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184_notes:dipole_sup [2018/05/15 14:49] – curdemma | 184_notes:dipole_sup [2018/05/29 14:51] – [Dipole Superposition Example] curdemma | ||
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=== Electric Field between a Dipole === | === Electric Field between a Dipole === | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
We will start by finding the net electric field at the location of Point P (shown in the figure to the right) using superposition. Here we have P positioned a height h above the two charges in the dipole and centered between the positive and negative charge horizontally. From the superposition principle, we know that the total electric field at Point P ($\vec{E}_{net}$) should be equal to the electric field from the positive charge at Point P ($\vec{E}_{+}$) plus the electric field from the negative charge at Point P ($\vec{E}_{-}$): | We will start by finding the net electric field at the location of Point P (shown in the figure to the right) using superposition. Here we have P positioned a height h above the two charges in the dipole and centered between the positive and negative charge horizontally. From the superposition principle, we know that the total electric field at Point P ($\vec{E}_{net}$) should be equal to the electric field from the positive charge at Point P ($\vec{E}_{+}$) plus the electric field from the negative charge at Point P ($\vec{E}_{-}$): | ||
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First, we will find the electric field from the positive charge, which is given by: | First, we will find the electric field from the positive charge, which is given by: | ||
$$ E_{+}=\frac{1}{4\pi\epsilon_0}\frac{q_{+}}{(r_{+ \rightarrow P})^3}\vec{r}_{+ \rightarrow P}$$ | $$ E_{+}=\frac{1}{4\pi\epsilon_0}\frac{q_{+}}{(r_{+ \rightarrow P})^3}\vec{r}_{+ \rightarrow P}$$ | ||
- | where $\vec{r}_{+ \rightarrow P}= \langle d/2, h,0 \rangle $ because it points from the positive charge to the location of Point P. In this equation, $r_{+ \rightarrow P}$ is the magnitude of $\vec{r}_{+ \rightarrow P}$ so | + | FIXME where $\vec{r}_{+ \rightarrow P}= \langle d/2, h,0 \rangle $ because it points from the positive charge to the location of Point P. In this equation, $r_{+ \rightarrow P}$ is the magnitude of $\vec{r}_{+ \rightarrow P}$ so |
$$r_{+ \rightarrow P}=\sqrt{(d/ | $$r_{+ \rightarrow P}=\sqrt{(d/ | ||
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=== Electric Field far away from a Dipole === | === Electric Field far away from a Dipole === | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
- | Since dipoles occur frequently in nature (we can model any atom as dipole), it is often useful to have simplified equation for the electric field of a dipole. We can start with the equation that we found for the electric field of the dipole above: | + | Since dipoles occur frequently in nature (we can model any atom as a dipole), it is often useful to have simplified equation for the electric field of a dipole. We can start with the equation that we found for the electric field of the dipole above: |
$$\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{q}{(\sqrt{(d/ | $$\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{q}{(\sqrt{(d/ | ||
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=== Electric Field on Axis of the Dipole === | === Electric Field on Axis of the Dipole === | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
- | We could also follow a similar process to find the electric field for points far away but on the same axis as the dipole. This would consist of finding the electric field from both the positive and negative charge, adding those fields together through superposition, | + | We could also follow a similar process to find the electric field for points far away but on the same axis as the dipole. This would consist of finding the electric field from both the positive and negative charge, adding those fields together through superposition, |
$$|\vec{E}_{axis}|=\frac{1}{4\pi\epsilon_0}\frac{2qd}{r^3}$$ | $$|\vec{E}_{axis}|=\frac{1}{4\pi\epsilon_0}\frac{2qd}{r^3}$$ | ||
- | where r is the distance from the middle of the dipole to the point of interest, d is separation between the positive and negative charges, and q is the magnitude of one of the charges. | + | where $r$ is the distance from the middle of the dipole to the point of interest, |
Note this is only the magnitude of the electric field. The direction depends on which side of the dipole you are considering. (If you are closer to the negative charge, the electric field will point toward the negative charge. If you are closer to the positive charge, the electric field will point away from the positive charge.) | Note this is only the magnitude of the electric field. The direction depends on which side of the dipole you are considering. (If you are closer to the negative charge, the electric field will point toward the negative charge. If you are closer to the positive charge, the electric field will point away from the positive charge.) |