Problem 12 - University Physics with Modern Physics

Question

A negative charge \(-0.550 \mu C\) 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

(a) The unknown charge is +3.65 μC. (b) The force is 0.200 N downward.

Step 1: Understand Coulomb's Law

Coulomb's law states that the force between two point charges is given by \( F = k \frac{|q_1 q_2|}{r^2} \), where \( F \) is the force, \( k = 8.988 \times 10^9 \, \mathrm{N\,m^2/C^2} \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

Step 2: Plug Known Values into Coulomb's Law

We know \( F = 0.200 \, \mathrm{N} \), \( q_1 = -0.550 \, \mu \mathrm{C} = -0.550 \times 10^{-6} \, \mathrm{C} \) and \( r = 0.300 \, \mathrm{m} \). Plug these values into Coulomb's law:\[ 0.200 = 8.988 \times 10^9 \frac{|-0.550 \times 10^{-6} \cdot q_2|}{0.300^2} \]

Step 3: Solve for Unknown Charge Magnitude

Rearrange the equation to solve for \( q_2 \):\[ q_2 = \frac{0.200 \times 0.300^2}{8.988 \times 10^9 \times 0.550 \times 10^{-6}} \]Calculate \( q_2 \):\[ q_2 = \frac{0.200 \times 0.090}{8.988 \times 10^9 \times 0.550 \times 10^{-6}} \approx 3.65 \times 10^{-6} \, \mathrm{C} \]

Step 4: Determine the Sign of the Unknown Charge

Since a negative charge exerts an upward force on an unknown charge below it, and the force is attractive (between opposite signs), the unknown charge must be positive. Therefore, \( q_2 = +3.65 \times 10^{-6} \, \mathrm{C} \).

Step 5: Determine the Force Exerted by the Unknown Charge

According to Newton's third law, the force exerted by the unknown charge on the \(-0.550 \, \mu \mathrm{C} \) charge will have the same magnitude but opposite direction as the force exerted by the \(-0.550 \, \mu \mathrm{C} \) charge. Thus, the force magnitude is 0.200 N and the direction is downward, towards the \(-0.550 \, \mu \mathrm{C} \) charge.

Key Concepts

Electric Force

Electric force is a fundamental interaction between charged objects. It's a type of force exerted by charged particles on one another. This force can either attract or repel based on the nature of the charges involved. An important equation, known as Coulomb's Law, helps calculate this force.Coulomb's Law gives us:\[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) stands for the force between the charges,
• \( k \) is the Coulomb's constant, approximately \( 8.988 \times 10^9 \, \mathrm{N\,m^2/C^2} \),
• \( q_1 \) and \( q_2 \) are the charges in question,
• \( r \) is the distance separating these charges.

Under this law, the electric force decreases as the distance between charges increases. This is akin to the inverse square law observed in gravity.

Point Charges

Point charges are idealized charges that are thought of as existing at a single point in space. In the context of Coulomb's Law, these charges are assumed to have no physical size, allowing calculations to be made without considering the complexities of spatial dimensions.In many physics problems, we treat charges as point charges to simplify calculations and focus on the interaction itself rather than the physical properties of the charge holder.In the given exercise:

• The \(-0.550 \, \mu \mathrm{C} \) charge is considered a point charge exerting an upward force.
• Another charge lies directly below it at a known distance, further simplifying the setup as they can be modeled in a straight vertical line, making calculations straightforward.

This simplification allows us to use Coulomb's Law effectively as it assumes point charge interactions.

Charge Interaction

Charge interaction refers to the behavior and effect charges have on each other, particularly in terms of attraction or repulsion. Charges are generally classified into two types: positive and negative.Here are the rules for charge interaction:

• Like charges (both positive or both negative) repel each other.
• Opposite charges (one positive, one negative) attract each other.

In the problem, the given charge \(-0.550 \, \mu \mathrm{C} \) is pulling the unknown charge upward, indicating an attractive force.Since attraction occurs between opposite charges, the unknown charge must be positive. This conclusion is critical for further analysis, including solving for unknowns using Coulomb's Law.

Newton's Third Law

Newton's third law is often phrased as "for every action, there is an equal and opposite reaction." This principle is vital in understanding mutual force interactions between charges.In the context of electric forces:

• If Charge A exerts a force on Charge B, Charge B exerts an equal and opposite force on Charge A.

In the exercise, if the known \(-0.550 \, \mu \mathrm{C} \) charge exerts a force of 0.200 N upwards on the unknown charge, then according to Newton's third law, the unknown charge must exert a 0.200 N force back on the \(-0.550 \, \mu \mathrm{C} \) charge, but in the opposite direction (downwards).This reflection of force helps ensure consistency across calculations and reinforces the nature of equal and opposite interactions found in physics.