In Fig. 21-32, particles 1 and 2 of charge q1=q2=+3.20 ×10^-19 C are on a y axis at distance d=1

In Fig. 21-32, particles 1 and 2 of charge q1=q2=+3.20...

In Fig. $21-32,$ particles 1 and 2 of charge $q_{1}=q_{2}=+3.20 \times 10^{-19} \mathrm{C}$ are on a $y$ axis at distance $d=17.0 \mathrm{cm}$ from the origin. Particle 3 of charge $q_{3}=+6.40 \times 10^{-19} \mathrm{Cis}$ moved gradually along the $x$ axis from $x=0$ to $x=$ $+5.0 \mathrm{m} .$ At what values of $x$ will the magnitude of the electrostatic force on the third particle from the other two particles be (a) minimum and (b) maximum? What are the (c) minimum and (d) maximum magnitudes?

In Fig. $21-32,$ particles 1 and 2 of charge $q_{1}=q_{2}=+3.20 \times 10^{-19} \mathrm{C}$
are on a $y$ axis at distance $d=17.0 \mathrm{cm}$
from the origin. Particle 3 of charge
$q_{3}=+6.40 \times 10^{-19} \mathrm{Cis}$ moved gradually along the $x$ axis from $x=0$ to $x=$
$+5.0 \mathrm{m} .$ At what values of $x$ will the magnitude of the electrostatic force on
the third particle from the other two
particles be (a) minimum and (b) maximum? What are the (c) minimum and (d) maximum magnitudes?

Key Concepts

Geometric Considerations in Charge Configurations

The arrangement of charges in space significantly influences the resulting forces. Understanding how distances and relative positions (e.g., along the x-axis or y-axis) affect the magnitude and direction of the forces is crucial when setting up and solving problems. This includes analyzing symmetry, relative distances, and vector orientations to accurately compute the net force.

Coulomb's Law

This fundamental principle governs the electric force between two point charges. The law states that the magnitude of the electrostatic force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It serves as the basis for analyzing forces in any charge configuration, regardless of the specific values provided.

Superposition Principle

In systems with multiple charges, the net force on any single charge is determined by vectorially summing the individual forces exerted by all other charges. This principle allows the complex interactions in multi-charge systems to be decomposed into simpler, pairwise interactions that can be analyzed independently and then added together.

Optimization of Force Magnitudes

When examining the variation of the net electrostatic force as one of the charges moves, it becomes necessary to identify conditions where this force is minimized or maximized. This typically involves setting up an expression for the net force as a function of position and then applying calculus techniques, such as differentiation, to find the critical points that correspond to these extreme values.

Transcript

00:01 So we have two charges q1 and q2 of the same magnitude, 3 .2 times 10 per minus 19 volumes.

00:07 Let's call this charge q.

00:10 And the other charge q3 has twice that much.

00:13 So this is 2 here.

00:15 We'll move this from this point, x equal to 0 to x equal to 5 meters.

00:19 We know that t is equal to 17 centimeters, which is this.

00:32 Now we want to find the maximum and minimum forces.

00:35 Now, let's say the distance is x and this is d.

00:42 So the force due to q1 will be in this direction and the force due to the other one will be in this direction.

00:54 Now, we'll consider the angle theta and because these two forces will have the equal magnitudes as these two charges have the same, as these two charges are the same, the net force will be 2 f pos theta, where f is the force you to each charge.

01:16 Now because each one is f, we can divide it into two components.

01:22 The total magnitude is f.

01:24 This will be f cost theta.

01:26 So we have f sine theta, this one.

01:30 And the two f sine thetas cancel each other out while the two f cost thetas add up to give us two f cost theta.

01:39 And that is it.

01:41 So we know that cost theta is equal to x by square root of x squared plus d squared and we use this so the net force on the charge q3 is simply 2f cost theta so that's 2 times f if you remember is k q by r squared so that's k q1 q3 so q3 is 2 q by r squared and notice that r is the distance of this hypotenous so r is equal to square root of x squared t squared so that's square root of x squared plus d squared times cos theta and cost theta is x by square root of x squared plus b so this is 4 k q squared x by x squared plus t squared whole power 3 by and that is it.

02:49 To see where the net maximum is and the net minimum is, we'll simply need to minimize and maximize this function...

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