To calculate the force between two small charged spheres, we'll utilize Coulomb's Law. Coulomb's law states that the electrostatic force [tex]\( F \)[/tex] between two charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] in a vacuum (or air, which is nearly the same) is given by:
[tex]\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( k \approx 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \)[/tex],
- [tex]\( |q_1 \cdot q_2| \)[/tex] is the absolute value of the product of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Let's proceed step-by-step:
1. Identify the charges (in Coulombs):
[tex]\[
q_1 = +1 \times 10^{-7} \, \text{C}
\][/tex]
[tex]\[
q_2 = -1 \times 10^{-7} \, \text{C}
\][/tex]
2. Distance between the charges (in meters):
Given distance, [tex]\( r = 30 \, \text{cm} \)[/tex]. Converting to meters:
[tex]\[
r = 0.3 \, \text{m}
\][/tex]
3. Coulomb's constant:
[tex]\[
k \approx 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2}
\][/tex]
4. Absolute value of the product of the charges:
[tex]\[
|q_1 \cdot q_2| = |(+1 \times 10^{-7} \, \text{C}) \cdot (-1 \times 10^{-7} \, \text{C})| = 1 \times 10^{-14} \, \text{C}^2
\][/tex]
5. Apply Coulomb's law:
[tex]\[
F = k \frac{|q_1 \cdot q_2|}{r^2}
\][/tex]
6. Substitute the known values into the equation:
[tex]\[
F = 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \times \frac{1 \times 10^{-14} \, \text{C}^2}{(0.3 \, \text{m})^2}
\][/tex]
7. Calculate the distance squared:
[tex]\[
(0.3)^2 = 0.09 \, \text{m}^2
\][/tex]
8. Substitute this value back into the formula:
[tex]\[
F = 8.99 \times 10^9 \frac{1 \times 10^{-14}}{0.09}
\][/tex]
9. Simplify the fraction:
[tex]\[
\frac{1 \times 10^{-14}}{0.09} = 1.1111 \times 10^{-13}
\][/tex]
10. Multiply by Coulomb's constant:
[tex]\[
F = 8.99 \times 10^9 \times 1.1111 \times 10^{-13}
\][/tex]
11. Final multiplication:
[tex]\[
F \approx 0.0009988888888888888 \, \text{N}
\][/tex]
Thus, the force between the two small charged spheres is approximately [tex]\( 0.0009988888888888888 \, \text{N} \)[/tex].
