The mutual inductance between the rectangular | vedclass

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Question

(B) The magnetic field $B$ at a distance $x$ from the long straight wire carrying current $i$ is given by $B = \frac{\mu_0 i}{2\pi x}$.Consider a smal

Vedclass · Accepted Answer

$M = \frac{\mu_0 a}{2\pi} \ln \left( 1 + \frac{b}{c} \right)$

Vedclass · Answer

(B) The magnetic field $B$ at a distance $x$ from the long straight wire carrying current $i$ is given by $B = \frac{\mu_0 i}{2\pi x}$.Consider a small rectangular strip of width $dx$ at a distance $x$ from the wire. The area of this strip is $dA = a \cdot dx$.The magnetic flux $d\phi$ through this strip is $d\phi = B \cdot dA = \left( \frac{\mu_0 i}{2\pi x} \right) a \cdot dx$.To find the total flux $\phi$ through the loop,we integrate from $x = c$ to $x = c + b$:$\phi = \int_{c}^{c+b} \frac{\mu_0 i a}{2\pi x} dx = \frac{\mu_0 i a}{2\pi} [\ln x]_{c}^{c+b} = \frac{\mu_0 i a}{2\pi} \ln \left( \frac{c+b}{c} \right) = \frac{\mu_0 i a}{2\pi} \ln \left( 1 + \frac{b}{c} \right)$.Since $\phi = M i$,the mutual inductance is $M = \frac{\mu_0 a}{2\pi} \ln \left( 1 + \frac{b}{c} \right)$.

The mutual inductance between the rectangular loop and the long straight wire as shown in the figure is $M$.

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