Figure (a) shows charged particles 1 and 2 that are fixed in place on an x axis. Particle 1 has a charge with a magnitude of |q1| = 12.0e. Particle 3 of charge q3 = +13.0e is initially on the x axis near particle 2. Then particle 3 is gradually moved in the positive direction of the x axis. As a result, the magnitude of the net electrostatic force on particle 2 due to particles 1 and 3 changes.
Figure (b) gives the x component of that net force as a function of the position x of particle 3. The scale of the x axis is set by xs = 1.70 m. The plot has an asymptote of F2,net = 1.370 × 10-25 N as x → ∞.
As a multiple of e and including the sign, what is the charge q2 of particle 2?
Here are the two figures:
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To find the charge q2 of particle 2, we can consider the relationship between the electrostatic force and the charge involved. The electrostatic force between two charged particles is given by Coulomb's Law:
F = k * |q1| * |q2| / r^2,
where F is the force between the particles, k is the Coulomb constant (8.99 x 10^9 N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
In Figure (b), we see that the x-component of the net force on particle 2 is plotted as a function of the position x of particle 3. As x increases, the force initially decreases rapidly and then approaches a constant value as x goes to infinity.
The force on particle 2 due to particle 1 and 3 is given by:
F2,net = F1 + F3 = k * |q1| * |q2| / r1^2 + k * |q3| * |q2| / r3^2,
where r1 is the distance between particle 2 and 1, and r3 is the distance between particle 2 and 3.
From the plot, we can see that F2,net tends to a constant value as x approaches infinity, i.e., F2,net = 1.370 x 10^-25 N.
This means that the second term in the equation above, k * |q3| * |q2| / r3^2, must approach zero as x goes to infinity, leaving only the first term.
Therefore,
F2,net = F1 = k * |q1| * |q2| / r1^2.
We can rearrange this equation to solve for |q2|:
|q2| = (F2,net * r1^2) / (k * |q1|).
Substituting the given values:
|q1| = 12.0 e,
F2,net = 1.370 x 10^-25 N,
xs = 1.70 m,
k = 8.99 x 10^9 N m^2/C^2,
we find:
|q2| = (1.370 x 10^-25 N * (1.70 m)^2) / (8.99 x 10^9 N m^2/C^2 * 12.0 e).
Simplifying this expression, we can multiply the numerator and denominator by e to obtain:
|q2| = (1.370 x 10^-25 N * (1.70 m)^2 * e) / (8.99 x 10^9 N m^2/C^2 * 12.0).
Evaluating this expression will give us the charge on particle 2, including the sign.