Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve this problem, we need to find the individual forces exerted on charge [tex]\( q_3 \)[/tex] by charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex], and then compute the net force on [tex]\( q_3 \)[/tex]. Here are the given values:
- Charge [tex]\( q_1 = 2.0 \times 10^{-6} \)[/tex] C
- Charge [tex]\( q_2 = -3.0 \times 10^{-6} \)[/tex] C
- Charge [tex]\( q_3 = 1.21 \times 10^{-6} \)[/tex] C
- Distance between [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex], [tex]\( r_{13} = 0.05 \)[/tex] m
- Distance between [tex]\( q_2 \)[/tex] and [tex]\( q_3 \)[/tex], [tex]\( r_{23} = 0.03 \)[/tex] m
Coulomb's constant: [tex]\( k = 8.99 \times 10^9 \)[/tex] N m[tex]\(^2\)[/tex] C[tex]\(^{-2}\)[/tex]
### Step-by-Step Solution
1. Calculate the force [tex]\( \vec{F}_1 \)[/tex] exerted by [tex]\( q_1 \)[/tex] on [tex]\( q_3 \)[/tex]:
The formula for the electrostatic force [tex]\( F \)[/tex] is given by Coulomb’s law:
[tex]\[ F = k \frac{|q_1 q_3|}{r_{13}^2} \][/tex]
Plugging in the values:
[tex]\[ F_1 = 8.99 \times 10^9 \frac{(2.0 \times 10^{-6} \times 1.21 \times 10^{-6})}{(0.05)^2} \][/tex]
By computing this value, we get:
[tex]\[ F_1 = 8.702319999999999 \text{ N} \][/tex]
Since both charges [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex] are positive, the force [tex]\( F_1 \)[/tex] is repulsive and directed to the right, so [tex]\( \vec{F}_1 \)[/tex] is positive.
2. Calculate the force [tex]\( \vec{F}_2 \)[/tex] exerted by [tex]\( q_2 \)[/tex] on [tex]\( q_3 \)[/tex]:
Again, using Coulomb’s law:
[tex]\[ F_2 = k \frac{|q_2 q_3|}{r_{23}^2} \][/tex]
Plugging in the values:
[tex]\[ F_2 = 8.99 \times 10^9 \frac{(3.0 \times 10^{-6} \times 1.21 \times 10^{-6})}{(0.03)^2} \][/tex]
By computing this value, we get:
[tex]\[ F_2 = 36.25966666666666 \text{ N} \][/tex]
Since [tex]\( q_2 \)[/tex] is negative and [tex]\( q_3 \)[/tex] is positive, the force [tex]\( F_2 \)[/tex] is attractive and hence also directed to the right, so [tex]\( \vec{F}_2 \)[/tex] is positive.
3. Calculate the net force [tex]\( \vec{F} \)[/tex] on [tex]\( q_3 \)[/tex]:
The net force is the sum of the individual forces [tex]\( \vec{F}_1 \)[/tex] and [tex]\( \vec{F}_2 \)[/tex]:
[tex]\[ F_{\text{net}} = F_1 + F_2 \][/tex]
Substituting the values:
[tex]\[ F_{\text{net}} = 8.702319999999999 \text{ N} + 36.25966666666666 \text{ N} \][/tex]
By adding these values, we get:
[tex]\[ F_{\text{net}} = 44.96198666666666 \text{ N} \][/tex]
Therefore, the net force [tex]\( \vec{F} \)[/tex] on [tex]\( q_3 \)[/tex] is [tex]\( 44.96198666666666 \)[/tex] N directed to the right.
