AwesomeTrains
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1. The problem statement, all variables and given/known data
Hello PF,
I could use some help calculating the electric field of a static quadrupole with side lengths a. The four charges are each placed in the corners of a square. (See attached picture).
2. Relevant equations
We've been given these as a help:
[itex]\Phi_{Q}(\vec{r})=\vec{a}\cdot\nabla\Phi_{D}+O(\left\vec{a}^{3}\right)[/itex] Were [itex]O(\left\vec{a}^{3}\right)[/itex] can be left out.
And [itex]\Phi_{D}(\vec{r})=\frac{1}{4\pi\epsilon_{0}}\frac{\vec{p}\cdot\vec{r}}{r^{3}}[/itex]
Were [itex]\vec{p}=q\cdot\vec{d}[/itex] is the dipole moment, [itex]q[/itex] is the charge and [itex]\vec{d}[/itex] is the distance vector.
3. The attempt at a solution
[itex]\Phi_{Q}[/itex] is the potential of the quadrupole. Therefore I thought I could plug in [itex]\Phi_{D}[/itex] and then use Gauss's law [itex]\vec{E}=\nabla\Phi_{Q}[/itex]
But how do I do that? I have problems using nabla on that function, and [itex]\vec{d}[/itex], [itex]\vec{a}[/itex] confuses me. (Are they the same?)
Any hints are very appreciated :)
Hello PF,
I could use some help calculating the electric field of a static quadrupole with side lengths a. The four charges are each placed in the corners of a square. (See attached picture).
2. Relevant equations
We've been given these as a help:
[itex]\Phi_{Q}(\vec{r})=\vec{a}\cdot\nabla\Phi_{D}+O(\left\vec{a}^{3}\right)[/itex] Were [itex]O(\left\vec{a}^{3}\right)[/itex] can be left out.
And [itex]\Phi_{D}(\vec{r})=\frac{1}{4\pi\epsilon_{0}}\frac{\vec{p}\cdot\vec{r}}{r^{3}}[/itex]
Were [itex]\vec{p}=q\cdot\vec{d}[/itex] is the dipole moment, [itex]q[/itex] is the charge and [itex]\vec{d}[/itex] is the distance vector.
3. The attempt at a solution
[itex]\Phi_{Q}[/itex] is the potential of the quadrupole. Therefore I thought I could plug in [itex]\Phi_{D}[/itex] and then use Gauss's law [itex]\vec{E}=\nabla\Phi_{Q}[/itex]
But how do I do that? I have problems using nabla on that function, and [itex]\vec{d}[/itex], [itex]\vec{a}[/itex] confuses me. (Are they the same?)
Any hints are very appreciated :)
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