Question

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 , a diameter , and its ends, plated with material of infinite conductivity, are connected to the rest of the circuit. when red hot, the rod's conductivity varies with position measured from the center of the rod as

Answer 1

a)The total resistance of the rod R = 2.1 * 10^(3)Ω

b) current Density when voltage ΔV is applied between two ends is J(x) = $=\frac{12\sigma_o\Delta V}{L}$

c) Steady-state electric field is E(x) = $\frac{12\Delta V}{L^3} x^2\hat{i}$

a) Consider a small slice of the rod of thickness dx situated at a distance x from the center of the rod.

L = length of rod = 2cm, d= diameter = 0.5cm,  

Given $\sigma(x)=\sigma_oL^2/x^2$     where σ is the rod's conductivity.

$\sigma_o = 4 * 10^{-2}(\Omega.m)^{-1}$        (given)

The resistance dR of this small section of the rod is

dR=$\frac{\rho_rdx}{A}=\frac{dx}{\sigma_rA} = \frac{4x^{2} dx}{\sigma_o L^2\pi d^2}$

The total resistance of the rod can be found by the integral of these smalls sections over the whole rod

$R = \int\limits^{L/2}_{-L/2} \, dR = \int\limits^{L/2}_{-L/2} \,\frac{4x^2dx}{\sigma_oL^2\pi d^2}=\frac{4x^3}{3\sigma_oL^2\pi d^2} |_{-L/2}^{L/2} = \frac{L}{3\sigma_o\pi d^2}$

⇒R = (2*10^(-2)) / (3 * 4 * 10 ^(-2)* π * 0.5 * 10^(-2))

⇒R = 2.1 * 10^(3) Ω

Thus the total resistance of the rod R = 2.1 * 10^(3) Ω

b)The current density when voltage ΔV is applied between two ends is given by

J(x) $=\frac{I}{A}=\frac{4\Delta V}{R\pi d^2} =\frac{12\sigma_o\Delta V}{L}$

c) the steady-state electric field is  

$E(x) = \frac{J(x)}{\sigma} =\frac{12\sigma_o \Delta V / L}{\sigma_o L^2 / x^2} \hat{i} = \frac{12\Delta V}{L^3} x^2\hat{i}$

The question is incomplete. The full question is probably " 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 of L=2.0cm, and a diameter of d= 0.5cm, and its ends, plated with the material of infinite conductivity, are connected to the rest of the circuit. When red hot, the rod’s conductivity varies with position x measured from the center of the rod as $\sigma(x)=\sigma_oL^2/x^2$ with $\sigma_o = 4 * 10^{-2}(\Omega.m)^{-1}$  a)What is the resistance of the glass rod?  Express your answer both symbolically and as a value in ohms. b)When a voltage ΔV is applied between the two ends, what is the current density J(x)?  c)What is the steady-state electric field E(x)? "

Problem on surface current density:

brainly.com/question/13958826

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