In summary:
[tex]\[ \vec{F}_1 = 8.702319999999999 \text{ N} \quad \text{(to the right)} \][/tex]
[tex]\[ \vec{F}_2 = 36.25966666666666 \text{ N} \quad \text{(to the right)} \][/tex]
[tex]\[ \vec{F}_{\text{net}} = 44.96198666666666 \text{ N} \quad \text{(to the right)} \][/tex]
- Charge [tex]\( q_1 = 2.0 \times 10^{-6} \)[/tex] C
- Charge [tex]\( q_2 = -3.0 \times 10^{-6} \)[/tex] C
- Charge [tex]\( q_3 = 1.21 \times 10^{-6} \)[/tex] C
- Distance between [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex], [tex]\( r_{13} = 0.05 \)[/tex] m
- Distance between [tex]\( q_2 \)[/tex] and [tex]\( q_3 \)[/tex], [tex]\( r_{23} = 0.03 \)[/tex] m
Coulomb's constant: [tex]\( k = 8.99 \times 10^9 \)[/tex] N m[tex]\(^2\)[/tex] C[tex]\(^{-2}\)[/tex]
### Step-by-Step Solution
1. Calculate the force [tex]\( \vec{F}_1 \)[/tex] exerted by [tex]\( q_1 \)[/tex] on [tex]\( q_3 \)[/tex]:
The formula for the electrostatic force [tex]\( F \)[/tex] is given by Coulomb’s law:
[tex]\[ F = k \frac{|q_1 q_3|}{r_{13}^2} \][/tex]
Plugging in the values:
[tex]\[ F_1 = 8.99 \times 10^9 \frac{(2.0 \times 10^{-6} \times 1.21 \times 10^{-6})}{(0.05)^2} \][/tex]
By computing this value, we get:
[tex]\[ F_1 = 8.702319999999999 \text{ N} \][/tex]
Since both charges [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex] are positive, the force [tex]\( F_1 \)[/tex] is repulsive and directed to the right, so [tex]\( \vec{F}_1 \)[/tex] is positive.
2. Calculate the force [tex]\( \vec{F}_2 \)[/tex] exerted by [tex]\( q_2 \)[/tex] on [tex]\( q_3 \)[/tex]:
Again, using Coulomb’s law:
[tex]\[ F_2 = k \frac{|q_2 q_3|}{r_{23}^2} \][/tex]
Plugging in the values:
[tex]\[ F_2 = 8.99 \times 10^9 \frac{(3.0 \times 10^{-6} \times 1.21 \times 10^{-6})}{(0.03)^2} \][/tex]
By computing this value, we get:
[tex]\[ F_2 = 36.25966666666666 \text{ N} \][/tex]
Since [tex]\( q_2 \)[/tex] is negative and [tex]\( q_3 \)[/tex] is positive, the force [tex]\( F_2 \)[/tex] is attractive and hence also directed to the right, so [tex]\( \vec{F}_2 \)[/tex] is positive.
3. Calculate the net force [tex]\( \vec{F} \)[/tex] on [tex]\( q_3 \)[/tex]:
The net force is the sum of the individual forces [tex]\( \vec{F}_1 \)[/tex] and [tex]\( \vec{F}_2 \)[/tex]:
[tex]\[ F_{\text{net}} = F_1 + F_2 \][/tex]
Substituting the values:
[tex]\[ F_{\text{net}} = 8.702319999999999 \text{ N} + 36.25966666666666 \text{ N} \][/tex]
By adding these values, we get:
[tex]\[ F_{\text{net}} = 44.96198666666666 \text{ N} \][/tex]
Therefore, the net force [tex]\( \vec{F} \)[/tex] on [tex]\( q_3 \)[/tex] is [tex]\( 44.96198666666666 \)[/tex] N directed to the right.
In summary:
[tex]\[ \vec{F}_1 = 8.702319999999999 \text{ N} \quad \text{(to the right)} \][/tex]
[tex]\[ \vec{F}_2 = 36.25966666666666 \text{ N} \quad \text{(to the right)} \][/tex]
[tex]\[ \vec{F}_{\text{net}} = 44.96198666666666 \text{ N} \quad \text{(to the right)} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
