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## Main Question or Discussion Point

Hi, I have some confusion about the jump conditions for an electric field across an interface between two materials with different properties. In general, we have the two jump conditions across an interface:

Here, + and - subscripts denote the properties outside and inside the interface respectively.

If we define an electric potential V, then electric field

div(grad(V)) = ρ/ɛ

where ρ is the volume density of charge

Now, if we use the boundary conditions shown above, we will have two equations for V at the interface, which will overconstrain the system of equations. (We only need one boundary condition at the interface for a scalar quantity). I have seen that people normally use only the jump condition for the normal direction. Does it mean that if the jump condition in the normal direction for the potential V is satisfied, then the tangential boundary condition will be automatically fulfilled? Can anyone provide a logical explanation?

Thanks!

**n**.(ɛ**E**)_{+}**.(ɛ****- n****E**)_{-}**=**σ (Normal direction)**where σ is the surface charge density on the interface****;****n**x**E**_{+}**-****n**x**E**_{-}= 0 (Tangential direction)Here, + and - subscripts denote the properties outside and inside the interface respectively.

If we define an electric potential V, then electric field

**E**= grad(V) where V satisfies the equation:div(grad(V)) = ρ/ɛ

where ρ is the volume density of charge

Now, if we use the boundary conditions shown above, we will have two equations for V at the interface, which will overconstrain the system of equations. (We only need one boundary condition at the interface for a scalar quantity). I have seen that people normally use only the jump condition for the normal direction. Does it mean that if the jump condition in the normal direction for the potential V is satisfied, then the tangential boundary condition will be automatically fulfilled? Can anyone provide a logical explanation?

Thanks!

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