Respuesta :
Answer:
[tex]q_{1} \approx 2.66\, q_{2}[/tex].
Explanation:
The charge [tex](+Q)[/tex] is between [tex]q_{1}[/tex] and [tex]q_{2}[/tex]. For the overall electrostatic force on [tex](+Q)\![/tex] to be zero, [tex]q_{1}\![/tex] and [tex]q_{2}\![/tex] need to either both attract or both repel [tex]\!(+Q)[/tex]. Hence, [tex]\!q_{1}[/tex] and [tex]\!q_{2}[/tex] should have the same sign.
For the overall force from [tex]q_{1}[/tex] and [tex]q_{2}[/tex] on [tex]\!(+Q)[/tex] to be balanced, the electrostatic force from each charge should be equal in magnitude.
Let [tex]r_{1}[/tex] denote the distance between [tex]q_{1}[/tex] and [tex](+Q)[/tex]. Let [tex]r_{2}[/tex] denote the distance between [tex]q_{2}[/tex] and [tex](+Q)\![/tex]. It is given that [tex]q_{1} = 0.62\, d[/tex] and [tex]q_{2} = (1 - 0.62)\, d = 0.38\, d[/tex].
Let [tex]k[/tex] denote Coulomb's Constant. By Coulomb's Law, the magnitude of the electrostatic force between [tex]q_{1}[/tex] and [tex]Q[/tex] would be:
[tex]\displaystyle F_{1} = \frac{k\, |q_{1}|\, |Q|}{{r_{1}}^{2}}[/tex].
Similarly, the magnitude of the electrostatic force between [tex]q_{2}[/tex] and [tex]Q[/tex] would be:
[tex]\displaystyle F_{2} = \frac{k\, |q_{2}|\, |Q|}{{r_{2}}^{2}}[/tex].
Set the magnitude of the two forces to be equal and solve for [tex]q_{1}[/tex] in terms of [tex]q_{2}[/tex]:
[tex]\displaystyle \frac{k\, |q_{1}|\, |Q|}{{r_{1}}^{2}} = \displaystyle \frac{k\, |q_{2}|\, |Q|}{{r_{2}}^{2}}[/tex].
[tex]\displaystyle |q_{1}| = \frac{{r_{1}}^{2}}{{r_{2}}^{2}}\, |q_{2}|[/tex].
Eliminate the absolute value since [tex]q_{1}[/tex] and [tex]q_{2}[/tex] are of the same sign:
[tex]\displaystyle|q_{1} = \frac{{r_{1}}^{2}}{{r_{2}}^{2}}\, q_{2} = \frac{(0.62\, d)^{2}}{(0.38\, d)^{2}}\, q_{2} \approx 2.66\, q_{2}[/tex].
In other words, [tex]q_{1} \approx 2.66\, q_{2}[/tex].
Final answer:
To find the relationship between q1 and q2, we can set up an equation using Coulomb's law and the fact that the net electrostatic force experienced by +Q is zero. Solving for q2, we find q2 = -(q1 * r1^2) / (r2^2).
Explanation:
To find the relationship between q1 and q2, let's analyze the situation. We have two point charges, q1 and q2, placed at x = 0 and x = d, respectively, on the x-axis. A third charge, +Q, is placed at x = 0.62d. The net electrostatic force experienced by +Q is zero, which means the forces exerted by q1 and q2 on +Q must cancel each other out.
Using Coulomb's law, the force between +Q and q1 is given by F1 = (k * q1 * Q) / (r1^2), where k is the electrostatic constant and r1 is the distance between the charges.
Similarly, the force between +Q and q2 is given by F2 = (k * q2 * Q) / (r2^2), where r2 is the distance between the charges. Since the net force is zero, we have F1 = F2.
Equating the two forces and solving for q2, we get q2 = -(q1 * r1^2) / (r2^2).
