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#3: Post edited by user avatar deleted user · 2022-01-02T15:28:45Z (9 months ago)
  • Let's start with displacement field equation
  • $$\vec D = \epsilon_0 \vec E + \vec P$$
  • We know that $$-\vec \nabla \cdot \vec P = \rho_b$$
  • Here $\rho_b$ is surface charge density.
  • $$\implies -\int_V (\vec \nabla \cdot \vec P)\mathrm d\tau=\int_V \rho_b \mathrm d\tau$$
  • $$\implies \vec P \cdot \hat n = \sigma_b$$
  • The OP found $\vec D = \sigma_f \cdot \hat n$
  • $\sigma_b$ is used for surface charge density while $\sigma_f$ is used for free surface charge density if you look at the displacement field equation then you will get
  • $$\implies\sigma_f = \epsilon_0 \vec E+\sigma_b$$
  • $$\implies\vec E = \frac{\sigma_f-\sigma_b}{\epsilon_0}$$
  • $$=\frac{\sigma}{\epsilon_0}$$
  • >The free charge density serves as a useful simplification in Gauss's law for electricity
  • > Take a stationary atom and apply an electric field externally to the electron then the free particles align themselves with the field and create an electric dipole $\vec p = q\vec d$ where d is distance between those free particles.
  • >If you take a look at electric field generally then you will get the same expression $\vec E = \frac{1}{4\pi\epsilon_0}\int\frac{\sigma}{r^2}da\\\\ \implies \vec E = \frac{\sigma}{\epsilon_0}$ so both equation agrees on each other.
  • >So polarization is **surface charge density** and electric displacement is **free surface charge density** and direction of polarization and electric displacement depends on electric field.
  • Simply, free particles being moved for electric and field and attraction and distraction of other particles
  • References :
  • 1 : https://en.wikipedia.org/wiki/Charge_density
  • 2 : https://www.youtube.com/watch?v=-NjJiQN_cyA
  • Let's start with displacement field equation
  • $$\vec D = \epsilon_0 \vec E + \vec P$$
  • We know that $$-\vec \nabla \cdot \vec P = \rho_b$$
  • Here $\rho_b$ is surface charge density.
  • $$\implies -\int_V (\vec \nabla \cdot \vec P)\mathrm d\tau=\int_V \rho_b \mathrm d\tau$$
  • $$\implies \vec P \cdot \hat n = \sigma_b$$
  • The OP found $\vec D = \sigma_f \cdot \hat n$
  • $\sigma_b$ is used for surface charge density while $\sigma_f$ is used for free surface charge density if you look at the displacement field equation then you will get
  • $$\implies\sigma_f = \epsilon_0 \vec E+\sigma_b$$
  • $$\implies\vec E = \frac{\sigma_f-\sigma_b}{\epsilon_0}$$
  • $$=\frac{\sigma}{\epsilon_0}$$
  • >The free charge density serves as a useful simplification in Gauss's law for electricity
  • > Take a stationary atom and apply an electric field externally to the electron then the free particles align themselves with the field and create an electric dipole $\vec p = q\vec d$ where d is distance between those free particles.
  • >If you take a look at electric field generally then you will get the same expression $\vec E = \frac{1}{4\pi\epsilon_0}\int\frac{\sigma}{r^2}da\\\\ \implies \vec E = \frac{\sigma}{\epsilon_0}$ so both equation agrees on each other.
  • >So polarization is **surface charge density** and electric displacement is **free surface charge density** and direction of polarization and electric displacement depends on electric field.
  • Simply, free particles being moved for electric and field and attraction and distraction of other particles
  • References :
  • 1 : https://en.wikipedia.org/wiki/Charge_density
  • 2 : [Electromagnetism: Displacement Field and Gauss's Law (YT video)](https://www.youtube.com/watch?v=-NjJiQN_cyA)
  • Further reading :
  • [What is electric displacement](https://qr.ae/pG62QB)
#2: Post edited by user avatar deleted user · 2022-01-01T11:59:08Z (9 months ago)
  • Let's start with displacement field equation
  • $$\vec D = \epsilon_0 \vec E + \vec P$$
  • We know that $$-\vec \nabla \cdot \vec P = \rho_b$$
  • Here $\rho_b$ is surface charge density.
  • $$\implies -\int_V (\vec \nabla \cdot \vec P)\mathrm d\tau=\int_V \rho_b \mathrm d\tau$$
  • $$\implies \vec P \cdot \hat n = \sigma_b$$
  • The OP found $\vec D = \sigma_f \cdot \hat n$
  • $\sigma_b$ is used for surface charge density while $\sigma_f$ is used for free surface charge density if you look at the displacement field equation then you will get
  • $$\implies\sigma_f = \epsilon_0 \vec E+\sigma_b$$
  • $$\implies\vec E = \frac{\sigma_f-\sigma_b}{\epsilon_0}$$
  • $$=\frac{\sigma}{\epsilon_0}$$
  • >The free charge density serves as a useful simplification in Gauss's law for electricity
  • > Take a stationary atom and apply an electric field externally to the electron then the free particles align themselves with the field and create an electric dipole $\vec p = q\vec d$ where d is distance between those free particles.
  • >If you take a look at electric field generally then you will get the same expression $\vec E = \frac{1}{4\pi\epsilon_0}\int\frac{\sigma}{r^2}da\\\\ \implies \vec E = \frac{\sigma}{\epsilon_0}$ so both equation agrees on each other.
  • >So polarization is **surface charge density** and electric displacement is **free surface charge density** and direction of polarization and electric displacement depends on electric field.
  • Simply, free particles being moved for electric and field and attraction and distraction of other particles
  • Let's start with displacement field equation
  • $$\vec D = \epsilon_0 \vec E + \vec P$$
  • We know that $$-\vec \nabla \cdot \vec P = \rho_b$$
  • Here $\rho_b$ is surface charge density.
  • $$\implies -\int_V (\vec \nabla \cdot \vec P)\mathrm d\tau=\int_V \rho_b \mathrm d\tau$$
  • $$\implies \vec P \cdot \hat n = \sigma_b$$
  • The OP found $\vec D = \sigma_f \cdot \hat n$
  • $\sigma_b$ is used for surface charge density while $\sigma_f$ is used for free surface charge density if you look at the displacement field equation then you will get
  • $$\implies\sigma_f = \epsilon_0 \vec E+\sigma_b$$
  • $$\implies\vec E = \frac{\sigma_f-\sigma_b}{\epsilon_0}$$
  • $$=\frac{\sigma}{\epsilon_0}$$
  • >The free charge density serves as a useful simplification in Gauss's law for electricity
  • > Take a stationary atom and apply an electric field externally to the electron then the free particles align themselves with the field and create an electric dipole $\vec p = q\vec d$ where d is distance between those free particles.
  • >If you take a look at electric field generally then you will get the same expression $\vec E = \frac{1}{4\pi\epsilon_0}\int\frac{\sigma}{r^2}da\\\\ \implies \vec E = \frac{\sigma}{\epsilon_0}$ so both equation agrees on each other.
  • >So polarization is **surface charge density** and electric displacement is **free surface charge density** and direction of polarization and electric displacement depends on electric field.
  • Simply, free particles being moved for electric and field and attraction and distraction of other particles
  • References :
  • 1 : https://en.wikipedia.org/wiki/Charge_density
  • 2 : https://www.youtube.com/watch?v=-NjJiQN_cyA
#1: Initial revision by user avatar deleted user · 2022-01-01T11:55:06Z (9 months ago)
Let's start with displacement field equation 
$$\vec D = \epsilon_0 \vec E + \vec P$$

