To find the force acting on a dust particle with a charge of [tex]\(-2.30 \times 10^{-8}\)[/tex] C located 0.250 m away from a dust cloth carrying a charge of [tex]\(+5.20 \times 10^{-7}\)[/tex] C, we can use Coulomb's Law. Coulomb's Law is given by the formula:
[tex]\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the magnitude of the force between the two charges.
- [tex]\( k \)[/tex] is Coulomb's constant, which is 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 magnitudes of the two charges.
- [tex]\( r \)[/tex] is the distance between the centers of the two charges.
Given values:
- [tex]\( q_1 = -2.30 \times 10^{-8} \)[/tex] C
- [tex]\( q_2 = +5.20 \times 10^{-7} \)[/tex] C
- [tex]\( r = 0.250 \)[/tex] m
- [tex]\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex]
First, we calculate the magnitude of the product of the charges:
[tex]\[ |q_1 \cdot q_2| = |-2.30 \times 10^{-8} \cdot 5.20 \times 10^{-7}| \][/tex]
[tex]\[ = 2.30 \times 10^{-8} \cdot 5.20 \times 10^{-7} \][/tex]
[tex]\[ = 1.196 \times 10^{-14} \, \text{C}^2 \][/tex]
Now, we use Coulomb's Law to find the magnitude of the force:
[tex]\[ F = 8.99 \times 10^9 \frac{1.196 \times 10^{-14}}{(0.250)^2} \][/tex]
[tex]\[ = 8.99 \times 10^9 \frac{1.196 \times 10^{-14}}{0.0625} \][/tex]
[tex]\[ = 8.99 \times 10^9 \times 1.9136 \times 10^{-13} \][/tex]
[tex]\[ = 1.7203264 \times 10^{-3} \, \text{N} \][/tex]
Thus, the force acting on the dust particle from the cloth is approximately [tex]\( 1.72 \times 10^{-3} \)[/tex] N. Given the charges, the force is attractive (since one charge is positive and the other is negative), but we are asked for the magnitude, which does not consider the sign.
So, the closest answer from the given options in magnitude is:
[tex]\[ -1.72 \times 10^{-3} \, \text{N} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{-1.72 \times 10^{-3} \, \text{N}} \][/tex]
