Calculate the resistance (in Ω ) of the conducting strip xyzw having uniform conductivity σ = 6 π mho / met , thickness t = ℓ n 2 1 m ., current flows along the strip as shown in figure :

Question

Calculate the resistance (in Ω ) of the conducting strip xyzw having uniform conductivity σ = 6 π ​ mho / met , thickness t = ℓ n 2 1 ​ m ., current flows along the strip as shown in figure :

TestHub · Accepted Answer

Current flows along the circular arcs. A thin arc shell at radius r and width dr has length $\frac{π r}{6}$ and cross-section $t d r$ ; all such shells are in parallel between the straight edges.

Conductance: $d G = \frac{σ t d r}{π r /6}$ , so $G = \frac{6 σ t}{π} \int_{a}^{2 a} \frac{d r}{r} = \frac{6 σ t}{π} ln 2$ .

With $σ = \frac{π}{6} mho/m$ and $t = \frac{1}{l n 2} m$ : $G = \frac{6}{π} \cdot \frac{π}{6} \cdot \frac{1}{l n 2} \cdot ln 2 = 1 mho$ , hence $R = 1 Ω$ .

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