Please explain steps as best as you can. Thank you.
Non-uniform Conductivity
0.0/2.0 points (graded)
A cylindrical glass rod is heated with a torch until it conducts enough current to cause a light bulb to glow. The rod has a length L, a cross sectional area A, and its
ends, plated with material of very high conductivity, are connected to the rest of the circuit including a voltage supply. When red hot, the rod's conductivity varies
with position x measured from the center of the rod as sigma (x)=sigma _(0)(L^(2))/(x^(2)).
(Part a) Consider a small slice of the rod of thickness dx located a distance x from the center of the rod as indicated in the figure.
What is dR, the resistance of the small slice of rod? Express your answer in terms of A,x,dx,L, and sigma_ 0 for sigma _(0) as needed.
dR=
(Part b) Calculate the total resistance of the rod. Express your answer in terms of A,x,dx,L, and sigma_0 for sigma _(0) as needed.
R=
(Part c) When a voltage Delta V is applied between the two ends, what is the current density vec(J) ? Express your answer using some or all of the following: sigma_0 for sigma _(0),
DeltaV for Delta V, A, and L.
|vec(J)|=
(Part d) What is the magnitude of the steady-state electric field |vec(E)(x)| as a function of x ? Express your answer using some or all of the following: DeltaV for Delta V,
x,A, and L.
|vec(E)|=
L14Q6: Non-Uniform Conductivity
Non-uniform Conductivity
0.0/2.0 points (graded)
A cylindrical glass rod is heated with a torch until it conducts enough current to cause a light bulb to glow. The rod has a length L, a cross sectional area A, and its ends, plated with material of very high conductivity, are connected to the rest of the circuit including a voltage supply. When red hot, the rod's conductivity varies with position x measured from the center of the rod as (x) = L2/x2.
(Part a) Consider a small slice of the rod of thickness dx located a distance x from the center of the rod as indicated in the figure
small slice of rod
x=0
x=-L/2
x=+L/2
What is dR, the resistance of the small slice of rod? Express your answer in terms of A, x, dx, L, and sigma_o for o as needed.
dR=
Part bCalculate the total resistance of the rod.Express your answer in terms of A,x,dxLand sigma_o for o as needed
R
(Part c) When a voltage V is applied between the two ends, what is the current density J? Express your answer using some or all of the following: sigma_0 for Go, DeltaV for V,A, and L.
1J1
(Part d) What is the magnitude of the steady-state electric field |E (x) I as a function of x? Express your answer using some or all of the following: DeltaV for V, x, A, and L.
E|