We know that $$-\vec \nabla \cdot \vec P = \rho_b$$
Here $\rho_b$ is surface charge density.

$$\implies -\int_V (\vec \nabla \cdot \vec P)\mathrm d\tau=\int_V \rho_b \mathrm d\tau$$
$$\implies \vec P \cdot \hat n = \sigma_b$$
The OP found $\vec D = \sigma_f \cdot \hat n$

$\sigma_b$ is used for surface charge density while $\sigma_f$ is used for free surface charge density if you look at the displacement field equation then you will get

$$\implies\sigma_f = \epsilon_0 \vec E+\sigma_b$$
$$\implies\vec E = \frac{\sigma_f-\sigma_b}{\epsilon_0}$$
$$=\frac{\sigma}{\epsilon_0}$$

>The free charge density serves as a useful simplification in Gauss's law for electricity

> Take a stationary atom and apply an electric field externally to the electron then the free particles align themselves with the field and create an electric dipole $\vec p = q\vec d$ where d is distance between those free particles. 

>If you take a look at electric field generally then you will get the same expression $\vec E = \frac{1}{4\pi\epsilon_0}\int\frac{\sigma}{r^2}da\\\\ \implies \vec E = \frac{\sigma}{\epsilon_0}$ so both equation agrees on each other. 

>So polarization is **surface charge density** and electric displacement is **free surface charge density** and direction of polarization and electric displacement depends on electric field. 

Simply, free particles being moved for electric and field and attraction and distraction of other particles