To calculate the force between two charges, we can use Coulomb's Law, which is given by the formula:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant \( (8.99 \times 10^9 , \text{N m}^2/\text{C}^2) \),
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.
Given:
- \( q_1 = 5.0 \times 10^{-5} , C \)
- \( q_2 = 2.0 \times 10^{-6} , C \)
- \( r = 0.040 , m \)
Substituting these values into the formula:
\[ F = (8.99 \times 10^9) \frac{(5.0 \times 10^{-5})(2.0 \times 10^{-6})}{(0.040)^2} \]
Calculating \( (0.040)^2 \):
\[ (0.040)^2 = 0.0016 , m^2 \]
Now substituting this back into the equation for force:
\[ F = (8.99 \times 10^9) \frac{(5.0 \times 10^{-5})(2.0 \times 10^{-6})}{0.0016} \]
Calculating the product of the charges:
\[ (5.0 \times 10^{-5})(2.0 \times 10^{-6}) = 1.0 \times 10^{-10} \]
Now substituting:
\[ F = (8.99 \times 10^9) \frac{1.0 \times 10^{-10}}{0.0016} \]
This simplifies to:
\[ F = (8.99 \times 10^9) \frac{1.0 \times 10^{-10}}{1.6 \times 10^{-3}} \]
Calculating \( \frac{1.0 \times 10^{-10}}{1.6 \times 10^{-3}} \):
\[ \frac{1.0 \times 10^{-10}}{1.6 \times 10^{-3}} = 0.625 \times 10^{-7} = 6.25 \times 10^{-8} \]
Now calculating the force:
\[ F = (8.99 \times 10^9)(6.25 \times 10^{-8}) \]
Calculating this product:
\[ F = 8.99 \times 6.25 \times 10^{9 - 8} = 56.1875 , N \approx 5.6 \times 10^1 N \]
Thus, the force between the charges is approximately 5.6 × 10^1 N.
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