The charge density within a charged sphere of radius R is given by ρ=ρ0-a r^2, where ρ0 and a ar

step 1 So we have a sphere. Sorry, this should be solid, dashed lines back here, with radius capital R.

The charge density within a charged sphere of radius R is...

Key Concepts

Gauss' Law

Gauss's law is a fundamental principle in electromagnetism which states that the electric flux through any closed surface is proportional to the total charge enclosed within the surface. This principle is used to analyze electric fields generated by various charge distributions and is central in determining conditions where the field can vanish outside a charged object.

Charge Distribution Integration

Integrating a volume charge density over the entire volume of an object allows the calculation of the total charge contained within. This process is especially important for non-uniform charge distributions, where the density varies with position, by summing the contributions from each infinitesimal volume element.

Spherical Symmetry

Spherical symmetry implies that a charge distribution, as well as the resulting electric field, depends only on the radial distance from the center and not on the direction. This simplification is critical when applying Gauss's law because it ensures that the electric field, at a given radial distance, is uniform over the surface of a sphere, greatly simplifying the calculations.

Net Charge Cancellation

Ensuring that the electric field outside a sphere is zero involves setting the total net charge of the sphere to zero. This condition requires that the integrated charge over the entire volume of the sphere vanishes, which in turn imposes specific relationships among the parameters defining the charge distribution. This concept is essential when designing or analyzing systems with precise electrical properties.

Transcript

00:01 So we have a sphere.

00:06 Sorry, this should be solid, dashed lines back here, with radius capital r.

00:14 And charge density rho is equal to rho -naught minus a times r -squared for some parameters rho -naught and a.

00:23 We want to find a value of a, such that the electric field outside the sphere is equal to zero.

00:31 What that means, by gauss's law, is that the total charge on the sphere has to be zero.

00:37 So we set up an expression for the total charge.

00:40 It's going to be the integral over the sphere of rho -dv, right? which is the integral over the sphere of rho -naught minus a times r -squared dv, which we can set up in spherical coordinates.

00:59 Theta goes from zero to two pi, phi goes from zero to pi, r is going to go from zero to big r, of our rho, rho -naught minus a times r -squared, times r times sine of phi, dr dphi dtheta.

01:26 We want this to be equal to zero.

01:30 First things first, theta is not in the terms, so we can treat all this as a constant with respect to theta, just multiply it by two pi.

01:36 And since we're setting it equal to zero, divide by two pi as well.

01:40 So great, you've dealt with theta all at once.

01:44 Let's now pull sine phi out of the r integral.

01:48 So get from zero to pi, sine of phi, integral from zero to capital r, rho -naught times r minus a times r -cubed, dr dphi.

02:06 So that's great.

02:09 We can also go ahead and pull phi out of this, because all of this rho integral is constant with respect to phi.

02:20 We can pull it out of the phi integral as a constant.

02:25 This also needs to be zero.

02:30 And then so the integral from zero to pi, sine of phi, dphi, which is just a constant, some constant times integral from zero to r, of rho -naught times r minus a times r -cubed, dr is equal to zero.

02:53 Divide through by this constant, which i think is one, or pi, it doesn't matter, it's a constant, it goes away.

03:00 And we just need to solve this.

03:02 Now it's a polynomial, we can antidifferentiate...

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