What is this form of a vector field called, and how does one interpret it?

Question

I am working on an assignment and my professor has written a question regarding a vector field as:

$$\vec{F}(\vec{x})=\frac{\vec{x}}{|\vec{x}|^3}$$

But I have not seen this in our lectures and cannot find it used elsewhere. Is this telling me in shorthand to use,

$$\vec{F}(\vec{x}) = \langle \frac{x}{|x|^{3}},\frac{y}{|y|^3},\frac{z}{|z|^3}\rangle$$

as the vector field? If not, how should I interpret this so that I can do things like calculate the divergence and flux?

Thank you!

Answer 1

If you consider $\vec{x'} = \left<x,y,z\right>$ ($'$ is used to indicate difference),

then, $\dfrac{\vec{x'}}{|\vec{x'}|^3} = \dfrac{\left<x,y,z\right>}{(\sqrt{x^2+y^2+z^2})^3} = \dfrac{\left<x,y,z\right>}{r^3} = \left<\dfrac{x}{r^3},\dfrac{y}{r^3},\dfrac{z}{r^3}\right>$

where $r = \sqrt{x^2+y^2+z^2}$.

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In short, you shouldn't divide by the cube of magnitude of individual components, rather by the cube of magnitude of the whole vector.

Answer 2

This form of vector field is common for inverse-square field equations, such as electric and gravitational fields. For example, the force on a test charge $q$ by electric field $\vec{E}$ created by source charge $Q$ is: $$\vec{F}=\frac{kqQ}{|\vec{r}|^2}\hat r$$ (I used $\vec{r}$ instead of $\vec{x}$)

$\hat r$ is a unit vector, so its magnitude is $1$, it really is only there to specify a direction.

You can rewrite this using the definition of a unit vector:

$$\vec{F}=\frac{kqQ}{|\vec{r}|^2}\hat r=\frac{kqQ}{|\vec{r}|^2}\frac{\vec r}{|\vec r|}=\frac{kqQ\vec{r}}{|\vec{r}|^3}$$

Remember that the the magnitude of $\vec{r}$ has to use all three components, so $|\vec{r}|=\sqrt{x^2+y^2+z^2}$, and this goes into each component.