(q_3\) is positive, \(q_1\) must be negative for the force to be attractive and directed towards the origin.

Step 4: Solve for \(q_1\) Using Force Equality

Set the magnitudes of the forces equal:\[ k \cdot \frac{|q_3 \, q_2|}{(0.04)^2} = k \cdot \frac{|q_3 \, q_1|}{(0.02)^2} \]Simplify by canceling \(k\) and \(q_3\):\[ \frac{|-3.00\, \mathrm{nC}|}{0.0016} = \frac{|q_1|}{0.0004} \]Cross-multiply and solve for \(q_1\):\[ |q_1| = \frac{3.00\, \mathrm{nC} \times 0.0004}{0.0016} = 0.75\, \mathrm{nC} \]Since \(q_1\) must be negative, \(q_1 = -0.75\, \mathrm{nC}\).

Key Concepts

Point Charges

Point charges are simply individual or discrete amounts of electric charge located at a specific point in space. Think of them as tiny charged particles that can interact with other charges due to their electric field. These interactions are fundamental to understanding electrostatic phenomena.

In the problem we're handling, three point charges are aligned on a straight line. Each charge, whether positive or negative, creates an invisible field in the space around it. The strength and direction of the forces these charges exert depend on their magnitude and sign.

• Magnitude of Charge: This is the size or amount of charge. It's expressed in units like nanoCoulombs (nC).
• Sign of Charge: Charges can be either positive or negative, which affects how they interact—either attracting or repelling each other.

If you imagine these point charges as part of a game where invisible force fields exert pushes or pulls on each other, you have a good concept of how point charges operate in electrostatics.

Net Force

Net force is the overall force on an object after summing up all individual forces acting on it. In terms of electrostatics, it refers to the total force one charge experiences because of other surrounding charges.

When we talk about the net force on a charge being zero, it means that all the forces acting on it are perfectly balanced. So, in the exercise, our goal is to ensure that the combined forces from other charges on the point charge at the origin (referred as charge \(q_3\)) cancel each other out.

• Force Cancellation: To achieve net zero force, forces exerted by individual charges on \(q_3\) must be equal in magnitude but opposite in direction.
• Balancing Forces: This often involves adjusting the magnitude or position of one of the charges involved, in our case, figuring out the exact charge \(q_1\) to ensure balance.

Envision net force as a tug-of-war where balanced teams ensure neither side wins over the other, causing no overall movement.

Coulomb's Law

Coulomb's Law is a fundamental principle that quantifies the amount of force between two point charges. Named after Charles-Augustin de Coulomb, it provides insight into how electrostatic forces work.

The law is given by the equation:\[F = k \, \frac{|q_1 \, q_2|}{r^2} \]

• \(F\): The force between the charges. It's measured in Newtons (N).
• \(k\): The Coulomb's constant, approximately \(8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2}\).
• \(q_1, q_2\): The magnitudes of the two charges, in Coulombs (C).
• \(r\): The distance between the charges, in meters (m).

In simpler terms, Coulomb's Law tells us that the farther apart charges are, the weaker the force between them. Similarly, the larger the charges, the stronger the force. It's a key law in electrostatics, allowing us to predict the force interactions between charges based on their sizes and distances apart.

Charge Distribution

Charge distribution refers to how electric charge is arranged in a given system or space. In this exercise, we have three distinct point charges distributed along a line at specified positions.

The positioning of charges (or charge distribution) plays a critical role in determining the resultant forces and interactions between them.

• Position: Where each point charge is located along the line, such as at the origin or other points specified by coordinates (e.g., \(x=+2.00 \, \mathrm{cm}\)).
• Distribution: The overall layout or arrangement of charges affects how forces develop between them.

This distribution can be visualized like a set of magnets on a ruler. Depending on where you place each magnet, the resulting interactions, be it attraction or repulsion, can vary significantly.