Problem 14 - University Physics with Modern Physics

Question

A negative charge of \(-0.550 \mu C\) exerts an upward \(0.200-\mathrm{N}\) force on an unknown charge 0.300 \(\mathrm{m}\) directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the \(-0.550-\mu \mathrm{C}\) charge?

Step-by-Step Solution

Answer

The unknown charge is +3.64 nC; it exerts a 0.200 N downward force.

Step 1: Understand Coulomb's Law

Coulomb's Law states that the electrostatic force between two point charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between them. The formula is given by:\[ F = k \frac{|q_1 q_2|}{r^2} \]where \( F \) is the force, \( k = 8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2} \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

Step 2: Identify Known Values

From the problem statement, we know the following:- The negative charge \( q_1 = -0.550 \, \mu C = -0.550 \times 10^{-6} \, C \).- The force exerted is \( F = 0.200 \, N \).- The distance between the charges \( r = 0.300 \, m \).Our task is to find the magnitude and sign of the unknown charge \( q_2 \).

Step 3: Rearrange Coulomb's Law Formula

We need to solve for the unknown charge \( q_2 \). Rearrange Coulomb's Law formula to isolate \( |q_2| \):\[ |q_2| = \frac{F \cdot r^2}{k \cdot |q_1|} \]

Step 4: Substitute Known Values

Substitute the known values into the rearranged formula:\[ |q_2| = \frac{0.200 \, N \times (0.300 \, m)^2}{8.99 \times 10^9 \, N\cdot m^2/C^2 \times 0.550 \times 10^{-6} \, C} \]

Step 5: Calculate |q_2|

Perform the calculations:\[ |q_2| = \frac{0.200 \times 0.09}{8.99 \times 10^9 \times 0.550 \times 10^{-6}} \approx 3.64 \times 10^{-9} \mathrm{C} \]Because the force is upward and the known charge is negative, the unknown charge must be positive to result in an attractive force.

Step 6: Determine the Direction of the Force

According to Newton's Third Law, every action has an equal and opposite reaction. Thus, the force that the unknown charge \( q_2 \) exerts on the \(-0.550 \, \mu C\) charge is equal in magnitude and opposite in direction to the force that \(-0.550 \, \mu C\) exerts on \( q_2 \). This means the force is still \( 0.200 \, N \), but directed downward.

Step 7: Summarize the Results

The unknown charge has a magnitude of approximately \( 3.64 \, nC \) (Coulombs) and is positive. The force exerted by the unknown charge on the \(-0.550 \, \mu C\) charge is \( 0.200 \, N \) directed downward.

Key Concepts

Electrostatic Force

Electrostatic force is a fundamental interaction that occurs between charged particles. It is a non-contact force, which means it can act over a distance without the charges needing to touch each other. This force is responsible for the attraction or repulsion between electric charges. The magnitude of the electrostatic force can be calculated using Coulomb's Law, which is encapsulated in the formula: \[ F = k \frac{|q_1 q_2|}{r^2} \]Here, \( F \) represents the force between two charges, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, \( r \) is the distance separating them, and \( k \) is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2} \). The force is attractive if the charges are of opposite signs, and repulsive if they have the same sign.

• Charges of opposite sign attract each other.
• Charges of the same sign repel each other.

Coulomb's law provides a way to quantify these interactions mathematically, allowing us to predict how charged particles will behave in various circumstances.

Point Charges

In the context of electrostatics, point charges are hypothetical charges located at a single point in space. These are idealized models, used to simplify complex interactions between charges by assuming that all of their charge is concentrated into an infinitely small space. Despite their simplicity, point charges are a crucial concept for understanding electrostatic interactions and are often used in Physics problems and experiments.

• The concept helps in easily calculating forces between charges.
• All charge effects are considered to be emanating from just one point.

When dealing with point charges, as in the provided exercise, it becomes straightforward to apply Coulomb's Law to determine the force acting between them. This abstraction is useful for visualizing and solving complex physical problems without getting into the messy details of the physical size and shape of charged objects.

Newton's Third Law

Newton's Third Law is a principle that underpins much of classical physics, and it states: "For every action, there is an equal and opposite reaction." In the context of electrostatic forces, this law applies quite clearly. When one charge exerts a force on another charge, the second charge simultaneously exerts a force of equal magnitude and opposite direction on the first charge.This principle ensures that forces between charges are mutual. If one charge pulls another charge upwards with a certain force, the second charge pulls the first charge downward with the same force.

• Action and reaction forces are equal and opposite.
• This symmetry is key in analyzing force interactions between charges.

In the specific exercise problem, when the charge of \(-0.550 \, \mu C\) exerts a force of \(0.200 \, \mathrm{N}\) upwards on the unknown charge, the unknown charge must exert a force of \(0.200 \, \mathrm{N}\) downwards on the \(-0.550 \, \mu C\) charge, aligning perfectly with Newton's Third Law. This helps us understand not only how forces are distributed but also how they ensure the consistency of motion as predicted by classical mechanics.