top of page
Boundary condition for tangential component of E (electrostatics case)
Here, we derive the boundary condition for the tangential component of E in a more formal way than in Griffiths, which will be particularly useful in deriving similar conditions for the magnetostatics (Ch. 5.4.2) and electrodynamics cases (Ch. 7.3.6).
Consider a figure below showing a region around a boundary with surface charge density σ(r).
![boundary.png](https://static.wixstatic.com/media/fe3e97_bdca63f6cecf4bc8b1051fc9d88fbeee~mv2.png/v1/fill/w_396,h_195,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/boundary.png)
For a box whose top and bottom surfaces are above and below the boundary and whose area and height approach zero, we apply the following identity (Problem 1.61b in Griffiths) with v = E:
![integral identity.png](https://static.wixstatic.com/media/fe3e97_257b01ac360140a6b2a6ffa8da799913~mv2.png/v1/fill/w_350,h_72,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/integral%20identity.png)
For the left hand side,
![LHS.png](https://static.wixstatic.com/media/fe3e97_204fc2bb214347c29c806a5f2c9e01bc~mv2.png/v1/fill/w_440,h_75,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/LHS.png)
For the right hand side,
![RHS_1.png](https://static.wixstatic.com/media/fe3e97_c664d466b21c43d2917da1cbfa0ef690~mv2.png/v1/fill/w_575,h_75,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/RHS_1.png)
Equating the left and right sides,
![Eq_1.png](https://static.wixstatic.com/media/fe3e97_0ef7b314f6ba483e8709ec26256cf5dd~mv2.png/v1/crop/x_5,y_0,w_1475,h_86/fill/w_60,h_4,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/Eq_1.png)
![bac_cab.png](https://static.wixstatic.com/media/fe3e97_459fe12133cb4ab3bd353ea838b4ed44~mv2.png/v1/fill/w_57,h_3,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/bac_cab.png)
![bac_cap_2.png](https://static.wixstatic.com/media/fe3e97_c8068ec295b448379f0cceb9abb888b8~mv2.png/v1/fill/w_47,h_4,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/bac_cap_2.png)
![boundary_condition.png](https://static.wixstatic.com/media/fe3e97_9d46828a6d974f189726a234ce5e5807~mv2.png/v1/fill/w_77,h_5,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/boundary_condition.png)
bottom of page