Electrostatics Study Notes Part 2 for Electrical Engineering

In this article, you will find the Study Notes on Electrostatics Part 2 which will cover the topics such as Introduction to Gauss’s law for a conductor and example on Gauss’s Law Applications and Divergence of the flux density, electric field and potential.

• The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.
• Total electric flux y through any closed surface is equal to the total charge enclosed by that surface.
• Gauss’s Law helps us understand the behaviour of electric fields inside the conductors.
• The Gauss law also helps us understand the distribution of electric charge placed on a conductor.
• The area integral of the electric field over any closed surface is equal to the net charge enclosed in the surface divided by the permittivity of space.
• Gauss’ law is a form of one of Maxwell’s equations.

 

(Maxwell’s first equation)

• Net flux through a surface is equal to the net charge enclosed by the volume occupied by the surface.

Where, r_V = Volume charge density

• Total charge enclosed:

 

• Gauss’s law is an alternative statement of Coulomb’s law. Proper application of the divergence theorem to Coulomb’s law results in Gauss’s law.
• Coulomb’s law is applicable in finding the electric field due to any charge configuration but Gauss’s law is applicable when charge distribution is symmetrical.

Example-1: Find the flux through a spherical Gaussian surface of radius a = 1 m surrounding a charge of 8.85 pC.  Answer: The flux thru the Gaussian surface is the charge located inside the surface.

Let E be a simple solid region and S is the boundary surface of E with positive orientation.

Let D be a vector field whose components have continuous first-order partial derivatives.

Then, the divergence of a vector field D is defined at any point as

(Gauss’s law in differential form)

(Gauss’s law in integral form)

Energy Associated with Charge Distribution: The total energy of the system becomes

• Where, n = Number of point charges, Q_i = Charge of each point charge, V_i = Potential at location Q_i due to all the other charges except that of charge Q_j itself.
• Due to continuous charge distribution, the energy density is obtained by

Continuity Equation of Current: According to this, the point form of the continuity equation is

• Boundary Conditions for Perfect Dielectric Materials

• The two dielectrics having permittivities ε₁ and ε₂.
• Here, the tangential component of the electric field is continuous E_t1 = E_t2
• The normal component of electric flux density is continuous D_N1 = D_N2
• D₂ and E₂ can be given by

• Electric flux density at point 2

• Electric field intensity at point 1

Electric Fields in Material Space: Materials for which conductivity (σ) is greater than 1 are insulators. Semiconductors conductivity lies in between the conductivity of conductors and insulators.

• The electric current in terms of current density

where J is current density:

 

• Convection currents do not involve conductors and do not obey the Ohm’s law, convection current density is J = ρ_Vu; where, u is velocity and ρ_v are volume charge density.
• Conduction currents involve conductors and obey Ohm’s law conduction current density is J = σ E; where, σ is conductivity and E is electric field intensity,
• For perfect conductor: 
• The resistance of a conductor is:

• Dipole moment p of the electric dipole can be given as p = Q d; where d is the distance vector –Q to +Q.
• Polarisation vector p is the net dipole moment per unit volume of the dielectric.

• If a dielectric has in-built dipole ten it is known as a polar dielectric.
• If in a dielectric dipole is resulted as the effect of the external electric field, the dielectric is known as a non-polar dielectric.
• Polarisation surface charge density:

• Polarisation volume charge density:

• Hence, the effect of dielectric on the electric field is to increase the flux density by an amount p.
• p is proportional to the applied electric field E

where,  = Electric susceptibility

All the Best.

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