To determine the magnitude and direction of the electrical force [tex]\( F_e \)[/tex] that [tex]\( q_1 \)[/tex] applies on [tex]\( q_2 \)[/tex], we will use Coulomb's Law. Coulomb's Law is given by:
[tex]\[ F_e = k \frac{|q_1 q_2|}{r^2} \][/tex]
Where:
- [tex]\( k \)[/tex] is Coulomb's constant, approximately [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 charges
- [tex]\( r \)[/tex] is the distance between the charges
Given:
- [tex]\( q_1 = -6 \, \mu C = -6 \times 10^{-6} \, C \)[/tex]
- [tex]\( q_2 = 3 \, \mu C = 3 \times 10^{-6} \, C \)[/tex]
- [tex]\( r = 0.002 \, m \)[/tex]
First, we calculate the magnitude of the force:
[tex]\[ F_e = k \frac{|q_1 q_2|}{r^2} \][/tex]
[tex]\[ F_e = 8.99 \times 10^9 \frac{|-6 \times 10^{-6} \times 3 \times 10^{-6}|}{(0.002)^2} \][/tex]
Calculate the product of the charges:
[tex]\[ |-6 \times 10^{-6} \times 3 \times 10^{-6}| = 18 \times 10^{-12} \][/tex]
[tex]\[ = 18 \times 10^{-12} \, \text{C}^2 \][/tex]
Now, square the distance [tex]\( r \)[/tex]:
[tex]\[ (0.002)^2 = 4 \times 10^{-6} \, \text{m}^2 \][/tex]
Substituting these values into Coulomb's Law:
[tex]\[ F_e = 8.99 \times 10^9 \frac{18 \times 10^{-12}}{4 \times 10^{-6}} \][/tex]
Calculate the division:
[tex]\[ \frac{18 \times 10^{-12}}{4 \times 10^{-6}} = 4.5 \times 10^{-6} \][/tex]
Now multiply by Coulomb's constant:
[tex]\[ F_e = 8.99 \times 10^9 \times 4.5 \times 10^{-6} \][/tex]
[tex]\[ F_e = 40455 \, \text{N} \][/tex]
So, the magnitude of the force is approximately [tex]\( 4.0455 \times 10^4 \, N \)[/tex].
Next, let's determine the direction of the force. Since [tex]\( q_1 \)[/tex] is negative and [tex]\( q_2 \)[/tex] is positive, [tex]\( q_1 \)[/tex] will attract [tex]\( q_2 \)[/tex]. Given that [tex]\( q_1 \)[/tex] is located 0.002 m north of [tex]\( q_2 \)[/tex], the positive charge [tex]\( q_2 \)[/tex] will experience a force towards the north.
Therefore, the magnitude and direction of the electrical force [tex]\( F_e \)[/tex] applied by [tex]\( q_1 \)[/tex] on [tex]\( q_2 \)[/tex] are:
- Magnitude: [tex]\( 4 \times 10^4 \, N \)[/tex]
- Direction: North
So the correct answer is:
- Magnitude: [tex]\( 4 \times 10^4 \, N \)[/tex]
- Direction: North
