Question

A particle (charge \(=+19.0 \mu C)\) is located on the \(x\) -axis at \(x=-10.0 \mathrm{~cm},\) and a second particle (charge \(=-57.0 \mu \mathrm{C})\) is placed on the \(x\) -axis at \(x=+20.0 \mathrm{~cm} .\) What is the magnitude of the total electrostatic force on a third particle (charge = \(-3.80 \mu \mathrm{C})\) placed at the origin \((x=0) ?\)

Short answer

Answer: The magnitude of the total electrostatic force on the third particle is \(9.56 \times 10^{-2} N\).

Step-by-step solution

Identify the charges and their positions

We have three charged particles: 1. First particle: Charge = \(+19.0 \mu C\), Position = \(-10.0 cm\) 2. Second particle: Charge = \(-57.0 \mu C\), Position = \(+20.0 cm\) 3. Third particle: Charge = \(-3.80 \mu C\), Position = \(0 cm\) (origin)

Set up the Coulomb's Law formula for the forces

Coulomb's Law formula is: \(F = k \frac{|q_1q_2|}{r^2}\), where k is Coulomb's constant (\(k = 8.99 \times 10^9 Nm^2/C^2\)), \(q_1\) and \(q_2\) are the charges, and r is the distance between the charges. We will apply Coulomb's Law for two pairs: between the first and third particles and between the second and third particles.

Calculate the distance between each pair of particles

For the first pair (particles 1 and 3) the distance is: \(r_{13} = |-10.0 - 0| = 10.0 cm = 0.1 m\) For the second pair (particles 2 and 3) the distance is: \(r_{23} = |20.0 - 0| = 20.0 cm = 0.2 m\)

Apply Coulomb's Law to find the force between each pair

Now we calculate the forces between each pair of particles: Force between particles 1 and 3: \(F_{13} = k \frac{|q_1q_3|}{r_{13}^2} = 8.99 \times 10^9 \frac{|(19.0 \times 10^{-6 })(-3.80 \times 10^{-6})|}{(0.1)^2} = 1.02 \times 10^{-2} N\) Force between particles 2 and 3: \(F_{23} = k \frac{|q_2q_3|}{r_{23}^2} = 8.99 \times 10^9 \frac{|(-57.0 \times 10^{-6})(-3.80 \times 10^{-6})|}{(0.2)^2} = 8.54 \times 10^{-2} N\) Note that both forces are attractive because the charges of the particles are opposite.

Use vector addition to find the total force on the third particle

Since both forces are acting on the third particle along the x-axis, we can add them directly: \(F_{total} = F_{13} + F_{23} = 1.02 \times 10^{-2} N + 8.54 \times 10^{-2} N = 9.56 \times 10^{-2} N\) The magnitude of the total electrostatic force on the third particle is \(9.56 \times 10^{-2} N\).

Key concepts

Coulomb's Law

Understanding Coulomb's Law is essential to calculate the electrostatic force between two charged particles. This fundamental principle of electromagnetism is defined by the equation:\[F = k \frac{|q_1q_2|}{r^2}\], where:\

• \(F\) represents the magnitude of the electrostatic force between the particles.
• \(k\) is the Coulomb's constant, which has a value of approximately \(8.99 \times 10^9 Nm^2/C^2\).
• \(|q_1q_2|\) is the absolute value of the product of the two charges involved.
• \(r\) is the distance between the centers of the two charges.

Coulomb's Law tells us that the electrostatic force changes in proportion to the product of the two charges and inversely with the square of the distance between them. The force is attractive if the charges have opposite signs, and repulsive if they have the same sign.

When applying this law to our textbook problem, we use the charges of the particles and their distances from the third particle to calculate the forces exerted on it. It is important to convert distances from centimeters to meters because the SI unit for distance in the Coulomb's Law formula is meters. The product of the charge magnitudes and the inverse square of the distance directly provides the magnitude of force they exert on 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 charges: positive and negative. Like charges repel each other, while unlike charges attract.

In the context of the electrostatic force calculation, charges are typically measured in coulombs (C). However, in many practical situations, including our textbook problem, charges might be presented in microcoulombs (\(\mu C\)), where one microcoulomb equals \(1 \times 10^{-6}\) coulombs. It is crucial to convert microcoulombs to coulombs when applying Coulomb's Law, to ensure that the units match those prescribed by the international system of units (SI).

In our problem, we have three particles each with a given charge: the first particle has a positive charge, the second has a negative charge, and the third one, placed at the origin, also has a negative charge. The interaction between these charges is governed by the sign and magnitude of each charge, determining the direction and strength of the electrostatic force exerted on the third particle.

Vector Addition

Vector addition is a mathematical operation used to combine two or more vectors. In physics, especially when dealing with forces, vector addition allows us to find the resultant force when multiple forces are acting on an object. Vectors have both magnitude and direction, which means that simply adding the magnitudes will not suffice when the forces are not aligned.

In our exercise, however, the situation is simplified since all the particles are lined up along the x-axis, meaning the forces have a one-dimensional character. Therefore, we can add the magnitudes of the forces directly, taking into account their direction (sign). In a more complex scenario, where forces are not aligned, it would be necessary to use the principles of vector addition, such as breaking down the vectors into their components and then adding these components separately.

For the third particle in our problem, we find the total force by simply summing the forces exerted on it by the first and second particles, because they are all along the same line. This calculation yields the total electrostatic force experienced by the third particle at the origin.