Consider a spherical plastic shell with inner radius r=1cm, r2=2cm, and a dielectric constant K=4. The center is hollow. Free charge is bound within the plastic. The charge distribution is spherically symmetric but decays with radius, following a charge density:p_f=K/r where K=20uC/m^2 is constant.

a) what is the total amount of free charge embedded in the plastic in coulombs?
b)What bound surface charge is induced on the outer surface of the plastic, in C/m^2?

To find the total amount of free charge embedded in the plastic, we need to integrate the charge density over the volume of the plastic shell.

a) The charge distribution is given by p_f = K/r, where K = 20 μC/m² is the constant charge density.

To find the total amount of charge, we integrate the charge density over the volume of the plastic shell:

Q = ∫ p_f dV

Since the charge density is spherically symmetric and decays with radius, we can express the volume element dV as dV = 4πr²dr. Substituting this into the integral, we get:

Q = ∫(0→R) K/r * 4πr² dr,

where R is the outer radius of the plastic shell.

Simplifying the integral:

Q = 4πK ∫(0→R) r dr,

Q = 4πK [ 1/2r² ] (0→R),

Q = 4πK (1/2R² - 1/2(1cm)²),

Substituting the given values r = 1cm (0.01m) and K = 20 μC/m²:

Q = 4π(20 μC/m²) (1/2R² - 1/2(0.01m)²),

Now we have to substitute the values of the outer radius R = 2cm (0.02m):

Q = 4π(20 μC/m²) (1/2(0.02m)² - 1/2(0.01m)²),

Calculating this expression will give you the total amount of free charge embedded in the plastic shell in coulombs (C).

b) To find the bound surface charge induced on the outer surface of the plastic, we need to consider the dielectric constant (K) and the charge density (p_f) at the inner surface of the plastic shell.

The bound surface charge induced on the outer surface (σ_s) is related to the free charge density (p_f) by the following equation:

σ_s = -K(p_f),

where K is the dielectric constant.

Since p_f = K/r, substituting this into the equation for σ_s gives:

σ_s = -K(K/r) = -K²/r.

Substituting the given values K = 4 and r = 2cm into the equation, we have:

σ_s = - (4)²/(0.02m).

Calculating this expression will give you the bound surface charge induced on the outer surface of the plastic shell in C/m².