a. To find the constant A, we can use the given total charge in the inner layer of the onion, which is Q = 2C. The volumetric charge density in the inner layer is given by rho = A * e ^ (- (2m ^ - 3) * r ^ 3).
Since we want the total charge in the inner layer to be 2C, we can integrate the volumetric charge density over the volume of the inner layer and set it equal to 2C:
∫[0 to R] (4/3) * π * r^2 * A * e ^ (- (2m ^ - 3) * r ^ 3) dr = 2C
Here, R represents the radius of the inner layer. Since we don't know the exact limits of integration without more information about the onion's dimensions, we cannot solve for A. The integral will depend on the specific geometry of the onion.
b. To find the electric field at r = 3.5m, we can use Gauss's