Question

3. In the 2022 film Glass Onion, there is a scene in which the glass onion at the top of a building is filled with hydrogen and detonated. As Shrek observes, onions, just like ogreshave layers This glass onion is no exception. In the instant before it explodes, the onion has an inner layer of plasma (made from the exploding hydrogen) surrounded by a fairly thin metal shell. Around this is a layer of outward-shooting electrons that acts as an insulator with volumetric charge density rho = A * e ^ (- (2m ^ - 3) * r ^ 3) where A is an unknown constant. The inner layer likewise has a volumetric charge density rho = B * e ^ (- (2m ^ - 3) * r ^ 3) with B as an unknown constant. The total charge in the inner layer of the onion is Q = 2C so the total charge in the outer layer (the electron layer) must be Q = - 2C; rho = A * e ^ (- (2m ^ - 3) * r ^ 3)

a. Find the constant A
b. Find the electric field at r = 3.5m (you'll need to find the function first ).
c. BONUS : find the electric field everywhere else, too.

Answer 1

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