Consider the charge distribution shown in Figure P 23.59 . (a) Show that the magnitude of the el

Consider the charge distribution shown in Figure P 23.59 ....

Key Concepts

Superposition Principle

The superposition principle states that the net electric field caused by a group of charges is the vector sum of the fields produced by each charge independently. This concept allows complex problems involving multiple charges to be broken down into simpler problems by considering each charge’s contribution separately before summing them.

Coulomb's Law

Coulomb's law is the fundamental principle that determines the electric force or field between point charges. It states that the magnitude of the electric field produced by a point charge is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge, with the constant k? setting the proportionality. This law is essential for calculating the fields from individual charges in any configuration.

Electric Field

The electric field is a vector field that quantifies the force per unit positive charge at any point in space due to other charges. It is calculated by summing (or integrating) the contributions from all charges present, taking into account both magnitude and direction. Understanding how to determine the electric field is crucial for analyzing how charged objects interact.

Symmetry in Charge Distributions

Symmetry in a charge distribution can simplify electric field calculations significantly. In systems with geometric symmetry—like a cube—the cancellation of field components due to symmetric placement of charges allows for easier calculation of the net field at specific locations, such as the center of a face, by reducing the number of unique contributions that must be analyzed.

Vector Analysis

Vector analysis is important in determining the magnitude and direction of the net electric field. It involves decomposing individual field contributions into components and applying vector addition principles. This is especially useful when the geometry of the charge configuration leads to fields that do not align along a single common axis.

Transcript

00:01 So in this question, we are asked to consider the charge distribution in 19 .74, which is basically a cube with little charges placed on each vertex of the cube.

00:14 We're asked to show that the magnitude of the electric field at the center of the face of any face on the cube is a certain value.

00:22 And then we need to comment on the direction of the electric field.

00:26 So what i've done is i've kind of just re - sketched the cube so we can speak a little bit more clearly about what's going on.

00:37 And what i'd like to do is i'd like to focus on calculating the electric field in the center of the top face here first.

00:45 Okay.

00:46 So the first thing that we should realize is that the electric field coming from the four charges that are closest to this point.

00:57 Those four charges that are in the top face, you could say, of the cube, they are going to, their electric fields are going to sum up to zero, right? because if you've got, let's say, you know, a contribution from this point, let's call this point a, if you've got a contribution that's going in this direction from a, then you've got an equal.

01:28 Opposite contribution from this point b over here and they're going to cancel out.

01:33 Same with this point and this point we could call them c and d, right? so we don't need to worry about those top four charges and then the bottom four charges those have a similar symmetry argument, except there is going to be a component that doesn't cancel out.

01:53 So let's say that this is point a and this is point b, the electric field from point a, is going to recharge a is going to point this way the electric field from point b is going to point this way so we've got some cancellation in the xy plane but in the z plane you can see that they are adding to create something that is non -zero so what we need to do here is calculate the electric field from one of these charges in the z direction and then we just multiply by four because their contribution, all of those charges are going to have the same contribution to the electric field in the z direction.

02:36 And we don't need to worry about x and y because everything cancels out in that direction.

02:41 So let's focus on, let's say, charge a here.

02:45 And let's calculate the electric field in the z direction from that charge.

02:51 So one thing that you need to be careful about is the radius.

02:55 So what is the separation between charge a and the point that we are considered? so you do need to use pythagoras ' theorem in three dimensions.

03:06 So the separation here is going to be equal to s2 squared plus s2 squared plus s squared.

03:21 So this is just using pythagoras ' theorem in three dimensions.

03:25 It's kind of hard to imagine.

03:26 But basically it's something like this, right? s over 2, s over 2, and s.

03:34 And then we're drawing a vector from a to the point that we're considering and calculating the length of that.

03:43 So that's going to be s over 2 squared plus s over 2 squared plus s squared.

03:48 And so that will give s over 4 plus s over 4.

03:53 That's s over 2 plus s squared.

03:57 Plus s squared.

03:59 So we've got the square root of 3 over 2 s squared, when you add all of those together.

04:12 So that's our radius.

04:14 And then we can easily calculate what's the electric field from a...

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