Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
The flux of [tex]\vec E = -x\,\vec\imath + y\,\vec\jmath[/tex] is given by the surface integral
[tex]\displaystyle \iint_S \vec E \cdot d\vec\sigma[/tex]
where [tex]S[/tex] is the given square region, which we can parameterize by
[tex]\vec s(x, z) = x\,\vec\imath + z\,\vec k[/tex]
with [tex]0\le x\le 1[/tex] and [tex]0\le z\le 1[/tex]. The area element is
[tex]d\vec\sigma = \vec n \, dx\,dz[/tex]
where [tex]\vec n[/tex] is the normal vector to [tex]S[/tex]. Depending on the orientation of [tex]S[/tex], this vector could be
[tex]\vec n = \dfrac{\partial\vec s}{\partial x} \times \dfrac{\partial\vec s}{\partial z} = -\vec\jmath[/tex]
or [tex]-\vec n = \vec \jmath[/tex]; either way, the integral reduces to
[tex]\displaystyle \iint_S \vec E \cdot d\,\vec\sigma = \int_0^1 \int_0^1 (-x\,\vec\imath + z\,\vec k) \cdot (\pm\vec\jmath) \, dx\,dz = \boxed{0}[/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
