Use your result in Prob. 2.7 to find the field inside and outside a sphere of radius R, which ca

Use your result in Prob. 2.7 to find the field inside and...

Key Concepts

Electric Field Behavior in Spherical Geometry

The behavior of the electric field in spherical distributions is characterized by two distinct regions: inside and outside the sphere. Inside the sphere, the field increases linearly with the distance from the center because only a portion of the total charge contributes to the field at any given point. Outside the sphere, the field decreases as the inverse square of the distance, reflecting the way that the net enclosed charge acts as if it were concentrated at a point at the center. This conceptual framework is crucial in understanding and graphing the dependence of the electric field magnitude on the distance from the center.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism stating that the total electric flux out of a closed surface is proportional to the charge enclosed by that surface. This law is especially useful for calculating electric fields when the charge distribution exhibits high symmetry, as it simplifies the integral form of Maxwell's equations into an algebraic equation relating field strength to enclosed charge.

Spherical Symmetry

Spherical symmetry implies that the properties of a system are invariant with respect to rotations about its center. In the context of electric fields, this symmetry justifies the use of spherical Gaussian surfaces, enabling the conclusion that the electric field at any point depends only on the radial distance from the center. This greatly simplifies the determination of the field both inside and outside the sphere.

Uniform Charge Distribution

A uniform charge distribution means that the charge density is constant throughout the volume of the sphere. This consistent distribution ensures that the total charge of the sphere can be expressed as the product of the charge density and the volume, and it leads to a predictable variation of the electric field: inside the sphere the field increases linearly with distance from the center, whereas outside it behaves as if all the charge were concentrated at the center.

Transcript

00:01 Hi there, so for this problem, we are told that you need to use the result in problem 2 .7 to find the field inside and upside a sphere of radius capital art, which carries an uniform volume charge density row, and we need to express your answer in terms of the total charge of this sphere.

00:25 And also we need to draw a graph of the electric field as a function of the distance from the center.

00:34 So we know that according to problem 2 .7, sorry, all shells interior to the point at a small radius are, contribute as though their charge were concentrated at the center, while all exterior shells contribute nothing.

00:59 So therefore, we are going to have that, the electric field for art, for a smaller, add smaller art, is going to be four times pi times, xelm sub 0 times the interior charge divided by the distance r square in a radial direction of course.

01:31 Now where the interior queue is the total charge interior to the point.

01:38 Now upside the sphere all the charge is interior.

01:44 So for the upside, upside, we're going to have that.

01:53 Field is just simply 1 over 4 times 5 and epsilon sub 0 times the total charge q divided by the separation distance r to the square in the radial direction.

02:11 Now with that we can conclude that inside the sphere only that fraction of the total which is interior to the pound counts.

02:24 So the interior charge is equal to 4 over 3 times pi times the radius to the 3 where this radius is less than the radius of the sphere divided by 4 times 3 times 3 times the total radius of the sphere to this to the 3 and this times the total charge q so we can cancel some terms in here and then the interior charge is 3, r to the 3 divided by the radius of the sphere to the 3 times the charge q.

03:09 So with this, we can use this equation right here and obtain that for inside the sphere, the electric field is equal to 1 over 4 times pi times epsilon sub 0 times the charge q divide by the charge of the, well, in here, this is for inside...

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