Consider the space region bounded below by the right angled cone z=sqrt(x^2+y^2), and above by the sphere
x^2+y^2+z^2=2.
These two surfaces intersect in a horizontal circle, let T be the horizontal disk having this circle as boundary, S the spherical cap forming the upper surface, and U the cone forming the lower surface.

Orient S,T,U "upwards", so the normal vector has a positive k-component.

QUESTION 1)For each of the three surfaces, determine geometrically (without calculation) whether the flux of the vector field F= -x*i-y*j is positive or negative.

QUESTION 2) Calculate the flux of F ACROSS EACH SURFACE (with the upwards orientation) .

Solution Preview

1.)
Because, S, T and U are upward, therefore, the area vector for cone (U)
will be diverging from the axis of cone (z-axis) and normal to the surface, for sphere (S) will be converging towards the origin and for circle (T) it will be in only in +ve z direction.
F = -x*i-y*j will always be in a plane parallel to x-y plane and directed towards the z axis, therefore for cone (U) the flux (F.A) will be -ve, for sphere (S) it will be +ve and for circle (T) it will be zero. --Answer

2.)
For ...

Solution Summary

This shows how to work with the flux of given vector fields.

Static Magnetic Field - Flux, direction of flux, equivalent reluctance of the core, permeability of the core material. ... Q4) What direction of flux Φ1? ...

... If theta = 24°, find the ratio phi xz/phi xy of the magnetic fluxes that pass through the surfaces. Magnetic flux is the product of the average magnetic field ...

Flux Analysis. ... Note: A flux analysis is also known as an analytical review which is comparing one period to the next, or to budget or to another company. ...

China State in Flux. ... 150 words. 2. The state of flux - argue what is in 'flux' and the points you believe that prove this position - 150 words. ...

Four problems on electric field, charge, flux, charge density, electron velocity in an electric field. ... Hence, electric flux through each of these sides is zero. ...