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

step 1 Hello. So here we have the setup of the problem. Particles Q1 and Q2 are both the distance D fro

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 $q_{3}=+6.40 \times 10^{-19} \mathrm{C}$ is 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

Vector Addition of Forces

Vector addition is the process of combining forces that have both magnitude and direction to obtain a resultant force. In the context of electrostatics, this is important because the forces exerted by different charges may not be aligned along the same line, so their components must be carefully added to find the net force acting on a charge.

Coulomb's Law

Coulomb's Law states that the magnitude of the electric force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This fundamental law is essential for determining the force between any two isolated charges in an electrostatic system.

Superposition Principle

The superposition principle in electrostatics asserts that when more than two charges are present, the total force on any one charge is the vector sum of the forces exerted on it by each of the other charges individually. This principle is crucial for solving problems where multiple forces interact, as it allows each force to be calculated separately before combining them.

Distance Dependence in Electrostatics

The electrostatic force between charges decreases rapidly with distance, as specified by the inverse-square law. This distance dependence means even small changes in separation between charges can lead to significant variations in the force magnitude, a fact that is vital when calculating variations in force as a charge moves within an electric field.

Optimization in Physics

Optimization in physics involves determining the conditions that maximize or minimize a physical quantity, such as force magnitude. In problems like these, optimization techniques—often involving calculus to find the extrema of a function—are used to identify the positions at which the resultant force reaches its minimum or maximum value.

Transcript

0:00 Hello.

00:01 So here we have the setup of the problem.

00:04 Particles q1 and q2 are both the distance d from the origin on the y axis and particle 3 is on the x -axis anywhere between 0 and 5 meters.

00:13 And these are the equations for the coolant force from particle 1 on 3 and from particle 2 on 3.

00:20 For part 1 to find the minimum amount of force on the particle, well on particle 3, since q1 and q2 have the same charge and they are the same.

00:30 Distance away from particle three.

00:33 The minimum distance will occur when x equals zero.

00:40 This is because q3 will be, q1 and q2 will be directly in line with q3 and so the force on q1 on q3 and the force from q2 on q3 will cancel out because they're all the chargers are the same.

00:56 They're all positive and so q1 push q3 away and q2 will also push q3 away in such a case.

01:05 So like this, basically, q1, q2, 3, and so there will be opposing forces, and so it will cancel.

01:21 For part b, we need an equation for the force for any distance x, for which particle 3 is at.

01:32 And so for that, we need.

01:33 You basically add the forces.

01:34 Remember, forces a vector, and so there are x and y directions for which the force acts upon.

01:41 In this case, the y direction always cancels out, because each particle is the same y distance or d away, like q1 and q2, or both the distance d away from q3.

01:55 And of course they're the same charge, so like q1 and q2 are the same, so it will cancel in the y direction.

02:00 But in the x direction, if you can tell from this picture, the forces will actually add because f23 and f1, 2 are both pointing in the x direction, but yeah, both pointing in x direction.

02:14 So you can just add the forces.

02:16 So the net force in the x direction is this.

02:23 So it's f particle 1 on 3x plus f particle 2 on 3x...

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