Consider the task of calculating the electric field inside a material. How could you calculate the electric field there. The book Electricity and Magnetism by Purcell has nicely illustrated and derivate the way to do this. Consider a parallel plate capacitor and lets assume some material is placed inside the parallel plate capacitor as shown in the following figure.

To calculate the electric field inside a material, the trick it turns out is to calculate the potential at a point inside the material. Then using the fact that electric field is the gradient of potential we can know the electric field inside the material. Consider the following diagram again. In this diagram you can imagine that point A is any point within the material. The balled figure at the origin are molecules or atoms which should be taken as the parallel plate capacitor plate.

If we let ρ be the volume charge density at place distance r'(x',y',z') from the origin, R be the distance from origin, dv' be the small volume at distance r' then the potential at point A can be calculated using the following formula.

To calculate the electric field inside a material, the trick it turns out is to calculate the potential at a point inside the material. Then using the fact that electric field is the gradient of potential we can know the electric field inside the material. Consider the following diagram again. In this diagram you can imagine that point A is any point within the material. The balled figure at the origin are molecules or atoms which should be taken as the parallel plate capacitor plate.

If we let ρ be the volume charge density at place distance r'(x',y',z') from the origin, R be the distance from origin, dv' be the small volume at distance r' then the potential at point A can be calculated using the following formula.

Now using the gradient of this potential we can calculate the electric field at A.

In this steps we can calculate the electric field inside a material. The detailed derivation and further analysis is given in Purcell Electromagnetism pdf book.

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