Certainly! Let's walk through this problem step-by-step to determine the forces [tex]\(\vec{F}_1\)[/tex] and [tex]\(\vec{F}_2\)[/tex], and then find the net force [tex]\(\vec{F}_{\text{net}}\)[/tex] on [tex]\(q_3\)[/tex].
### Step-by-Step Solution:
1. Given values and constants:
- Charge [tex]\(q_1 = 12 \, \text{C}\)[/tex]
- Charge [tex]\(q_2 = -10 \, \text{C}\)[/tex]
- Charge [tex]\(q_3 = 5 \, \text{C}\)[/tex]
- Coulomb's constant [tex]\(k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2\)[/tex]
- Distance between [tex]\(q_1\)[/tex] and [tex]\(q_3\)[/tex], [tex]\(d_{13} = 0.3 \, \text{m}\)[/tex]
- Distance between [tex]\(q_2\)[/tex] and [tex]\(q_3\)[/tex], [tex]\(d_{23} = 0.5 \, \text{m}\)[/tex]
2. Calculate the force [tex]\(\vec{F}_1\)[/tex]:
[tex]\[
\vec{F}_1 = \frac{k |q_1 q_3|}{d_{13}^2}
\][/tex]
Plugging in the values:
[tex]\[
\vec{F}_1 = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \times |12 \, \text{C} \times 5 \, \text{C}|}{(0.3 \, \text{m})^2}
\][/tex]
The calculated value for [tex]\(\vec{F}_1\)[/tex] is:
[tex]\[
\vec{F}_1 = 5993333333333.334 \, \text{N}
\][/tex]
3. Calculate the force [tex]\(\vec{F}_2\)[/tex]:
[tex]\[
\vec{F}_2 = \frac{k |q_2 q_3|}{d_{23}^2}
\][/tex]
Plugging in the values:
[tex]\[
\vec{F}_2 = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \times |-10 \, \text{C} \times 5 \, \text{C}|}{(0.5 \, \text{m})^2}
\][/tex]
The calculated value for [tex]\(\vec{F}_2\)[/tex] is:
[tex]\[
\vec{F}_2 = 1798000000000.0 \, \text{N}
\][/tex]
4. Determine the net force on [tex]\(q_3\)[/tex]:
- Since [tex]\(q_1\)[/tex] and [tex]\(q_3\)[/tex] have opposite signs, [tex]\(\vec{F}_1\)[/tex] is attractive, meaning [tex]\(q_3\)[/tex] is attracted towards [tex]\(q_1\)[/tex].
- Since [tex]\(q_2\)[/tex] and [tex]\(q_3\)[/tex] also have opposite signs, [tex]\(\vec{F}_2\)[/tex] is attractive, meaning [tex]\(q_3\)[/tex] is attracted towards [tex]\(q_2\)[/tex].
- Assume the direction towards [tex]\(q_1\)[/tex] is positive and towards [tex]\(q_2\)[/tex] is negative.
The net force on [tex]\(q_3\)[/tex] then is:
[tex]\[
\vec{F}_{\text{net}} = \vec{F}_1 - \vec{F}_2
\][/tex]
Plugging in the calculated values:
[tex]\[
\vec{F}_{\text{net}} = 5993333333333.334 \, \text{N} - 1798000000000.0 \, \text{N}
\][/tex]
The calculated net force is:
[tex]\[
\vec{F}_{\text{net}} = 4195333333333.334 \, \text{N}
\][/tex]
### Conclusion:
- The force exerted on [tex]\(q_3\)[/tex] by [tex]\(q_1\)[/tex] is [tex]\(\vec{F}_1 = 5993333333333.334 \, \text{N}\)[/tex].
- The force exerted on [tex]\(q_3\)[/tex] by [tex]\(q_2\)[/tex] is [tex]\(\vec{F}_2 = 1798000000000.0 \, \text{N}\)[/tex].
- The net force on [tex]\(q_3\)[/tex] is [tex]\(\vec{F}_{\text{net}} = 4195333333333.334 \, \text{N}\)[/tex].
These values represent the magnitudes and respective directions of the forces acting on [tex]\(q_3\)[/tex].
