Question

part 2 -- questions: after completing the reading, answer the following questions. any data, calculations, or models you use to support your thinking should be explained.

1. use coulomb’s law to find the force between a particle with a charge of +1 e and a particle with a charge of -1 e. the charges are 2.15 μm apart. after you calculate the force, determine if the force is attractive or repulsive.
2. make a prediction: if the first particle is replaced with a particle with a +4 e charge, will the force between the particles increase, decrease, or stay the same?

Explanation:

Question 1

Step 1: Recall Coulomb's Law formula

Coulomb's Law is given by \( F = k\frac{|q_1q_2|}{r^2} \), where \( k = 8.988\times 10^{9}\ \text{N·m}^2/\text{C}^2 \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them. The elementary charge \( e = 1.602\times 10^{-19}\ \text{C} \). Here, \( q_1 = + 1e \), \( q_2=- 1e \), and \( r = 2.15\ \mu\text{m}=2.15\times 10^{-6}\ \text{m} \).

Step 2: Substitute values into the formula

First, calculate \( |q_1q_2|=|(1e)(- 1e)|=e^{2}=(1.602\times 10^{-19}\ \text{C})^2 = 2.566404\times 10^{-38}\ \text{C}^2 \). Then, \( r^{2}=(2.15\times 10^{-6}\ \text{m})^2 = 4.6225\times 10^{-12}\ \text{m}^2 \). Now substitute into \( F \):

\( F=8.988\times 10^{9}\times\frac{2.566404\times 10^{-38}}{4.6225\times 10^{-12}} \)

Step 3: Calculate the force magnitude

First, calculate the fraction \( \frac{2.566404\times 10^{-38}}{4.6225\times 10^{-12}}\approx5.552\times 10^{-27} \). Then multiply by \( 8.988\times 10^{9} \):

\( F\approx8.988\times 10^{9}\times5.552\times 10^{-27}\approx4.99\times 10^{-17}\ \text{N} \) (approx \( 5.0\times 10^{-17}\ \text{N} \))

Step 4: Determine force nature

Since \( q_1 \) is positive and \( q_2 \) is negative, the force is attractive (opposite charges attract).

Step 1: Analyze Coulomb's Law dependence on charge

From Coulomb's Law \( F = k\frac{|q_1q_2|}{r^2} \), the force \( F \) is directly proportional to the product of the magnitudes of the charges (\( |q_1q_2| \)) when \( r \) is constant.

Step 2: Compare initial and new charge product

Initially, \( |q_1q_2| = |(1e)(- 1e)|=e^{2} \). After replacing \( q_1 \) with \( + 4e \), the new product \( |q_1'q_2|=|(4e)(- 1e)| = 4e^{2} \), which is 4 times the initial product (\( e^{2} \)). Since \( r \) and \( k \) are constant, the force will be 4 times the initial force, so the force increases.

Answer:

(for Question 1):
The magnitude of the force is approximately \( \boldsymbol{4.99\times 10^{-17}\ \text{N}} \) (or \( 5.0\times 10^{-17}\ \text{N} \)) and the force is attractive.

Question 2