A cube of side a has an inhomogeneous conductivity sigma(x) = sigma_0(1 + x/a),

Question

A cube of side a has an inhomogeneous conductivity sigma(x) = sigma_0(1 + x/a), where the axes of a Cartesian coordinate system are aligned with three edges of the cube according to the figure below. The positive electrode of a battery is connected to a metal plate of side a and the negative electrode to another square metal plate of the same size. a) Calculate the resistance of the cube when the battery is connected to the surfaces of the conducting cube that have constant x-coordinates. b) Calculate the resistance of the cube when the battery is connected to the surfaces of the conducting cube that have constant y-coordinates. The answers should be a) R = ln 2/sigma a b) R = 2/3 sigma a

Explanation / Answer

given conductivity , sigma(x) = sigma-o(1 + x/a)
now,
a. consider the surfaces having constant x coordinates
so consider any surface at a coordinate x such tat x > 0 and x < a
so, considering a small volume of cross section a^2
resistance of this volume
dR = dx/a^2*sigma-o(1 + x/a)
net resistance
R = integrate [ dx/a^2*sigma-o*(1 + x/a)] from x = 0 to x = a
so R = ln(2)/a*sigma-o
b. when connected to surfqaces having constant y coordinates
consider a thickness dx at coordinate x, then resistance of this small strip is
dR = a/dx*a*sigma(1 + x/a)
current through this element, di = V/dR [ ohms law]
so, di = V*sigma(1 + x/a)dx
i = V*sigma(a + a^2/2a) = V*(sigma*3a/2)
so, Reff = V/i = 2/3asigma

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.