Find the electric field a distance z above the center of a flat circular disk of radius Find the

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

Key Concepts

Electric Field from Continuous Charge Distributions

This concept involves calculating the electric field generated by a charge distribution by integrating the contributions of infinitesimal charge elements using Coulomb’s law. When the charge is continuously distributed over a region, the net electric field is obtained by summing (integrating) the vector contributions from all such elements, taking into account their distance and orientation relative to the point where the field is being evaluated.

Uniform Surface Charge Density

Uniform surface charge density refers to a constant amount of charge per unit area distributed over a surface. This uniformity simplifies the integration process by allowing the use of symmetry in the calculations, as every part of the surface contributes in a predictable manner. It is a key parameter in determining the magnitude of the electric field from surfaces like disks or planes.

Use of Symmetry and Integration in Cylindrical Coordinates

Exploiting the symmetry of the problem, such as in a flat circular disk, allows the electric field calculation to be simplified by converting the problem into cylindrical coordinates. This method takes advantage of the radial symmetry so that only the vertical component of the electric field survives after integration, reducing the complexity of the integral.

Limiting Cases – Infinite Plane and Far-Field Approximation

Analyzing limiting cases is a crucial technique in physics. In the limit as the radius approaches infinity, the disk becomes an infinite plane, for which there is a known result for the electric field that is independent of the distance from the plane. Conversely, when the observation point is much further away than the size of the disk (z >> R), the disk’s charge distribution approximates that of a point charge, leading to an electric field that falls off as 1/z^2. These limits serve as consistency checks for the derived expression.

Transcript

00:01 Hi there, so for this problem, we need to find the electric field, a distant seat above the center of a flat disk of radius.

00:12 You need to find the of radius capital art, which carries a uniform surface charge stigma.

00:23 So the question in this is, what does your formula give in the limit when the radius tends to infinity? and also we need to check when seed is much greater than the radius art.

00:39 So first, we're going to put in here the image of this problem, this one in here.

00:52 So as you can see, this is a disk, a circular desk of radius art, and we are also the information that it has an uniform source.

01:05 Surface charge sigma and a radius art.

01:09 And we now we need to calculate the electric fuel at this point right here.

01:15 So what we can do is to break it into rings of radius art and thickness delta art.

01:26 So we use the solution for the problem 2 .5 to express the field of each ring.

01:34 So the total charge of a ring, we know that that is going to be sigma, times 2 times pi times the radius r times the differential in the thickness, and this is equal to lambda times two times pi times the radius art, where lambda is equal to sigma times the differential in the radius, and is the line charge of each ring.

02:07 So we can write the electric field of each ring as 1 over 4 times pi times epsilon sub 0 times the charge lambda times 2 times pi times r times seed.

02:30 And this divided by the distance, we know that this, this distance right here from each ring is going to be equal in this case to the radius r squared plus cid square and all of this elevated to 3 over 2.

02:53 And this now to find the electric field of the disk, what we need to do is to simply integrate the previous expression.

03:06 But first we're going to take out everything that do not depend on the radius.

03:17 So the integral starts at zero and ends at the total at the maximum radius.

03:25 This is the radius r divided by the radius r square plus seed square.

03:35 And this elevated to 3 over 2, integrated over the radius art.

03:41 So when we do this integral, we'll immediately obtain that the electric field of the desk is equal to 1 over 4 times pi times epsilon sub 0 times 2 times pi times sigma times sit.

03:59 And this times 1 over seed minus the 1 over the square root of the radius square plus seed square.

04:09 Of course, this after we have done the integral and evaluated at the point zero in art.

04:19 And of course, this is in the seed direction...

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