It's not hard to show that the function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]g = \frac{1}{2} (c \times r)[/tex]

is a "vector potential" function for the constant vector "c". That is, that:

[tex]\nabla \times g = c[/tex]

The calculation is straightforward to carry out in Cartesian coordinates, and I won't reproduce it here.

However, my question concerns the following. It is also a standard result in vector algebra that we have:

[tex]a \times (b \times c) = (a \cdot c) b - (a \cdot b) c[/tex]

My question is, when taking the "curl" of the vector potential above, why can't I just treat the "del" operator as though it were a vector, and write something like the following:

[tex]\nabla \times (c \times r) = (\nabla \cdot r) c - (\nabla \cdot c) r [/tex]

[tex]\nabla \times (c \times r) = 3c [/tex]

where the last equality follows because (a) the divergence of a constant vector is 0; and (b) the divergence of the radial vector (i.e., the vector [x,y,z]) is 3.

However, when factoring back in the factor of (1/2) to obtain the final "answer", the "answer" obtained using this method is (3/2)c, rather than the "correct" answer of c.

Why is this?

Is it because you can't treat the "del" operator as though it's a "vector" in this case?

Or am I doing something else wrong?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Vector Potentials

Loading...

Similar Threads for Vector Potentials |
---|

I Vector spaces and subspaces |

A Angular Moment Operator Vector Identity Question |

I How does the Vector Laplacian come about? |

I A question about Vector Analysis problems |

I Question about vector calculus |

**Physics Forums | Science Articles, Homework Help, Discussion**