The magnetic flux passing through the rectang | vedclass

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Question

(B) Consider a small strip of width $dx$ at a distance $x$ from the wire. The magnetic field $B$ at this distance is $B = \frac{\mu_0 I}{2 \pi (r+x)}$

Vedclass · Accepted Answer

$\phi \propto s$

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(B) Consider a small strip of width $dx$ at a distance $x$ from the wire. The magnetic field $B$ at this distance is $B = \frac{\mu_0 I}{2 \pi (r+x)}$.The area of this strip is $dA = s \cdot dx$.The magnetic flux $d\phi$ through this strip is $d\phi = B \cdot dA = \frac{\mu_0 I s}{2 \pi (r+x)} dx$.To find the total flux $\phi$,integrate from $x = 0$ to $x = t$:$\phi = \int_{0}^{t} \frac{\mu_0 I s}{2 \pi (r+x)} dx = \frac{\mu_0 I s}{2 \pi} [\ln(r+x)]_{0}^{t} = \frac{\mu_0 I s}{2 \pi} \ln\left(\frac{r+t}{r}\right)$.From the expression,it is clear that the magnetic flux $\phi$ is directly proportional to the side length $s$ of the loop.

The magnetic flux passing through the rectangular loop is proportional to:

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