184_notes:dipole_sup

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184_notes:dipole_sup [2018/05/29 14:48] – [Dipole Superposition Example] curdemma184_notes:dipole_sup [2018/05/29 14:54] – curdemma
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 === Electric Field far away from a Dipole === === Electric Field far away from a Dipole ===
 [{{  184_notes:dipolefar.png?100|Point P is very far away from the dipole}}] [{{  184_notes:dipolefar.png?100|Point P is very far away from the dipole}}]
-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:
  
 $$\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{q}{(\sqrt{(d/2)^2+h^2})^3}\langle d, 0,0 \rangle $$ $$\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{q}{(\sqrt{(d/2)^2+h^2})^3}\langle d, 0,0 \rangle $$
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 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, and making an approximation that the separation between the charges is much smaller than the distance to the point of interest. This will give you an //approximate// electric field on the axis of: 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, and making an approximation that the separation between the charges is much smaller than the distance to the point of interest. This will give you an //approximate// electric field on the axis of:
 $$|\vec{E}_{axis}|=\frac{1}{4\pi\epsilon_0}\frac{2qd}{r^3}$$ $$|\vec{E}_{axis}|=\frac{1}{4\pi\epsilon_0}\frac{2qd}{r^3}$$
-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, $d$ is separation between the positive and negative charges, and $q$ is the magnitude of one of the charges.
  
 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.)
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 Once we have the electric field from the dipole, it becomes relatively simple to find the force from that dipole on any other charge that we would have near by simply by using the electric force relationship: $\vec{F}=q\vec{E}$.  Once we have the electric field from the dipole, it becomes relatively simple to find the force from that dipole on any other charge that we would have near by simply by using the electric force relationship: $\vec{F}=q\vec{E}$. 
  
-{{  184_notes:dipoleandcharge.png}}+[{{  184_notes:dipoleandcharge.png|Problem setup: $r$ vectors between the charges in the dipole and point P}}]
  
 For example, if we had a positive charge $q_3$ that was centered a distance h above our dipole (as in the first example), we could find the force on $q_3$ by using the electric field that we calculated before: For example, if we had a positive charge $q_3$ that was centered a distance h above our dipole (as in the first example), we could find the force on $q_3$ by using the electric field that we calculated before:
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 If $q_3$ happened to be negative, then the force would instead point to left (but we would not have to rewrite $\vec{E}_{net}$ because we didn't change anything about the dipole): If $q_3$ happened to be negative, then the force would instead point to left (but we would not have to rewrite $\vec{E}_{net}$ because we didn't change anything about the dipole):
 $$\vec{F}_{q_3}=-q_3\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{-q_3*q}{(\sqrt{(d/2)^2+h^2})^3}\langle d, 0,0 \rangle$$ $$\vec{F}_{q_3}=-q_3\vec{E}_{net}=\frac{1}{4\pi\epsilon_0}\frac{-q_3*q}{(\sqrt{(d/2)^2+h^2})^3}\langle d, 0,0 \rangle$$
-This is often why we care about the electric field - **it doesn't matter what change about $q_3$, the electric field from the dipole is going to stay the same.** This idea will save you time when doing your calculations, especially for calculations involving more than three charges. (Since we already have the electric field for when Point P is far away from the dipole and when it is on the axis of the dipole, we could also easily find the force on $q_3$ at those locations too.)+This is often why we care about the electric field - **it doesn't matter what changes about $q_3$, the electric field from the dipole is going to stay the same.** This idea will save you time when doing your calculations, especially for calculations involving more than three charges. (Since we already have the electric field for when Point P is far away from the dipole and when it is on the axis of the dipole, we could also easily find the force on $q_3$ at those locations too.)
  
 ==== Superposition of Potential and Energy ==== ==== Superposition of Potential and Energy ====
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 Once you have the total electric potential around the dipole, it is then very easy to find the electric potential energy between a third charge ($q_3$) and the dipole: Once you have the total electric potential around the dipole, it is then very easy to find the electric potential energy between a third charge ($q_3$) and the dipole:
 $$U=q_3 V_{tot}$$ $$U=q_3 V_{tot}$$
-Again, this is often why we care about the electric potential - **it doesn't matter what change about $q_3$, the electric potential from the dipole is going to stay the same.**+Again, this is often why we care about the electric potential - **it doesn't matter what changes about $q_3$, the electric potential from the dipole is going to stay the same.**
  • 184_notes/dipole_sup.txt
  • Last modified: 2020/08/17 17:29
  • by dmcpadden