To answer this question, we will use Coulomb's Law, which states that the force [tex]\( F \)[/tex] between two charges is given by:
[tex]\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( k \)[/tex] is Coulomb's constant,
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Given that the force between two charges [tex]\( q \)[/tex] separated by a distance [tex]\( d \)[/tex] is [tex]\( F \)[/tex], we can write:
[tex]\[ F = \frac{k \cdot q \cdot q}{d^2} = \frac{k \cdot q^2}{d^2} \][/tex]
We will calculate each of the forces W, X, Y, and Z in terms of [tex]\( F \)[/tex].
1. Force [tex]\( W \)[/tex] :
[tex]\[ W = \frac{k \cdot (q) \cdot (2q)}{d^2} = \frac{2k \cdot q^2}{d^2} = 2F \][/tex]
2. Force [tex]\( X \)[/tex] :
[tex]\[ X = \frac{k \cdot (q) \cdot (9q)}{(2d)^2} = \frac{9k \cdot q^2}{4d^2} = \frac{9}{4} \cdot \frac{k \cdot q^2}{d^2} = \frac{9}{4} F \][/tex]
3. Force [tex]\( Y \)[/tex] :
[tex]\[ Y = \frac{k \cdot (3q) \cdot (q)}{d^2} = \frac{3k \cdot q^2}{d^2} = 3F \][/tex]
4. Force [tex]\( Z \)[/tex] :
[tex]\[ Z = \frac{k \cdot (q) \cdot (q)}{(3d)^2} = \frac{k \cdot q^2}{9d^2} = \frac{1}{9} \cdot \frac{k \cdot q^2}{d^2} = \frac{F}{9} \][/tex]
Now, let's list the forces:
- [tex]\( Y = 3F \)[/tex]
- [tex]\( W = 2F \)[/tex]
- [tex]\( X = \frac{9}{4}F \)[/tex]
- [tex]\( Z = \frac{F}{9} \)[/tex]
We need to order them from greatest to least:
- The highest force is [tex]\( Y = 3F \)[/tex].
- The next is [tex]\( W = 2F \)[/tex].
- The following is [tex]\( X = \frac{9}{4}F \)[/tex].
- The smallest is [tex]\( Z = \frac{F}{9} \)[/tex].
So, the list of other forces from greatest to least is:
[tex]\[ Y, W, X, Z \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{Y, W, X, Z} \][/tex]
