Answer to Question #88631 in Field Theory for Megha

Question #88631
Using Gauss’ Theorem calculate the flux of the vector field F = x i^ + y j^ + z k^ through the surface of a cylinder of radius A and height H, which has its axis along the z-axis and the base of the cylinder is on the xy-plane.
1
Expert's answer
2019-04-30T09:44:14-0400

The Gauss theorem states that the flux "\\oint_{\\partial V} \\left( \\boldsymbol{F} \\cdot \\boldsymbol{n} \\right) dS" of a vector field "\\boldsymbol{F}" over a boundary "\\partial V" of volume "V" is equal to the volume integral "\\int_V \\left( \\nabla \\cdot \\boldsymbol{F} \\right) d V". Calculating the divergence of our vector field, we have "\\nabla \\cdot \\boldsymbol{F} = 3", and the volume integral is

"\\int_V \\left( \\nabla \\cdot \\boldsymbol{F} \\right) d V = 3 \\int_V d V = 3 V = 3 \\pi A^2 H \\, ,"


where we have taken into account that "V = \\pi A^2 H".


Answer: "3 \\pi A^2 H".


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