Two particles are in a uniform electric field that points in the +x direction and has a magnitud

Two particles are in a uniform electric field that points in...

Key Concepts

Equal Acceleration Condition for Constant Separation

When two particles remain at a constant separation while accelerating, their individual accelerations must be equal. In such scenarios, any differences in external forces, such as those from an electric field, must be counteracted by internal forces (for instance, the electrostatic force between them) so that the net acceleration difference vanishes. This condition is critical for solving for unknown quantities like the separation distance when balancing the forces on each particle.

Uniform Electric Field

A uniform electric field is one in which the electric field strength and direction are constant at every point in space. This means that any charged particle placed in the field experiences the same magnitude and direction of force, assuming the charge remains constant. Uniform fields are often used in problems to simplify the dynamics of charged particles.

Electric Force on Charged Particles

The electric force acting on a charged particle in an electric field is given by F = qE, where q is the charge of the particle and E is the electric field. This force determines the initial acceleration of the particle as per its mass and plays a central role in the motion of charged bodies in an electric field.

Newton's Second Law

Newton's Second Law of Motion, summarized by F = ma, establishes the relationship between the net force acting on an object, its mass, and the resultant acceleration. In problems involving charged particles, this law is used to connect the forces due to the electric field and other interactions to the acceleration of the particles.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges, given by F = k|q1q2|/r^2, where k is Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the separation between them. This law is fundamental for analyzing interactions between charged particles that are in proximity, as it determines the mutual repulsive or attractive force they exert on each other.

Transcript

00:02 In this question, we have two particles that exist in a uniform electric field, and we are given the masses and charges of those particles, and we know that when they are released from rest, they maintain a constant separation.

00:23 And we want to find out what is that separation.

00:27 So let's start.

00:29 We're going to draw in...actually, you know something, since i almost never get to use yellow, we're going to draw our electric field in yellow.

00:37 So i'm going to allow my electric field to exist pointing to the right, because it does say positive x direction.

00:48 And we're then going to have charge 1, it says, is to the left of charge 2.

00:55 So i'm going to put my negative q1 here, and then i'm going to draw my positive q2 over here.

01:15 And we are going to specify that they are...actually, i want a different color for that.

01:22 We're going to specify that they are separated by a distance d.

01:29 So each of these charges is going to feel two forces.

01:33 One from the electric field.

01:36 I'm going to change my mind, we're going to use orange to worry about electric field things.

01:43 And they will each also feel a force from the other charge.

01:50 So, keeping in mind that the direction of an electric field is determined by the force that a positive charge feels, that tells me that the force of the electric field on charge 1 is going to point to the left.

02:06 And the force of the electric field on charge 2 will point to the right.

02:13 Charge 1 is going to be attracted to charge 2, so it will feel forced to the right.

02:22 And charge 2, similarly, will feel attracted to charge 1, so it's going to feel a pull back to the left.

02:39 So, here's a key concept.

02:43 Is that two...let's try writing this a little bit more legibly.

02:48 So, to maintain d, they must have the same magnitude and direction of their acceleration upon release.

03:19 Because that means that they will gain velocity in precisely the same way.

03:25 And that's what will allow them to maintain the distance.

03:29 So, we need to be able to say that the acceleration of charge 1 will be equal to the net force on charge 1 divided by its mass.

03:40 The acceleration of charge 2 will be the net force on charge 2 divided by its mass.

03:47 And then we will be able to set those two accelerations equal to each other.

03:56 One thing i'm going to do before i dive into that.

04:01 I'm going to go ahead, since by newton's third law, f12 equals f21, i'm going to go ahead and just calculate that right now.

04:14 With k, or you could use 1 over 4 pi epsilon naught, but kq1q2 over d squared.

04:23 So, we'll do our 9 times 10 to the 9th.

04:25 And then of course, using coulomb's law, we only ever use the positive values.

04:37 Oops, that should be 10 to the negative 6th, sorry.

04:40 We only ever use the positive values of our charges, the absolute values.

04:44 We let the signs determine the relative directions of those forces, left, right, etc.

04:54 And so, we are able to say that the force between charges 1 and 2, oh go figure, i didn't actually, oh yeah i did, there we go, sorry i just had to find it.

05:12 1 .134 over d squared.

05:17 And we will apply that appropriately for each one in determining its net force.

05:24 So let's, let's go do acceleration of charge 1.

05:29 It's going to be the net force on 1 divided by the mass of 1.

05:34 So i'm going to write it as 1 over the mass of charge 1, which is 1 .4 times 10 to the minus 5th kilograms.

05:45 And then it is going to feel in the positive.

05:52 Oh, i just realized i'm totally, hang on, i like my color coding guys.

05:56 I'm going to fix this real quick...