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

Ackbach
Gold Member
MHB
Messages
4,148
Reaction score
93
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?
 
Mathematics news on Phys.org
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.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top