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The table shows the charges and the distance between five different pairs of objects.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline Charge 1 & Charge 2 & Distance & Force \\
\hline $q$ & $q$ & $d$ & $F$ \\
\hline $q$ & $2q$ & $d$ & $W$ \\
\hline $q$ & $q$ & $2d$ & $X$ \\
\hline $3q$ & $q$ & $d$ & $Y$ \\
\hline $q$ & $q$ & $3d$ & $Z$ \\
\hline
\end{tabular}
\][/tex]

If the force between two charges, [tex]\( q \)[/tex], separated by a distance, [tex]\( d \)[/tex], is [tex]\( F \)[/tex], what is the list of the other forces, from greatest to least?

A. [tex]\( Y, Z, W, X \)[/tex]

B. [tex]\( Y, W, X, Z \)[/tex]

C. [tex]\( Z, Y, W, X \)[/tex]

D. [tex]\( Z, X, W, Y \)[/tex]

Sagot :

GinnyAnswer
To solve the problem, we need to use Coulomb's law, which states that the force [tex]\( F \)[/tex] between two charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] separated by a distance [tex]\( d \)[/tex] is given by:

[tex]\[ F = k \frac{q_1 q_2}{d^2} \][/tex]

where [tex]\( k \)[/tex] is Coulomb's constant. Let's start by calculating the forces in each scenario given in the table.

1. For charges [tex]\( q \)[/tex] and [tex]\( q \)[/tex] separated by distance [tex]\( d \)[/tex]:
[tex]\[ F = k \frac{q \cdot q}{d^2} = k \frac{q^2}{d^2} \][/tex]
This force is denoted as [tex]\( F \)[/tex].

2. For charges [tex]\( q \)[/tex] and [tex]\( 2q \)[/tex] separated by distance [tex]\( d \)[/tex]:
[tex]\[ W = k \frac{q \cdot 2q}{d^2} = k \frac{2q^2}{d^2} = 2F \][/tex]

3. For charges [tex]\( q \)[/tex] and [tex]\( q \)[/tex] separated by distance [tex]\( 2d \)[/tex]:
[tex]\[ X = k \frac{q \cdot q}{(2d)^2} = k \frac{q^2}{4d^2} = \frac{F}{4} \][/tex]

4. For charges [tex]\( 3q \)[/tex] and [tex]\( q \)[/tex] separated by distance [tex]\( d \)[/tex]:
[tex]\[ Y = k \frac{3q \cdot q}{d^2} = k \frac{3q^2}{d^2} = 3F \][/tex]

5. For charges [tex]\( q \)[/tex] and [tex]\( q \)[/tex] separated by distance [tex]\( 3d \)[/tex]:
[tex]\[ Z = k \frac{q \cdot q}{(3d)^2} = k \frac{q^2}{9d^2} = \frac{F}{9} \][/tex]

Now, let's rank these forces from greatest to least:

- [tex]\( 3F \)[/tex] (Y)
- [tex]\( 2F \)[/tex] (W)
- [tex]\( \frac{F}{4} \)[/tex] (X)
- [tex]\( \frac{F}{9} \)[/tex] (Z)

So, the list of forces from greatest to least is:

[tex]\[ Y, W, X, Z \][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{Y, W, X, Z} \][/tex]
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