The Physical Significance of the Second Term in Kirchhoff's Current Law

[Apteronotus]

Hi Everyone,

I'm told that the following formula represents Kirchhoff's current law
$$g_1 E_{1 n}+\varepsilon_1 \frac{\partial E_{1 n}}{\partial t}=g_2 E_{2 n}+\varepsilon_2 \frac{\partial E_{2 n}}{\partial t}$$
where the first term on each side is Ohm's law and the conductive current
and the second term is the 'displacement current'

To give you the complete picture, we are looking at the boundary of two regions (1 and 2) having different dielectric properties.
$$E_{i n}$$ is the normal component of the electric field in region i
$$\varepsilon_i$$ the dielectric constant there, and
$$g_i$$ the conductivity

Could someone please shed some light on this. What exactly is the second term$$\varepsilon_i\frac{\partial E_{i n}}{\partial t}$$? What's its physical significance? Is it the buildup of charge on the boundary? If so why do we consider the normal component $$E_{i n}$$?

Please make me understand :(

Thanks

 

[Studiot]

This might help, look at post#10 in aprticular.

https://www.physicsforums.com/showthread.php?t=401967&highlight=capacitor+field

 

[Delta2]

How could u ever not know the famous displacement current ? :) James Clerk Maxwell introduced it around 1860, because he knew that currents produce magnetic field and was wondering what is happening to the magnetic field between the plates of a capacitor since the conducting current vanishes there.

The answer was brilliant: There is a time varying electric field between the plates of a capacitor(which correlates to the charge buildup in the plates) which time varying electric field produces its own magnetic field. Since it produces a magnetic field that time varying electric field can be considered some sort of current, a displacement current !.

We take the normal component of the electric field cause that is the one responsible for the flow of current through a surface normal to it. Hm more specifically, Kirchoff current law is a special case of Maxwell-Ampere Law, which is a surface integral law which takes into account the field flux through a closed surface. Computing field flux through a surface u always consider the normal component of the field to that surface

 

[Apteronotus]

So tell me if I'm reading this right. Supposing I have two dielectrics attached to one another. If I apply a current to one of them, two things could happen:

1. The current will pass through the dielectric ( this is the first terms: $$g_k E_{kn}$$ )
or
2. charge will build up on the boundary of the two dielectrics (this is the second term $$\varepsilon_k \frac{\partial E_{k n}}{\partial t}$$ )?

 

[dgOnPhys]

What exactly is the second term$$\varepsilon_i\frac{\partial E_{i n}}{\partial t}$$? What's its physical significance? Is it the buildup of charge on the boundary? If so why do we consider the normal component $$E_{i n}$$?

$$g_1 E_{1 n}+\varepsilon_1 \frac{\partial E_{1 n}}{\partial t}=g_2 E_{2 n}+\varepsilon_2 \frac{\partial E_{2 n}}{\partial t}$$
is basically a (free) charge continuity equation on the boundary between regions
$$j_{2 n}-j_{1 n}+\frac{\partial (D_{2 n}-D_{1 n})}{\partial t}=0$$
$$j_{2 n}-j_{1 n}+\frac{\partial \sigma}{\partial t}=0$$
The variation of the current across the boundary takes the role of divergence.

The separate displacement currents are basically the contributions to charge variation on the interface from each side.