UY1: Electric Field Of An Electric Dipole



Consider a dipole($q_{1}$ and $q_{2}$) with a separation distance of d, where $q_{1}$ is the positive charge and $q_{2}$ is the negative charge. A point b lies a distance x away from $q_{1}$ such that b, $q_{1}$ and $q_{2}$ form a straight line. Find the electric field at b and what happens if x becomes very large. Another point c lies a distance r from both $q_{1}$ and $q_{2}$, such that c, $q_{1}$ and $q_{2}$ forms a triangle with sides of length r, r and d. Find the electric field at c.

 

Using the principle of superposition of electric fields,

$$\begin{aligned} E_{b} &= \frac{q}{4 \pi \epsilon_{0} \left( x \, – \frac{d}{2} \right)^{2}} – \frac{q}{4 \pi \epsilon_{0} \left( x + \frac{d}{2} \right)^{2}} \\ &= \frac{q}{4 \pi \epsilon_{0} x^{2}} \left[ \left(1 \, – \frac{d}{2x} \right)^{-2} – \left(1 + \frac{d}{2x} \right)^{-2} \right] \end{aligned} $$

Taking x to be very large. Using binomial expansion where $\left( 1 – \frac{d}{2x} \right)^{-2} \approx 1 \, – (-2) \left( \frac{d}{2x} \right)$ and $\left(1 + \left(\frac{d}{2x} \right) \right)^{-2} \approx 1 + (-2) \left( \frac{d}{2x} \right)$,

$$\begin{aligned} E_{b} &= \frac{q}{4 \pi \epsilon x^{2}} \frac{2d}{x} \\ &= \frac{p}{2 \pi \epsilon_{0} x^{3}} \end{aligned}$$

Note that p is the electric dipole moment ($p = qd$) and $\vec{E}_{b}$ is in the $\vec{i}$ direction.

Done for $E_{b}$. For point c, notice the contribution to the electric field by the positive charge and negative charge. The vertical component of both contributions will cancel each other as the magnitude of the two charges and distance are the same. The electric field at point c will end up with only a horizontal component:

$$\vec{E}_{c} = \frac{2q \text{cos} \, \alpha}{4 \pi \epsilon_{0} r^{2}}  \, \hat{i}$$

 

Next: Electric Field Of A Ring Of Charge

Previous: Electric Field Lines

Back to Electromagnetism (UY1)

Back To University Year 1 Physics Notes



Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.