Using the results of Problem 64, we can find the electric field at any radius for any sphericall

Using the results of Problem 64, we can find the electric...

Key Concepts

Gauss's Law

Gauss's law states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. This principle is vital for calculating electric fields in symmetrical charge distributions because it allows one to relate the field directly to the amount of charge contained within a Gaussian surface.

Spherical Symmetry

Spherical symmetry means that a physical system's properties, such as charge distribution, are uniform in every direction from the center. This symmetry simplifies the computation of the electric field since it depends only on the radial distance, making the use of Gaussian surfaces straightforward for both interior and exterior regions.

Uniform Charge Density

Uniform charge density indicates that the total charge is evenly distributed throughout the volume of the sphere. This constant distribution allows us to express the charge within a smaller spherical volume as a simple fraction of the total charge, facilitating the calculation of the electric field inside the sphere.

Electric Field Behavior Inside vs Outside the Sphere

For a uniformly charged sphere, the electric field inside (r < R) increases linearly with distance due to the proportionate increase in the enclosed charge, while outside (r ? R) the field decreases with the square of the distance, behaving as if all the sphere's charge were concentrated at its center. This differentiation arises directly from applying Gauss's law in the respective regions.

Graphical Representation of Electric Field

A graph of the electric field versus radius for a charged sphere typically shows a linear increase from the center up to the surface of the sphere, followed by a rapid decay proportionate to the inverse square of the distance outside the sphere. This visual representation reinforces the understanding of how the enclosed charge influences the field in different spatial regions.

Transcript

00:01 So we're looking at a sphere, solid sphere, that has the charge queue evenly spread out throughout all of its volume.

00:10 And we want to know the electric field for when our variable r is outside of the sphere.

00:20 Electric field for when our variable r is inside of the sphere.

00:26 And then we want to sketch a little graph of that.

00:30 Outside of the sphere, because this is based off the assumption of another problem, or the results of another problem, where we say that once we're outside, because of gauss's law, we can just treat it as if it's all concentrated as a point charge at the center.

00:46 So this is going to look exactly like the electric field of a point charge, which is charge q, 4 pi r squared, and in the radial direction.

00:59 But we're looking for magnitude, so we're not even going to worry about direction.

01:05 And our q is q, so we're just leaving it.

01:07 This is the electric field outside that sphere.

01:11 It gets a little trickier inside the sphere.

01:15 Because if it's evenly distributed, as we're integrating from the center and our little gaussian sphere is getting larger and larger, we're enclosing more and more charge.

01:26 We haven't enclosed the total q yet.

01:29 So let's introduce a volume charge density, where this is the total charge q divided by the volume of the sphere.

01:39 And so our q enclosed as this gaussian sphere gets larger and larger is going to be this volume charge density times the differential volume we're enclosing.

01:52 So on our other side of gauss's law, we have this e .d .a.

01:56 This area is getting larger because we're getting a larger sphere.

01:59 But it's enclosing a larger volume as well.

02:03 So that's going to be four thirds pi variable r cubed.

02:10 This is all over epsilon not.

02:11 And this is equal to that integral, which that integral is going to give us the 4 pi variable r squared.

02:21 Okay, we can already cancel it out...