Find the electric field a distance z above the center of a square loop (side a ) carrying unifor

step 1 In this poem, we're going to use example 2 .1 from your text. We have here a line of link 2L wit

Find the electric field a distance z above the center of a...

Key Concepts

Coordinate Geometry

Defining an appropriate coordinate system is essential to setting up and solving the problem, as it involves relating the spatial position where the field is calculated to the geometry of the charged object. A well-chosen coordinate system and geometry can simplify the integration process and help in evaluating the contributions from different charge elements.

Symmetry in Electrostatics

Symmetry plays a crucial role in simplifying the calculation of electric fields, as it can often indicate which components of the field cancel and which add constructively. In the context of a square loop, understanding the symmetry allows one to reduce the complexity of the integration by focusing on the non-cancelled field components, especially when considered above the center of the loop.

Superposition Principle

The superposition principle states that the total electric field due to a collection of charges is the vector sum of the fields produced by each individual charge. This concept is essential when calculating the field due to a distributed charge, such as a charged loop, by summing or integrating the contributions from each segment of the loop.

Coulomb's Law

Coulomb's law provides the basic relationship for the electric field generated by a point charge, indicating that the field is proportional to the charge and inversely proportional to the square of the distance from the charge. This law is the foundation for deriving the contributions of electric fields from small elements of a continuous charge distribution.

Integration of Continuous Charge Distributions

In scenarios involving continuous charge distributions, the electric field is calculated by integrating the infinitesimal contributions from each small element of charge along the distribution. This method adapts Coulomb's law to handle extended objects by considering the differential charge elements and integrating over the geometry of the object.

Transcript

00:01 In this poem, we're going to use example 2 .1 from your text.

00:08 We have here a line of link 2l with linear charge density lambda.

00:17 From the center of that line, the point p is a distance z.

00:22 And you'd find in that case that the electric field, oh, and this should have a c hat.

00:27 The electric field is the following.

00:30 Now, to do it here through superposition, we have to make some associations.

00:38 Just the length of the line now, l, in that formula will go to a over 2.

00:45 Now, z, z was the length from the center to the point p.

00:50 That's no longer that length.

00:53 Now we have to deal with this, which is a over 2, and we want that length there, the hypotenuse.

01:01 So z will go to the square root, z squared plus a over 2 squared.

01:10 So we get for that.

01:11 And instead of being z hat for the electric field, you have to be along, you've got to be along the line, this line here.

01:24 I'll call it r hat.

01:26 Z head goes to r hat.

01:29 So it's a unit vector from the center pointing to the point p.

01:32 Let me draw this out for the left and right edge.

01:42 I'm going to be looking down, you know, down into the page.

01:47 So here's the left edge.

01:50 Here's the right edge.

01:51 So my positions in a way, you've got to be kind of mindful of here would be.

02:04 So this is r square root of z squared plus a over two squared.

02:15 Thinking of this is r hat, so i call that r.

02:18 Not really of great importance.

02:26 This is a over 2, a over 2.

02:31 This is the angle theta, and this is z here.

02:36 Now, the electric field, based on what we've talked about for being an r hat, is going to be like this, for electric field, and like this.

02:49 So those are two electric fields for those two, and it'd be doubled for that because it's the same on the other side.

02:55 Notice anything in the horizontal cancels out only in the vertical.

03:01 So in the end, we will get a z hat out of this.

03:04 Now, sine theta will be z over square root, z squared plus a over two square.

03:13 We really don't need this r.

03:16 I did that just to identify the r hat.

03:20 There's the r hat.

03:21 So that's what we have there.

03:26 So putting it all together using these associations.

03:31 Now, we're going to need, this is going to give me two times e, 2e sine theta, but we have the front and back.

03:42 So it's twice that...

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