Find the electric field a distance z above the center of a circular loop of radius r (Fig. 2.9 )

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

Key Concepts

Coulomb's Law

Coulomb's law describes the electric force between charges and is the fundamental basis for calculating the electric field. When applied to continuous charge distributions, its differential form is used, where the contributions from each element of charge are integrated over the distribution to obtain the net field.

Line Charge and Linear Charge Density

A line charge is a distribution of electric charge along a one-dimensional path, and its density is described by the linear charge density (?), which represents the amount of charge per unit length. This concept is essential for modeling the charge distribution over objects like circular loops.

Electric Field from Continuous Charge Distributions

Calculating the electric field due to a continuous charge distribution involves breaking the distribution into infinitesimal charge elements and summing (integrating) their contributions to the electric field at the point of interest. This approach generalizes Coulomb's law to extended objects.

Symmetry in Electric Field Calculations

The exploitation of symmetry simplifies the computation of the electric field. In the case of a circular loop, symmetry ensures that horizontal components of the field cancel out, leaving only the vertical (axial) component, which streamlines the integration process.

Integral Calculus in Electromagnetism

Integral calculus is used to sum the contributions of infinitesimally small charge segments to determine the net electric field. This technique is crucial when dealing with continuous charge distributions, enabling the transformation of differential contributions into a closed-form solution.

Transcript

00:01 In this problem, we have a ring of charge, with linear charge, essentially lambda.

00:06 We're looking for the, and the origins at the center of that ring, and we're looking for the point, the electrophial at a point p, which is along the z -axis, and the radius of the ring is r, a little r.

00:18 Now, let's look at, and let's assume for a sake of argument that lambda is positive.

00:23 Anything we write, as long as we don't put absolute value signs around it, will work if lambda is negative.

00:28 We have negative charge.

00:29 So let's not worry about it.

00:32 So let's look at the two opposite dqs, infinitesimal charge elements.

00:39 And this would be r, which is equal to the square root, r squared plus e squared, and likewise in this side is totally symmetric, r.

00:54 Now everybody, just to remind you, everybody will, every dq will have a pair, will have a match.

01:01 E2 match, front and back match, everybody has a match.

01:04 Whatever i'm talking about here, whatever i have to talk about here will work for any pair, and so we'll be able to go through the whole sequence, whole set, infinite set of dqs.

01:14 I am going to need this angle theta in minutes, let me write it in.

01:17 All right.

01:18 So now let's look at the electric field due to the right dq.

01:23 That will be like this, and for the left, it'll be like this.

01:29 Notice anything in the horizontal? in this case, you think of this as the y axis and x coming out towards you and z.

01:37 The y components we cancel out, you only have z.

01:41 So what we're going to find, what we find then is that we have to just add up all the z components of all the dqs.

01:48 When we do our integration, we're just going to be adding up all the z components.

01:54 And before we get into doing that, we're going to need sine theta, z over capital r, which is r squared, or c squared.

02:07 Now we, dq, that's going to be lambda, times ds, ds, ds is an infinitesimal arc length, but this is a circle, lambda, r, d5.

02:19 Phi is the angle in the horizontal, taking you from one point in the circle all the way around every point.

02:26 It's got nothing to do with a point p.

02:29 This is for our integration.

02:32 Okay, so, d, e, z, this is the z component.

02:36 This is the electric field due to the right, dq, but this is the electric field due to the right and to the left.

02:46 This is the electric, the z component of electric field due to every dq.

02:55 So 1 over 4 pi epsilon 0, dq, capital r squared.

03:02 So that's going to be r squared plus z squared, time sign theta...

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