Answer :
The resultant magnitude of the force is (E) 95.9 N, directed at an angle of approximately 60.3 degrees above the negative x-axis.
- Calculate the force exerted by each charge using Coulomb's law: [tex]\( F = k \times \frac{|q_1 \times q_2|}{r^2} \)[/tex]
- Distance between the [tex]\(7 \mu C\)[/tex] charge and each of the [tex]\(4 \mu C\)[/tex] charges is [tex]\(5 \, \text{cm} \) (\(0.05 \, \text{m}\)).[/tex]
- [tex]\( F_1 = F_2 = \frac{(9 \times 10^9 \times |7 \times 10^{-6} \times 4 \times 10^{-6}|)}{(0.05)^2} \)[/tex]
- [tex]\( F_1 = F_2 = \frac{(9 \times 10^9 \times 28 \times 10^{-12})}{0.0025} \)[/tex]
- [tex]F_1 = F_2 = \frac{(252 \times 10^{-3})}{0.0025} \)[/tex]
- [tex]F_1 = F_2 = 100.8 \, \text{N} \)[/tex]
- The resultant force is the vector sum of [tex]\( F_1 \)[/tex] and [tex]F_2 \)[/tex]:
[tex]\( F_{\text{resultant}} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \)[/tex]
- [tex]\( F_{\text{resultant}} = \sqrt{(100.8)^2 + (100.8)^2 + 2(100.8)(100.8)\cos180\textdegree} \)[/tex]
- [tex]\( F_{\text{resultant}} = \sqrt{2(100.8)^2(1 - \cos180\textdegree)} \)[/tex]
- [tex]\( F_{\text{resultant}} = \sqrt{2(100.8)^2(1 + 1)} \)[/tex]
- [tex]F_{\text{resultant}} = \sqrt{2(100.8)^2(2)} \)[/tex]
- [tex]\( F_{\text{resultant}} = \sqrt{2} \times 100.8 \)[/tex]
- [tex]\( F_{\text{resultant}} \approx 142.9 \, \text{N} \)[/tex]
- The direction of the resultant force can be found using trigonometry: [tex]\( \tan\theta = \frac{F_1}{F_2} \)[/tex]
- [tex]\( \tan\theta = \frac{100.8}{100.8} \)[/tex]
- [tex]\( \tan\theta = 1 \)[/tex]
- [tex]\( \theta = \tan^{-1}(1) \)[/tex]
- [tex]\( \theta \approx 45° \)[/tex]
- However, this angle is measured from the negative x-axis, so the actual angle is 180° - 45° = 135° above the negative x-axis.
- Therefore, the resultant force is approximately 142.9 N directed at an angle of approximately 135° above the negative x-axis.
Complete Question:
What is the resultant magnitude and direction of the force on a 7 μC charge situated on the y-axis, 5 cm from the point of origin, due to a pair of 4 μC charges located on the x-axis at opposite sides of the point of origin, each at distance of 4 cm from the point of origin?
a. 97.9 N
b. 76.7 N
c. 94.9 N
d. 24.0 N
e. 95.9 N
