A long thin circular tube of radius 10cm carries a current $I = I_{0} \sin ω t A / m$ on it’s curved surface as shown in figure. $I_{0}$ is current per unit length perpendicular to flow on the surface. Current density vector at every point makes an angle $θ = 45^{\circ}$ with circumferential direction. A small wire loop having single turn of resistance R, and loop area A is placed inside the tube with it’s axis coinciding with axis of tube. Ignore the self inductance of loop. The average rate of Joule heating in loop is $\frac{x {(μ_{0} I_{0} ω A)}^{2}}{R}$ watt, find $x$

Question

A long thin circular tube of radius 10cm carries a current $I = I_{0} \sin ω t A / m$ on it’s curved surface as shown in figure. $I_{0}$ is current per unit length perpendicular to flow on the surface. Current density vector at every point makes an angle $θ = 45^{\circ}$ with circumferential direction. A small wire loop having single turn of resistance R, and loop area A is placed inside the tube with it’s axis coinciding with axis of tube. Ignore the self inductance of loop. The average rate of Joule heating in loop is $\frac{x {(μ_{0} I_{0} ω A)}^{2}}{R}$ watt, find $x$

Infinity Learn · Accepted Answer

Current in circumferential direction per unit axial length of tube is =I0cosθϕ=(μ0I0cosθsinωt)Aemf=μ0I0ωAcosθcosωtAverage rate of heating =e2rmsRP≡(μ0I0ωA)22Rcos2θ

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