MHB How do you find the direction of a displacement current?

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The displacement current, defined as i_d = ε₀ ∂Φ_e/∂t, has a direction despite being a scalar. To determine its direction, one can analyze the displacement current density, j_d, which incorporates the time derivatives of the electric field and polarization. By integrating j_d over a chosen surface, the displacement current can be expressed in terms of its components relative to the surface normals. Measuring i_d across three perpendicular surfaces provides the directional components of the displacement current. Identifying the surface with the highest i_d can also indicate the direction of the displacement current.
Ackbach
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We know that the so-called displacement current is defined as
$$i_d=\varepsilon_0 \, \frac{\partial\Phi_e}{\partial t}.$$
Like regular current which is the movement of charges, $i_d$ has a direction, even though it's technically a scalar. How do we find its direction?
 
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Ackbach said:
We know that the so-called displacement current is defined as
$$i_d=\varepsilon_0 \, \frac{\partial\Phi_e}{\partial t}.$$
Like regular current which is the movement of charges, $i_d$ has a direction, even though it's technically a scalar. How do we find its direction?

Hi Ackbach,

From wiki, the displacement current density is:
$$\boldsymbol{j}_d = \pd {\boldsymbol{D}} t = \varepsilon_0 \frac{\partial \boldsymbol{E}}{\partial t} + \frac{\partial \boldsymbol{P}}{\partial t}$$

If we pick some surface, through which we want to know the displacement current, this is:
$$i_d=\iint \boldsymbol{j}_d \cdot d\boldsymbol S
= \iint \Big(\varepsilon_0 \frac{\partial \boldsymbol{E}}{\partial t} + \frac{\partial \boldsymbol{P}}{\partial t}\Big) \cdot d\boldsymbol S
= \varepsilon_0 \iint\frac{\partial \boldsymbol{E}}{\partial t}\cdot d\boldsymbol S + \iint \frac{\partial \boldsymbol{P}}{\partial t} \cdot d\boldsymbol S
$$
For a fixed surface in a medium with constant polarization (such as vacuum), it simplifies to:
$$i_d = \varepsilon_0 \frac{\partial \Phi_e}{\partial t}$$

By doing this, we have "lost" the direction.
If we want to get the direction back, we can measure $i_d$ through 3 perpendicular surfaces of unit size.
The corresponding results are the components of the vector with respect to the normals of those surfaces.
Alternatively, we can search for the surface that has the greatest $i_d$. Its normal is the direction.

Btw, I think this topic belongs in Other Advanced Topics, so I've moved it there.
 
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