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(A) The circumference of the circular tube is $2\pi R$.The source $S$ and detector $D$ are at positions right-angled to each other.The length of the s

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$\pi R$

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(A) The circumference of the circular tube is $2\pi R$.The source $S$ and detector $D$ are at positions right-angled to each other.The length of the shorter path is $L_1 = \frac{1}{4} 	imes 2\pi R = \frac{\pi R}{2}$.The length of the longer path is $L_2 = \frac{3}{4} 	imes 2\pi R = \frac{3\pi R}{2}$.The path difference between the two waves is $\Delta x = L_2 - L_1 = \frac{3\pi R}{2} - \frac{\pi R}{2} = \pi R$.For constructive interference (maxima) at $D$,the path difference must be an integer multiple of the wavelength $\lambda$,so $\Delta x = n\lambda$,where $n = 1, 2, 3, \dots$.Thus,$\pi R = n\lambda$,which gives $\lambda = \frac{\pi R}{n}$.To find the maximum value of $\lambda$,we take the smallest integer $n = 1$.Therefore,the maximum value of $\lambda$ is $\pi R$.

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