Question

Charge \(q_{1}=1.4 \cdot 10^{-8} \mathrm{C}\) is placed at the origin. Charges \(q_{2}=-1.8 \cdot 10^{-8} \mathrm{C}\) and \(q_{3}=2.1 \cdot 10^{-8} \mathrm{C}\) are placed at points \((0.18 \mathrm{~m}, 0 \mathrm{~m})\) and \((0 \mathrm{~m}, 0.24 \mathrm{~m}),\) respec- tively, as shown in the figure. Determine the net electrostatic force (magnitude and direction) on charge \(q_{3}\)

Short answer

The net electrostatic force on charge \(q_{3}\) has a magnitude of approximately \(3.46 \times 10^{-7} N\) and a direction of approximately \(-111.77°\) from the x-axis.

Step-by-step solution

Determine magnitude of force between \(q_{1}\) and \(q_{3}\)

Coulomb's Law states that the magnitude of the electrostatic force F between two charges \(q_{1}\) and \(q_{2}\) is given by: \(F=\frac{k \cdot |q_{1} \cdot q_{2}|}{r^2}\) Where k is the electrostatic constant, \(\approx 8.99 \times 10^{9} Nm^2/C^2\). The distance between charges \(q_{1}\) and \(q_{3}\) is equal to the distance from the origin to \((0, 0.24m)\): \(r=\sqrt{0^2 + (0.24)^2}=0.24m\) Now, we can plug in the given values to find the force magnitude: \(F_{13}=\frac{8.99 \times 10^{9} Nm^2/C^2 \cdot |1.4 \times 10^{-8} C \cdot 2.1 \times 10^{-8} C|}{(0.24m)^2} \approx 1.82 \times 10^{-7}N\)

Determine magnitude of force between \(q_{2}\) and \(q_{3}\)

Follow the same procedure as step 1, but for charges \(q_{2}\) and \(q_{3}\). The distance between charges \(q_{2}\) and \(q_{3}\) is: \(r=\sqrt{(0.18)^2 + (0.24)^2} \approx 0.30 m\) Now, calculate the force magnitude: \(F_{23}=\frac{8.99 \times 10^{9} Nm^2/C^2 \cdot |-1.8 \times 10^{-8} C \cdot 2.1 \times 10^{-8} C|}{(0.30m)^2} \approx 1.68 \times 10^{-7} N\)

Find the components of the forces \(F_{13}\) and \(F_{23}\)

In order to add the forces as vectors, we need to find their respective x and y components. For \(F_{13}\), the force acts directly upward, so: \(F_{13x} = 0\) \(F_{13y} = 1.82 \times 10^{-7} N\) Now, we need to find the components of \(F_{23}\). Find the angle between the horizontal line and the line connecting charges \(q_{2}\) and \(q_{3}\): \(\theta = \arctan(\frac{0.24}{0.18}) \approx 53.13°\) Now, find the x and y components of \(F_{23}\) using trigonometry: \(F_{23x} = F_{23} \cos{\theta} \approx -1.28 \times 10^{-7} N\) \(F_{23y} = F_{23} \sin{\theta} \approx 1.39 \times 10^{-7} N\)

Calculate the net force on \(q_{3}\)

Now, add the x and y components of the forces to find the net force: \(F_{netx} = F_{13x} + F_{23x} = 0 + (-1.28 \times 10^{-7} N) = -1.28 \times 10^{-7} N\) \(F_{nety} = F_{13y} + F_{23y} = 1.82 \times 10^{-7} N + 1.39 \times 10^{-7} N = 3.21 \times 10^{-7} N\) The net force's magnitude is found using the Pythagorean theorem: \(F_{net} = \sqrt{(-1.28 \times 10^{-7} N)^2 + (3.21 \times 10^{-7} N)^2} \approx 3.46 \times 10^{-7} N\)

Calculate the direction of the net force

Now, we can find the direction of the net force: \(\theta_{net} = \arctan(\frac{F_{nety}}{F_{netx}}) = \arctan(\frac{3.21 \times 10^{-7} N}{-1.28 \times 10^{-7} N}) \approx -111.77°\) Since the angle is negative, it means the net force's direction is in the 4th quadrant. The net electrostatic force on charge \(q_{3}\) has a magnitude of approximately \(3.46 \times 10^{-7} N\) and a direction of approximately \(-111.77°\) from the x-axis.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes how the electrostatic force is calculated between two charged particles. You can think of this as the push or pull between objects with electric charge. The law is mathematically expressed as:
\[F = \frac{k \times |q_1 \times q_2|}{r^2}\]
where:

• \(F\) is the force between two charges
• \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \, \mathrm{Nm^2/C^2}\).
• \(q_1\) and \(q_2\) are the amounts of charge
• \(r\) is the distance between charges

Coulomb's Law gives us a way to calculate the force only by knowing the magnitudes of the charges and the distance between them. It tells us that the force is attractive if the charges are opposite, and repulsive if they are the same. This formula shows that the force gets weaker as the distance between charges increases.

Vector Addition

When dealing with forces in physics, especially electrostatic forces, it's important to remember they are vector quantities. Vector quantities have both magnitude and direction. Therefore, understanding vector addition becomes crucial when determining the net force acting on an object.
To add vectors:

• Break them down into components. Typically, these are horizontal (x-axis) and vertical (y-axis) components.
• Sum up all the components along the same axis. For instance, sum all x-components and y-components separately.
• Use the Pythagorean theorem to calculate the resultant vector, which is the combination of these components.
• The direction of the resultant vector can be calculated using trigonometric functions like \(\tan^{-1}\).

Vector addition is a method that allows us to combine forces acting at different angles. By applying this, we find the net electrostatic force - both its magnitude and direction, like in the original exercise with charge \(q_3\).

Electric Charge Interactions

Electric charge interactions happen when charged particles come into proximity and begin to experience forces due to their charges. Charges can be either positive or negative, influencing whether they attract or repel. This is best described by the following principles:

• Like charges repel each other. Two positive charges or two negative charges push away from one another.
• Opposite charges attract each other. A positive and a negative charge pull towards each other.

The strength of these interactions is governed by Coulomb's Law, as forces between charges depend on the distance between them and the magnitude of their charges. Understanding how these interactions work helps predict how charged objects, like atomic particles or even balloons rubbed against sweaters, will behave. In the context of the exercise, knowing whether the charges between \(q_1\), \(q_2\), and \(q_3\) attract or repel is critical in correctly determining the direction and magnitude of the net force on \(q_3\). These interactions underpin why the forces can be helpful or disruptive in various contexts – and why balancing charges is essential in electronics and chemistry.