Question

A particle (charge \(=+19.0 \mu \mathrm{C}\) ) 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 experienced by the third particle placed at the origin is 12564.70 N.

Step-by-step solution

Identify the given values

First, we need to identify the given values: - Particle 1: charge \(q_1 = +19.0 \mu C\), position \(x_1 = -10.0 cm\) - Particle 2: charge \(q_2 = -57.0 \mu C\), position \(x_2 = +20.0 cm\) - Particle 3: charge \(q_3 = -3.80 \mu C\), position \(x_3 = 0 cm\) (origin) Note that all charges are given in microcoulombs and positions in centimeters. Convert these values to SI units for calculations: \(q_1 = 19.0 \times 10^{-6} C\), \(q_2 = -57.0 \times 10^{-6} C\), \(q_3 = -3.80 \times 10^{-6} C\) \(x_1 = -0.10 m\), \(x_2 = 0.20 m\), \(x_3 = 0 m\)

Calculate the pairwise distances

Calculate the distances between particles 1 and 3, and particles 2 and 3 (they all lie on the x-axis): \(r_{13} = |x_1 - x_3| = |-0.10 - 0| = 0.10 m\) \(r_{23} = |x_2 - x_3| = |0.20 - 0| = 0.20 m\)

Calculate the forces on particle 3 using Coulomb's Law

Use Coulomb's Law to calculate the forces experienced by particle 3 due to particles 1 and 2: Coulomb's Law: \(F = k \frac{|q_1 q_2|}{r^2}\) where \(k = 8.99 \times 10^9 N m^2/C^2\) is the electrostatic constant. Force due to particle 1: \(F_{13} = k \frac{|q_1 q_3|}{r_{13}^2} = 8.99 \times 10^9 \frac{19.0 \times 10^{-6} \times 3.80 \times 10^{-6}}{(0.10)^2} = 10132.762 N\) Force due to particle 2: \(F_{23} = k \frac{|q_2 q_3|}{r_{23}^2} = 8.99 \times 10^9 \frac{57.0 \times 10^{-6} \times 3.80 \times 10^{-6}}{(0.20)^2} = 2431.9362 N\) The direction of \(F_{13}\) is positive (to the right) since like charges repel, and the direction of \(F_{23}\) is also positive (to the right) since opposite charges attract.

Calculate the total force on particle 3

Add the force vectors experienced by particle 3 due to particles 1 and 2: \(F_{total} = F_{13} + F_{23} = 10132.762 + 2431.9362 = 12564.6982 N\) The magnitude of the total electrostatic force on particle 3 placed at the origin is \(12564.70 N\) (rounded to two decimal places), and the direction of the force is positive (to the right).

Key concepts

Coulomb's Law

Coulomb's Law is foundational in understanding the interaction between charged particles. Named after Charles-Augustin de Coulomb who introduced the law in the 1780s, it describes the force of attraction or repulsion between two point charges. The formula is beautifully simple yet powerful:

\[\[\begin{align*}F = k \frac{|q_1 q_2|}{r^2}\tag{1}ewlineewlineewlineewlineewlineewlineewlineewlineewlinewhere:ewlineewline

\li\( F \) is the magnitude of the electrostatic force between two charges,
• \( k \) is the electrostatic constant (also known as Coulomb's constant),
• \( q_1 \) and \( q_2 \) are the amounts of electric charge on the two particles,
• \( r \) is the distance between the centers of the two charges.

ewlineIn our exercise, Coulomb's Law allows us to calculate the force experienced by a third particle due to the presence of two other charged particles along the same axis. We use it to find the forces between Particle 1 and 3, and between Particle 2 and 3 separately and then combine these forces to get the total force on Particle 3. Coulomb's Law is a vector equation, meaning that it provides both magnitude and direction of the force; however, in this problem, we are only considering the magnitudes since all particles lie on the same line.

Electrostatic Constant

The electrostatic constant, denoted by \(k\), is a proportionality factor that appears in Coulomb's Law. Its value is approximately \(8.99 \times 10^9 N m^2/C^2\), and it plays a key role in determining the strength of the electrostatic force between charges. The value of the electrostatic constant derives from the permittivity of free space and is a universal constant in electrostatics.

In simple terms, the electrostatic constant helps us translate the theoretical formula of Coulomb's Law into a practical calculation that we can use to predict the real-world forces between charges. For instance, in the exercise, the electrostatic constant is used to calculate the forces \(F_{13}\) and \(F_{23}\). It's important to keep mind that while the constant seems to be just a number, it conveys the relationship between the units involved in electrostatic calculations—newtons for force, meters for distance, and coulombs for electrical charge.

Electric Charge

Electric charge, symbolized as \(q\), is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. Charges come in two types: positive and negative. Like charges repel each other, while opposite charges attract, as described by Coulomb's Law.

In our example, we see charges denoted in microcoulombs (\(\mu C\)), which are a millionth of a coulomb. To work with Coulomb's Law, we convert these values into the standard unit of charge, the coulomb (\(C\)).

• Particle 1 has a positive charge of \(+19.0 \mu C \),
• Particle 2 has a negative charge of \(-57.0 \mu C\),
• Particle 3, for which we want to find the electrostatic force experienced, has a negative charge of \(-3.80 \mu C\).

The concept of electric charge is essential for explaining not only electrostatic phenomena but also electrical currents and electromagnetic fields in general. The movement of charges is what creates electricity. In the context of this problem, understanding the nature of electric charge and its measurement is crucial to correctly setting up and solving for the forces acting between the charges.