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Is it because it is perpendicular to the B field?

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Yes,you are right. – SchrodingersCat Sep 28 '15 at 13:43

Yes; as the integral has a dot product between magnetic field and length element. $$ B \hat{x} . dl \hat{y}=0$$

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it is because line integrals of magnetic field through these portions is zero.Yes, the vertical paths are perpendicular to magnetic field

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This is more a comment than an answer. Please visit the help center to know "How to write a good answer" at this – SchrodingersCat Sep 28 '15 at 13:51

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