An infinitely long cylinder with a radius of 1.0 cm has a charge density that is proportional to the distance from the axis of symmetry, $\rho$ = ar, where a = 10.0 C/m⁴. The charge per unit length within the cylinder is

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The charge per unit length is found by integrating the charge density over the cross-sectional area of the cylinder. The charge density is $\rho = ar$, so the charge per unit length is $\int_0^{0.01} ar \cdot 2\pi r \, dr = 2\pi a \int_0^{0.01} r^2 \, dr = 2\pi a \left[\frac{r^3}{3}\right]_0^{0.01} = 2\pi \cdot 10 \cdot \frac{(0.01)^3}{3} = 21 \, \mu C/m$.

An Infinitely Long Cylinder with a Radius of 1 ρ\rhoρ = Ar, Where a = 10

An Infinitely Long Cylinder with a Radius of 1 $\rho$ = Ar, Where a = 10