Question

Two point charges are fixed on the \(x\) -axis: \(q_{1}=6.0 \mu \mathrm{C}\) is located at the origin, \(O,\) with \(x_{1}=0.0 \mathrm{~cm},\) and \(q_{2}=-3.0 \mu \mathrm{C}\) is located at point \(A,\) with \(x_{2}=8.0 \mathrm{~cm} .\) Where should a third charge, \(q_{3},\) be placed on the \(x\) -axis so that the total electrostatic force acting on it is zero? a) \(19 \mathrm{~cm}\) c) \(0.0 \mathrm{~cm}\) e) \(-19 \mathrm{~cm}\) b) \(27 \mathrm{~cm}\) d) \(8.0 \mathrm{~cm}\)

Short answer

Answer: e) \(-19 \mathrm{~cm}\)

Step-by-step solution

Understand the problem and define the variables

We have two charges, \(q_1\) and \(q_2\), and their positions \(x_1\) and \(x_2\). We need to find the position \(x_3\) for a third charge, \(q_3\), so that the total electrostatic force acting on it is zero.

Find the force acting on \(q_3\) from \(q_1\)

By Coulomb's Law, the electrostatic force acting on \(q_3\) due to \(q_1\) is: $$F_{13} = k\frac{q_1 q_3}{(x_3-x_1)^2}$$ where k is Coulomb's constant, which is approximately \(8.99 \times 10^9 \frac{N m^2}{C^2}\).

Find the force acting on \(q_3\) from \(q_2\)

By Coulomb's Law, the electrostatic force acting on \(q_3\) due to \(q_2\) is: $$F_{23} = k\frac{q_2 q_3}{(x_3-x_2)^2}$$

Set the forces equal to each other

For the total force acting on the third charge to be zero, the force due to \(q_1\) must be equal and opposite to the force due to \(q_2\). Therefore, $$F_{13} = F_{23}$$ Substituting the expressions for \(F_{13}\) and \(F_{23}\) from Steps 2 and 3, we get: $$k\frac{q_1 q_3}{(x_3-x_1)^2} = k\frac{q_2 q_3}{(x_3-x_2)^2}$$

Solve for \(x_3\)

We can cancel out \(k\) and \(q_3\) from both sides, and rearrange the equation to solve for \(x_3\). Using the given values of \(q_1\), \(q_2\), \(x_1\) and \(x_2\), we get the following equation: $$\frac{6.0 \times 10^{-6}C}{(x_3-0.0)^2} = -\frac{3.0 \times 10^{-6}C}{(x_3-8.0)^2}$$ Solving the equation for \(x_3\), we get: $$x_3 = -19 cm$$ So, the correct option is: e) \(-19 \mathrm{~cm}\)

Key concepts

Coulomb's Law

Coulomb's Law helps us calculate the force between two charged objects. This fundamental law of electrostatics is expressed as: \[ F = k \frac{|q_1 q_2|}{r^2} \] where:

• \( F \) is the force between the charges, measured in Newtons (N).
• \( k \) is Coulomb's constant, approximately \(8.99 \times 10^9 \frac{N m^2}{C^2}\).
• \( q_1 \) and \( q_2 \) are the amounts of the charges, measured in Coulombs (C).
• \( r \) is the distance between the charges.

Coulomb's Law shows 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. Basically, the closer and more charged they are, the bigger the force.

Electrostatic Force

Electrostatic force is the interaction between charged particles. Opposite charges attract, while like charges repel each other. This attraction or repulsion is the essence of electrostatic force. In the context of the problem, the electrostatic force plays a crucial role because we want these forces to balance out. If they balance, the total force acting on the third charge is zero. Here’s why that matters:

• It allows us to place the charge such that its position results in no net movement.
• Understanding these balance points can help control where charges will stay stable in electric fields.

Getting the forces to be equal and opposite is key to solving problems involving electrostatic equilibrium.

Point Charges

Point charges are often used in Physics to simplify complex problems. They are assumed to have all their charge concentrated at a single point. This makes calculations and concepts like Coulomb's Law easier to work with.In the exercise, both charges \(q_1\) and \(q_2\) are considered point charges.

• They have specific positions along the x-axis.
• They help us understand the interaction of forces in this simple scenario.

Knowing how point charges interact provides a basis for understanding more complicated systems, such as distributions of charge where charge is spread out over an area or volume.

Charge Distribution

Charge distribution refers to how electric charge is spread out in a space. In this context, we look at how point charges are aligned on an axis.Even though point charges are singular, imagining them in different positions helps to solve problems:

• We assume point charges in a linear arrangement, like on the x-axis, to simplify the prediction of forces.
• This simplification helps in focusing on the essence of electrostatic forces without the complexities of real-world structures.

In the exercise, the charge \(q_3\) is placed at multiple potential points to find where the forces balance. This kind of analysis is known as finding an equilibrium position, crucial for understanding fields and forces around distributed charges.