- #1
dimensionless
- 462
- 1
I've got an equation (from Wikipedia)for the electrostatic potential of an electric dipole. It looks like this:
[tex]
\Phi (\mathbf{r}) = \frac {1} {4\pi\epsilon_0 r^2} (\mathbf{p}\cdot\hat{\mathbf{r}})
[/tex]
E is the electric field
r, r, r\hat are as above
p is the (vector) dipole moment
e0 is the primitivity of free space
To find the electric field I have to take the derivative as follows.
[tex]
\mathbf{E} = - \nabla \Phi
[/tex]
The derivative looks like this:
[tex]
\mathbf{E} = \frac {1} {4\pi\epsilon_0 r^3} \left(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\right)
[/tex]
I'm confused by the vector notation. Why do I have r\hat multiplied by r\hat? Why is the electric field simply not
[tex]
\Phi (\mathbf{r}) = \frac {1} {2\pi\epsilon_0 r^3} (\mathbf{p}\cdot\hat{\mathbf{r}})
[/tex]
[tex]
\Phi (\mathbf{r}) = \frac {1} {4\pi\epsilon_0 r^2} (\mathbf{p}\cdot\hat{\mathbf{r}})
[/tex]
E is the electric field
r, r, r\hat are as above
p is the (vector) dipole moment
e0 is the primitivity of free space
To find the electric field I have to take the derivative as follows.
[tex]
\mathbf{E} = - \nabla \Phi
[/tex]
The derivative looks like this:
[tex]
\mathbf{E} = \frac {1} {4\pi\epsilon_0 r^3} \left(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\right)
[/tex]
I'm confused by the vector notation. Why do I have r\hat multiplied by r\hat? Why is the electric field simply not
[tex]
\Phi (\mathbf{r}) = \frac {1} {2\pi\epsilon_0 r^3} (\mathbf{p}\cdot\hat{\mathbf{r}})
[/tex]