Answer:
a) 7232 N / C , 7232 N/C along x direction
b) 18459 N / C , 18459 N / C along x direction
Explanation:
Given:
a = 3 cm
b = 5 cm
p = 4 uC / m^3
d = 2
λ = -0.1 uC / m
ε = 8.85 * 10^-12
Step 1: Apply Gauss' Law for hollow cylinder at taking a surface between a < r < b.
ε_o Φ = ε_o*E*2*π*r*L
Q = π * p * L * ( r^2 - a^2) .... 1
2*π*r*L*ε_o*E = Q .... 2
Substitute 2 into 1
E (r) = (p / 2ε_o) * ( r^2 - a^2 / r ) .... 3
r = - b
E (b) = (4*10^-6 / 2 * 8.85*10^-12) * ( 0.05^2 - 0.03^2 / 0.05)
E(-b) = 7232 i
Step 2: Apply Gauss' Law for Infinite Line charge at taking a surface between d < r < d+b.
Q = λ*L .... 4
2*π*r*L*ε_o*E = Q .... 5
Substitute 2 into 1
E = ( λ / 2*π*ε_o) * ( 1 / (b + d) ) i ..... 6
Step 3 : Apply superposition to evaluate E net in x - direction @ (x,y) = (-b , 0)
Find component in x direction of hollow cylinder:
E_cyl = - (π * (b^2 - a^2) * p) / (2*π*ε_o*b) i .... 7
Use superposition principle 6 and 7:
E = (1 / 2*π*ε_o ) * ( ( λ / (b + d ) ) - (π * (b^2 - a^2) * p) / b ) i
Plug in the values:
E = (1 / 2*π*(8.85*10^-12) ) * ( 10^-7 / 0.07 ) - (π * (0.05^2 - 0.03^2) * 4*10^-6) / 0.05 )
E = 18459 i N / C