Final answer:
To find the total charge of the insulating sphere, integrate the charge density function over the volume of the sphere using the given formula λ(r) = ar^2. The total charge is Q = a * 4π/5 * R^5.
Step-by-step explanation:
To find the total charge of the insulating sphere, we need to integrate the charge density function over the volume of the sphere. Given that the charge density as a function of radial distance from the center is λ(r) = ar^2, where a is a constant, we can set up the integral as follows:
Q = ∫ λ(r) dV
Since the sphere is insulating, the charge is distributed throughout its volume, so the integral becomes:
Q = ∫ a*r^2 dV
We can solve this integral by considering the volume element dV = 4πr^2 dr, where dr is an infinitesimal shell of thickness dr. Substituting these values, we get:
Q = ∫ a*r^2 * 4πr^2 dr
Integrating over the limits of the radius from 0 to R, where R is the radius of the sphere, we obtain the total charge of the sphere:
Q = a * 4π * ∫[0]^R r^4 dr = a * 4π/5 * R^5
Therefore, the total charge of the sphere is Q = a * 4π/5 * R^5.