A point charge q1=+5.00 μC is held fixed in space. From a horizontal distance of 6.00 cm, a sma

A point charge q1=+5.00 μC is held fixed in space. From a...

A point charge $q_{1}=+5.00 \mu \mathrm{C}$ is held fixed in space. From a horizontal distance of $6.00 \mathrm{~cm},$ a small sphere with mass $4.00 \times 10^{-3} \mathrm{~kg}$ and charge $q_{2}=+2.00 \mu \mathrm{C}$ is fired toward the fixed charge with an initial speed of $40.0 \mathrm{~m} / \mathrm{s}$. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is $25.0 \mathrm{~m} / \mathrm{s} ?$

Key Concepts

Electrostatic Potential Energy

Electrostatic potential energy is the energy due to the configuration of two point charges. It is given by the expression k·q1·q2/r for two charges separated by a distance r. This concept is important in problems where the potential energy changes as the charges move relative to one another, and it is a key part of applying energy conservation to determine the distance (and hence the force) when the sphere's speed changes.

Energy Conservation

The principle of energy conservation states that the total energy in an isolated system remains constant. In this context, the initial kinetic energy of the sphere and its electrostatic potential energy relative to the fixed charge add up to its total mechanical energy. As the sphere moves, this energy conservation is used to relate changes in its speed to changes in position (and hence the distance between charges), which then affects the electrostatic force acting on it.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and the resulting acceleration by the equation F = m·a. This principle is applied once the force due to electrostatic interactions is determined, allowing for the calculation of the acceleration experienced by the charged sphere.

Coulomb's Law

Coulomb's Law quantifies the electrostatic force between two point charges. It states that the magnitude of the force is proportional to the product of the charges and inversely proportional to the square of the distance between them, with the constant of proportionality being Coulomb's constant. This concept is crucial when calculating the force that one charge exerts on another in an electrostatic system.

Transcript

00:01 So in this problem, we're looking at two point charges, one which is fixed, and one which is initially six centimeters away from the fixed charge.

00:13 And it's moving at a rate of 40 meters per second.

00:15 So we want to find what the acceleration would be when its velocity is 25 meters per second.

00:21 So what we need to know is we need to use a couple concepts here and are just to solve this problem.

00:27 The first being the conservation of energy.

00:29 So we know that the kinetic energy at the first point, which we'll call point a, just six centimeters away from the fixed charge, we know that the kinetic energy and the potential energy at point a should be equal to the kinetic energy and potential energy at point b.

00:51 So this is due to the conservation of energy, which is what will apply.

00:57 We also need to know kulam's law, which is describing an electrostatic force between two charged particles.

01:04 So we have f equal to k, which is kulam's constant, times q1, times q2, over r squared.

01:16 And that's the distance between the two charges.

01:19 And finally, from newton's second law, we know that force is equal to mass times acceleration.

01:27 Oh, and i'll just write down the mass of the second charge that we're looking at, because we'll lead that as well.

01:35 And we're given that, and we're told it's four times 10 to the negative third kilograms.

01:44 So now that we have all our constants here, we can go ahead and get started.

01:48 So the first thing that we want to do is we want to find the distance between the two charges when the second charge has reached its velocity of 25 meters per second.

01:57 So how we're going to do this is let's just write down how we find the kinetic energy and potential energy.

02:05 So kinetic energy can be described as one half times mass, times a velocity squared.

02:15 And for potential energy, we can use this equation.

02:18 So we have k times q1 times q2 over the distance.

02:25 So now that we have the formulas for potential energy and kinetic energy, we can go ahead and plug in and solve.

02:36 So we want to find the kinetic energy at point a.

02:41 So we're using one half times the mass, times the velocity at a, 40 meters per second, squared...

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