Find the electric field inside a sphere which carries a charge density proportional to the dista

Find the electric field inside a sphere which carries a...

Key Concepts

Non-uniform Charge Density

Non-uniform charge density refers to a situation where the amount of charge per unit volume varies with position rather than remaining constant. In electrostatics, this variation necessitates the use of integration to calculate the total charge enclosed within a given volume. A charge density that is a function of position complicates the calculation of the electric field since the standard formulas for uniform charge distributions no longer apply directly.

Spherical Symmetry

Spherical symmetry is a characteristic of systems where the physical properties are invariant under any rotation about the center. In problems involving electric fields in spheres, spherical symmetry simplifies the calculations because it allows one to assume that the electric field at any point inside or outside the sphere depends only on the distance from the center. This symmetry justifies the choice of a spherical Gaussian surface, making the integration process more straightforward.

Integration for Enclosed Charge

When dealing with a non-uniform charge density, one must integrate the charge density function over the volume to compute the total charge enclosed within a specified region. This process involves setting up an appropriate integral that accounts for the spatial variation of the charge density and often taking advantage of the symmetry of the system, such as spherical coordinates in a problem involving a sphere.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is a fundamental tool in electrostatics used to determine electric fields, particularly in cases with high symmetry such as spherical symmetry. By applying Gauss's Law, one can connect the integral of the electric field over a closed surface to the total charge enclosed, which is especially useful when dealing with continuous charge distributions.

Transcript

00:01 In this example, we're gonna be finding the electric field inside a sphere, okay? this sphere has a charge density that's proportional to the distance from the center, okay? so here's our sphere with radius r.

00:17 Here we wanna find the electric field somewhere inside the sphere.

00:24 Okay, so we'll call this radius little r, so we have r is less than big r.

00:30 Okay, we have a charge density that's given by rho equals kr, okay? so to find the enclosed charge for our gaussian surface, we're gonna need to integrate over three dimensions to find that enclosed charge.

00:53 Okay, so let's do that first.

00:54 So we have q enclosed equals, it's spherically symmetric so we can take those parts out.

01:04 So we have 4 pi times the integral from 0 to r, rho r squared dr, okay, so let's put our expression for rho in there.

01:20 K is a constant, so we can pull that out, so this equals 4 pi k, 0 to r, r cubed dr, okay, so let's do that integral.

01:34 And that becomes 4 pi k, 0 to r, r4 over 4, so that just evaluates to 4 pi k, r to the 4 over 4.

01:54 All right, so that's my q enclosed that we're gonna be using when we get to gauss's law, which we'll be doing right now.

02:00 Okay, so gauss's law says e .da, okay, where da is a surface integral over the area of our sphere.

02:11 That equals q enclosed over epsilon not the permittivity of free space.

02:20 Okay, we know that for a gaussian surface of radius r, spherical surface that da is just 4 pi r squared, the area of that sphere.

02:31 So we have e times 4 pi r squared equals q enclosed over epsilon not...

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