[amp_mcq option1=”$$\\mathop{{\\int\\!\\!\\!\\!\\!\\int}\\mkern-21mu \\bigcirc} {\\left( {\\nabla \\times \\overrightarrow {\\text{A}} } \\right) \\cdot \\overrightarrow {{\\text{ds}}} } $$ over the closed surface bounded by the loop” option2=”$$\\mathop{{\\int\\!\\!\\!\\!\\!\\int\\!\\!\\!\\!\\!\\int}\\mkern-31.2mu \\bigodot} {\\left( {\\nabla \\cdot \\overrightarrow {\\text{A}} } \\right){\\text{dv}}} $$ over the closed volume bounded by the top” option3=”$$\\iiint {\\left( {\\nabla \\cdot \\overrightarrow {\\text{A}} } \\right){\\text{dv}}}$$ over the open volume bounded by the loop” option4=”$$\\iint {\\left( {\\nabla \\times \\overrightarrow {\\text{A}} } \\right) \\cdot \\overrightarrow {{\\text{ds}}} }$$ over the open surface bounded by the loop” correct=”option1″]<\/p>\n
The correct answer is:<\/p>\n
$$\\oint {\\overrightarrow {\\text{A}} \\cdot \\overrightarrow {{\\text{d}}l} } = \\mathop{{\\int!!!!!\\int}\\mkern-21mu \\bigcirc} {\\left( {\\nabla \\times \\overrightarrow {\\text{A}} } \\right) \\cdot \\overrightarrow {{\\text{ds}}} } $$<\/p>\n
This is known as Stokes’ theorem. It states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of the vector field over any surface bounded by the loop.<\/p>\n
The curl of a vector field is a measure of how much the vector field rotates around a point. It is defined as the cross product of the gradient and the vector field.<\/p>\n
The surface integral of a vector field over a surface is the sum of the dot products of the vector field and the area element over the surface.<\/p>\n
In this case, the vector field is $\\overrightarrow {\\text{A}} \\left( {\\overrightarrow {\\text{r}} } \\right)$ and the surface is the closed surface bounded by the loop. The curl of the vector field is $\\nabla \\times \\overrightarrow {\\text{A}} $.<\/p>\n
The line integral of $\\overrightarrow {\\text{A}} \\cdot \\overrightarrow {{\\text{d}}l} $ around the loop is equal to the surface integral of $\\left( {\\nabla \\times \\overrightarrow {\\text{A}} } \\right) \\cdot \\overrightarrow {{\\text{ds}}} $ over the closed surface bounded by the loop.<\/p>\n
The other options are incorrect because they do not take into account the closed surface bounded by the loop.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”$$\\mathop{{\\int\\!\\!\\!\\!\\!\\int}\\mkern-21mu \\bigcirc} {\\left( {\\nabla \\times \\overrightarrow {\\text{A}} } \\right) \\cdot \\overrightarrow {{\\text{ds}}} } $$ over the closed surface bounded by the loop” option2=”$$\\mathop{{\\int\\!\\!\\!\\!\\!\\int\\!\\!\\!\\!\\!\\int}\\mkern-31.2mu \\bigodot} {\\left( {\\nabla \\cdot \\overrightarrow {\\text{A}} } \\right){\\text{dv}}} $$ over the closed volume bounded by the top” option3=”$$\\iiint {\\left( {\\nabla \\cdot \\overrightarrow {\\text{A}} } \\right){\\text{dv}}}$$ over the open volume bounded by … <\/p>\n