Question

(a) A vector field F(r) is called solenoidal if its divergence equals to zero, i.e. ? · F(r) = 0. Suppose that a 3-dimensional vector field F(r) has the form f(r)r, where r = xi + yj + zk and r = ||r|| = ?(x^2 + y^2 + z^2). Show that a(r) is solenoidal only if f(r) = const/r^3. (b) From the Maxwell equations, steady electric field E(r) = E(x,y,z) in a vacuum satisfies ? × E = 0, ? · E = 0. The first equation implies E is a gradient (also called conservative or irrotational) field, i.e. E(x,y,z) = -??(x,y,z) for some scalar field ?(x,y,z) (usually called electric potential). Using the fact that E(r) is also solenoidal (as seen from the second equation above) show that the general form of radial symmetric scalar field ?(r) = ?(r), where r = ||r|| = ?(x^2 + y^2 + z^2), is given by ?(r) = A/r + B with A, B arbitrary constants.

          (a) A vector field F(r) is called solenoidal if its divergence equals to zero, i.e. ? · F(r) = 0. Suppose that a 3-dimensional vector field F(r) has the form f(r)r, where r = xi + yj + zk and r = ||r|| = ?(x^2 + y^2 + z^2). Show that a(r) is solenoidal only if f(r) = const/r^3.

(b) From the Maxwell equations, steady electric field E(r) = E(x,y,z) in a vacuum satisfies ? × E = 0, ? · E = 0.

The first equation implies E is a gradient (also called conservative or irrotational) field, i.e. E(x,y,z) = -??(x,y,z) for some scalar field ?(x,y,z) (usually called electric potential). Using the fact that E(r) is also solenoidal (as seen from the second equation above) show that the general form of radial symmetric scalar field ?(r) = ?(r), where r = ||r|| = ?(x^2 + y^2 + z^2), is given by ?(r) = A/r + B with A, B arbitrary constants.

(a) A vector field F(r) is called solenoidal if its divergence equals to zero, i.e. ? · F(r) = 0. Suppose that a 3-dimensional vector field F(r) has the form f(r)r, where r = xi + yj + zk and r = ||r|| = ?(x^2 + y^2 + z^2). Show that a(r) is solenoidal only if f(r) = const/r^3.

(b) From the Maxwell equations, steady electric field E(r) = E(x,y,z) in a vacuum satisfies ? × E = 0, ? · E = 0.

The first equation implies E is a gradient (also called conservative or irrotational) field, i.e. E(x,y,z) = -??(x,y,z) for some scalar field ?(x,y,z) (usually called electric potential). Using the fact that E(r) is also solenoidal (as seen from the second equation above) show that the general form of radial symmetric scalar field ?(r) = ?(r), where r = ||r|| = ?(x^2 + y^2 + z^2), is given by ?(r) = A/r + B with A, B arbitrary constants.

Added by Lori J.

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