Question

Hollow Non-Conducting Cylinder and Infinite Line Charge Consider a non-conducting cylinder of infinite length with a hollow core. The inner radius is a, the outer radius is b, and the solid region in between carries a uniformly-distributed volume charge density.a = 3 cm b = 5 cmvolume charge density = +4 microC/m3d = 2 cmcharge per unit length= –0.1 microC/ma)What is the electric field at the position (x,y) =(–b,0) Be sure to give both magnitude and direction.b) Suppose we add an infinite line charge inside the hollow cylinder, but at an off-center location: the line runs along the line x = d where d < a. The line carries a charge per unit length .

Answer 1

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