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(A) The magnetic field $B$ at a distance $r$ from a long straight wire carrying current $i$ is given by $B = \frac{\mu_0 i}{2 \pi r}$.The area $A$ und

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$Al$

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(A) The magnetic field $B$ at a distance $r$ from a long straight wire carrying current $i$ is given by $B = \frac{\mu_0 i}{2 \pi r}$.The area $A$ under the $B-r$ curve between $r_1$ and $r_2$ is defined as $A = \int_{r_1}^{r_2} B \, dr = \int_{r_1}^{r_2} \frac{\mu_0 i}{2 \pi r} \, dr = \frac{\mu_0 i}{2 \pi} \ln\left(\frac{r_2}{r_1}\right)$.The magnetic flux $\phi$ through the rectangular loop of length $l$ is given by $\phi = \int B \, dA_{loop}$.Considering a strip of width $dr$ at a distance $r$ from the wire,the area of the strip is $dA_{loop} = l \, dr$.Thus,the flux is $\phi = \int_{r_1}^{r_2} B \, l \, dr = l \int_{r_1}^{r_2} B \, dr$.Since $A = \int_{r_1}^{r_2} B \, dr$,we substitute this into the flux equation:$\phi = l \cdot A = Al$.

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