We need to find the total charge enclosed in a cylindrical volume with a given charge density. The charge density is given as a function of the radial coordinate r in cylindrical coordinates: ρ(r)=3r+r3. The volume of interest is defined by the region 0≤r≤1, 0≤z≤1, and 0≤θ≤2π.
The total charge Q enclosed in the volume can be found by integrating the charge density over the specified volume. The differential volume element in cylindrical coordinates is given by dV=rdzdrdθ.
The integral to find the total charge is:
Q=∫02π∫01∫01ρ(r)rdzdrdθ
Substitute ρ(r)=3r+r3 into the integral:
Q=∫02π∫01∫01(3r+r3)rdzdrdθ
Simplify the integrand:
Q=∫02π∫01∫01(3r2+r4)dzdrdθ
First, integrate with respect to z:
Q=∫02π∫01(3r2+r4)[z]01drdθ=∫02π∫01(3r2+r4)drdθ
Next, integrate with respect to r:
Q=∫02π[33r3+5r5]01dθ=∫02π(r3+5r5)01dθ
=∫02π(1+51)dθ=∫02π56dθ
Finally, integrate with respect to θ:
Q=[56θ]02π=56×2π=512π
The total charge enclosed in the specified volume is 512π.