To find the value of the charge [tex]\( q_2 \)[/tex] that is repelling the [tex]\( +26.3 \mu C \)[/tex] charge [tex]\( q_1 \)[/tex] with a force of [tex]\( 0.615 \, \text{N} \)[/tex] at a distance of [tex]\( 0.750 \, \text{m} \)[/tex], we can use Coulomb's law:
[tex]\[ F = k \frac{|q_1 q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the force between the charges.
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 \cdot \text{C}^{-2} \)[/tex].
- [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:
- [tex]\( q_1 = 26.3 \mu \text{C} = 26.3 \times 10^{-6} \, \text{C} \)[/tex].
- [tex]\( F = 0.615 \, \text{N} \)[/tex].
- [tex]\( r = 0.750 \, \text{m} \)[/tex].
We need to solve for [tex]\( q_2 \)[/tex]:
1. First, rearrange Coulomb’s law to solve for [tex]\( q_2 \)[/tex]:
[tex]\[ q_2 = \frac{F \cdot r^2}{k \cdot |q_1|} \][/tex]
2. Substitute the known values into the equation:
[tex]\[ q_2 = \frac{0.615 \, \text{N} \cdot (0.750 \, \text{m})^2}{8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 \cdot \text{C}^{-2} \cdot 26.3 \times 10^{-6} \, \text{C}} \][/tex]
3. Calculate the numerator and the denominator separately:
- Calculate [tex]\( (0.750 \, \text{m})^2 \)[/tex]:
[tex]\[ (0.750)^2 = 0.5625 \, \text{m}^2 \][/tex]
- Calculate the numerator:
[tex]\[ 0.615 \, \text{N} \cdot 0.5625 \, \text{m}^2 = 0.346875 \, \text{N} \cdot \text{m}^2 \][/tex]
- Calculate the denominator:
[tex]\[ 8.99 \times 10^9 \cdot 26.3 \times 10^{-6} \, \text{C} = 8.99 \times 10^9 \cdot 26.3 \times 10^{-6} \][/tex]
[tex]\[ 8.99 \times 26.3 \times 10^{9-6} = 236.577 \times 10^3 = 236577.000 \, \text{N} \cdot \text{m}^2 \cdot \text{C}^{-1} \][/tex]
4. Divide the numerator by the denominator:
[tex]\[ q_2 = \frac{0.346875}{236577.000} \approx 1.4631275984723204 \times 10^{-6} \, \text{C} \][/tex]
Convert the result to microCoulombs:
[tex]\[ 1.4631275984723204 \times 10^{-6} \, \text{C} = 1.4631275984723204 \mu \text{C} \][/tex]
Since [tex]\( q_1 \)[/tex] is positive and the charges repel each other, [tex]\( q_2 \)[/tex] must also be positive.
Therefore, the value of [tex]\( q_2 \)[/tex] is approximately [tex]\( +1.4631275984723204 \times 10^{-6} \, \text{C} \)[/tex].
