Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To solve this problem, you need to apply Coulomb's law, which quantifies the electrostatic force between two point charges. Coulomb's law is given by the formula:
[tex]\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the magnitude of the force between the charges,
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex],
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Let's plug in the given values.
1. Convert the charges from microcoulombs (μC) to coulombs (C):
- [tex]\( q_1 = -6 \, \mu C = -6 \times 10^{-6} \, \text{C} \)[/tex]
- [tex]\( q_2 = 3 \, \mu C = 3 \times 10^{-6} \, \text{C} \)[/tex]
2. Substitute the known values into Coulomb's law:
- [tex]\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex]
- [tex]\( r = 0.002 \, \text{m} \)[/tex]
[tex]\[ F = 8.99 \times 10^9 \cdot \frac{|-6 \times 10^{-6} \cdot 3 \times 10^{-6}|}{(0.002)^2} \][/tex]
3. Calculate the product of the charges:
[tex]\[ |-6 \times 10^{-6} \cdot 3 \times 10^{-6}| = 18 \times 10^{-12} \][/tex]
4. Calculate the distance squared:
[tex]\[ (0.002)^2 = 4 \times 10^{-6} \][/tex]
5. Substitute these values into the formula:
[tex]\[ F = 8.99 \times 10^9 \cdot \frac{18 \times 10^{-12}}{4 \times 10^{-6}} \][/tex]
6. Simplify the fraction:
[tex]\[ \frac{18 \times 10^{-12}}{4 \times 10^{-6}} = 4.5 \times 10^{-6} \][/tex]
7. Calculate the final force magnitude:
[tex]\[ F = 8.99 \times 10^9 \cdot 4.5 \times 10^{-6} = 40455 \, \text{N} \][/tex]
The magnitude of the force is approximately [tex]\( 40455 \, \text{N} \)[/tex].
To determine the direction:
- [tex]\( q_1 \)[/tex] is negative.
- [tex]\( q_2 \)[/tex] is positive.
- Since opposite charges attract, the force on [tex]\( q_2 \)[/tex] due to [tex]\( q_1 \)[/tex] will be towards [tex]\( q_1 \)[/tex]. Since [tex]\( q_1 \)[/tex] is north of [tex]\( q_2 \)[/tex], the force on [tex]\( q_2 \)[/tex] will be directed south.
Therefore, the correct answer is:
- Magnitude: [tex]\( 4 \times 10^4 \, \text{N} \)[/tex]
- Direction: South
[tex]\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the magnitude of the force between the charges,
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex],
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Let's plug in the given values.
1. Convert the charges from microcoulombs (μC) to coulombs (C):
- [tex]\( q_1 = -6 \, \mu C = -6 \times 10^{-6} \, \text{C} \)[/tex]
- [tex]\( q_2 = 3 \, \mu C = 3 \times 10^{-6} \, \text{C} \)[/tex]
2. Substitute the known values into Coulomb's law:
- [tex]\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex]
- [tex]\( r = 0.002 \, \text{m} \)[/tex]
[tex]\[ F = 8.99 \times 10^9 \cdot \frac{|-6 \times 10^{-6} \cdot 3 \times 10^{-6}|}{(0.002)^2} \][/tex]
3. Calculate the product of the charges:
[tex]\[ |-6 \times 10^{-6} \cdot 3 \times 10^{-6}| = 18 \times 10^{-12} \][/tex]
4. Calculate the distance squared:
[tex]\[ (0.002)^2 = 4 \times 10^{-6} \][/tex]
5. Substitute these values into the formula:
[tex]\[ F = 8.99 \times 10^9 \cdot \frac{18 \times 10^{-12}}{4 \times 10^{-6}} \][/tex]
6. Simplify the fraction:
[tex]\[ \frac{18 \times 10^{-12}}{4 \times 10^{-6}} = 4.5 \times 10^{-6} \][/tex]
7. Calculate the final force magnitude:
[tex]\[ F = 8.99 \times 10^9 \cdot 4.5 \times 10^{-6} = 40455 \, \text{N} \][/tex]
The magnitude of the force is approximately [tex]\( 40455 \, \text{N} \)[/tex].
To determine the direction:
- [tex]\( q_1 \)[/tex] is negative.
- [tex]\( q_2 \)[/tex] is positive.
- Since opposite charges attract, the force on [tex]\( q_2 \)[/tex] due to [tex]\( q_1 \)[/tex] will be towards [tex]\( q_1 \)[/tex]. Since [tex]\( q_1 \)[/tex] is north of [tex]\( q_2 \)[/tex], the force on [tex]\( q_2 \)[/tex] will be directed south.
Therefore, the correct answer is:
- Magnitude: [tex]\( 4 \times 10^4 \, \text{N} \)[/tex]
- Direction: South
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
