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April 3

A query with triple products

Let $\mathbf {B}$ be a constant vector and let $\mathbf {r}$ be the position vector. Then is the integral $\oint (\mathbf {r\times B} )\times d\mathbf {l}$ zero?Sayan Ghosh 05:16, 3 April 2018 (UTC) — Preceding unsigned comment added by Sayan19ghosh99 (talk • contribs)

The integral is not generally zero. The vector field r × B is a rotational (cylindrical) flow around an axis that passes through the origin and that is parallel to B. This flow is everywhere perpendicular to B and has a magnitude proportional to the distance from the B-axis. Consider integration around a loop which lies in a plane that includes the B-axis (i.e. the axis lies in the plane). The flow r × B is everywhere normal to this plane and so is perpendicular to every element dl in the loop. Let the loop be a rectangle with two sides parallel to B and two sides radial to the B-axis. The integral over the two radial sides will cancel since they are similar except for the sign of dl. The integral over the two sides parallel with the axis will not cancel because the magnitude of the integrand is greater for the side further from the axis - and therefore the integral is non-zero. --catslash (talk) 11:42, 3 April 2018 (UTC)[reply]

You have a vector triple product under the integral. So expanding the triple product and using the Kelvin–Stokes theorem

\oint (\mathbf {r} \times \mathbf {B} )\times d\mathbf {l} =\mathbf {B} \oint \mathbf {r} d\mathbf {l} -\oint \mathbf {r} (\mathbf {B} d\mathbf {l} )=\mathbf {B} \int (\nabla \times \mathbf {r} )d\mathbf {S} -\mathbf {B} \times \int d\mathbf {S} =-\mathbf {B} \times \int d\mathbf {S} \neq \mathbf {0}

,

where it was taken into account that

\nabla \times \mathbf {r} =0

. Ruslik_Zero 20:38, 3 April 2018 (UTC)[reply]