Electric Field due to Continuous Charge Distribution


Electric Field due to Continuous Charge Distribution



Several problems of electrostatics asked in JEE and other competitive examinations involve continuous charge distributions. For example, students are expected to know the following formulae:

1.    Electric field due to a uniformly charged rod
2.    Electric field due to a uniformly charged cylinder
3.    Electric field due to a uniformly charged sphere
4.    Electric field due to a uniformly charged spherical shell
5.    Electric field due to a uniformly charged ring


So, it becomes important to understand the concept of electric field due to a continuous charge distribution.
For calculating electric field due to a continuous charge distribution, we use the very basic law – Coulomb's law. Students should be comfortable with Coulomb's law before jumping on to anything else in electrostatics.

Coulomb's law gives the expression of on elctrostatic force on a charge due to other charge in it's vicinity. It forms the basis of electrostatics. Students should clearly understand this law and the situations under which it is applicable as well as situations under which it isn't applicable.

After mastering Coulomb's law, students may move to charge distributions which are discrete. Discrete charge distributions involve simple mathematical summations and the problems of discrete charge distribution are usually straightforward applications of principle of superposition.

Once students are comfortable with discrete charge distributions, they should move to continuous charge distributions. Continuous charge distributions usually involve a single body which is either uniformly charged, or the charge density across the body varies as a function of the location of the point. This function is given in the problem statement and what is required is the expression of electric field at some point outside/inside the charge distribution.
Now, 2 cases arise here:

1.    There is some symmetry in the charge distribution
2.    There isn't any symmetry

When there is a symmetry in the charge distribution, it is usually a big hint for Gauss's law. Gauss's law exploits symmetry and helps us avoid cumbersome calculations to calculate electric field due to continuous charge distributions. It is widely asked in almost all competitive examinations. Students should be familiar with the application of Gauss's law to most common cases – line charge, cylinder, sphere, spherical shell, etc

When there isn't a symmetry, things become a bit complicated and the only option left is to use the very basic Coulomb's law. Since this process is challenging, we are mentioning the steps below:

1.    Choose a point on the body. Use parametrization to specify the coordinates of this point. Choose coordiinate system wisely to expoit even little amount of symmetry
2.    Find out the amount of charge present in this differential element. The charge present is simply the product of the charge density and the size of the element.
3.    Use Coulomb's law to write down the electric field at the required point. Make sure to write the electric field as a vector quantity.
4.    Integrate the expression obtained in step 3 over the entire body. Use mathematical techniques to simplify the process of integration. The properties of definite integrals help a lot here.

At the end, to check if everything went correctly, do dimensional analysis. Make sure that the dimension of the result matches with the dimension of electric field.



We hope that this article helped you. You can find a lot of good practice problems on Continuous charge distribution in Resnick Halliday Walker by Wiley Publishers.




About the Author


Aman Goel IIT B2 Aman Goel is B.Tech, computer science and engineering undergraduate student at Indian Institute of Technology, Bombay. Born in a business oriented family of Kanpur he secured an All India Rank 33 in JEE advanced 2013 and also scored 323/360 in JEE main 2013. He has also cleared Indian National Physics and Chemistry Olympiads, and KVPY. In his free time, he likes to write articles related to JEE preparation. He loves speedcubing (the art of speed solving a Rubik's cube) and also loves to play computer/mobile games.