Electromagnetic Fields - Electrostatics Complete Study Notes

In this article, you will find the Study Notes on Electrostatics which will cover the topics such as Force of interaction between two charged particles, Electric Field due to Infinite Line Charge, Electric Field due to Uniformly Charged Ring, Electric Field due to Infinite Sheet of Charge, Superposition Principle of Fields, Electric Flux, Electric Potential, Dipole Moment, Electric flux lines and Electric field due to Dipole etc

Coulomb’s Law

• Coulomb’s Law describes the electrostatic interaction between two charged particles.
• It can be derived by combining the equation for the electric field around a spherical charge.
• According to Coulomb's law, the force acting between two point charges is:
  • directly proportional to the magnitude of each charge,
  • inversely proportional to the square of the separation between their centres, and
  • directed along the separation vector connecting their centres.

• The force acting between two electric charges is radial, inverse-square, and proportional to the product of the charges.
• It is an inverse-square law, given by:

(OR)

F₁₂= (k_e Q₁ Q₂) / r²

• where Q₁ and Q₂ are the magnitudes of the two charges respectively and r is the distance between them, F₁₂ is the force on particle 1 from particle 2, k_e is called as Coulomb's constant (k_e = 8.99×109 N m² C^-2), and ε₀ is vacuum permittivity (8.85 × 10⁻¹² C²/Nm²).
• F₁₂ = –F₂₁ and |F₁₂| = |F₂₁|, it means that force acting on charge Q₂ due to Q₁ is always equal to the force acting on charge Q₁ due to Q₂ magnitude but opposite in direction.
• The signs of Q₁ and Q₂ must be taken into account means for the same polarity charges Q₁Q₂ > 0 and for opposite polarity charges Q₁Q₂ < 0.
• Coulomb’s force obeys the law of superposition.
• Note that when both particles have the same sign of charge then the force is in the same direction as the unit vector and the particle is repelled.
• The SI unit of electric charge is the coulomb (C)

Force per unit charge is called electric field intensity

 line charge : 

Where ρ_L is line charge density in C/m and r is radial distance.

 surface charge :

Where ρ_S is surface charge density in C/m2 and a_n is unit vector normal to the plane containing the sheet.

Electric Field due to Infinite Line Charge

Consider an infinitely long straight line carrying uniformly line charge having density ρ_L C/m.

The electric field intensity at point P.

Electric Field due to Uniformly Charged Ring

Consider a charged circular ring of radius r placed in XY plane with centre at the origin. Carrying a charge uniformly along its circumference. The charge density is ρ_L C/m.

Electric Field due to Infinite Sheet of Charge

Consider an infinite sheet of charge having uniform surface charge density ρ_s C/m², placed in XY plane. We want to find E at point P present at the z-axis.

Superposition Principle of Fields

The electric field of a point charge is a linear function of the value of the charge. The fields of more than one point charge are linearly superimposable by vector addition. This is the principle of superposition applied to the electric field and states that the total resultant field at a point is the vector sum of the individual components of the field at the point.

Electric Flux

It may be defined as 

Where D is electric flux density and it can be given as

D = ε₀ E = Electric flux density (C/m²)

 where, ε₀ = Permittivity of the vacuum = 8.854 × 10^–12 F/m

Electric Potential

The Work was done per unit charge or potential energy per unit charge is known as potential difference.

Electric field intensity can be given as the negative potential gradient of electric field i.e,

E = -∇V

The electric field is irrotational or conservative in nature i.e., ∇ × E = 0

Note: Electric dipole is formed when two points charged of equal magnitudes and opposite polarities are separated by a small distance.

 Dipole Moment

p = Q d Coulomb-meter

where, d is the distance vector from –Q to Q.

 Electric field Intensity at a Point due to a Dipole

An electric dipole or simply a dipole consists of two point charges of equal magnitude and opposite sign, separated by a very small distance.

Field due to Dipole

The field due to a dipole at a very large distance from the dipole.
                                              

where, p = Ql

and a_r, a_θ = Unit vectors

S.No.	Charge	Electric Field	Potential
1	One point charge (monopole)
2	Two-point charge (dipole)
3	Three-point charge (tripole)
4	Four-point charge (quadruple)

Electric Flux

The electric field at any distance r from a point charge in free space.

 Newton/Coulomb

With E as a vector in free space, ε₀E is designated by a symbol D; called electric flux density

D = ε₀ E

The integral of the normal component of the vector D over a surface is defined as the electric flux over the surface.

Electric Flux Lines

An electric flux line is an imaginary path or line which is drawn in such a way that its direction at any point is the same as the direction of the electric field at that point. The electric flux density (D) is always tangential to electric flux lines. Electric flux lines are also called electric lines of force.

Electric field emanates (or originates) from a positive and terminates (or ends) on a negative charge.

An equipotential surface is a surface on which potentially remains the same throughout the surface and there is no potential difference. The flux line or force line for an equipotential surface is known as an equipotent line on equipotential surfaces or equipotential lines, the potential difference between any points A and B is always zero.

Electrostatic Energy Density

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.
  
  

  Divergence of the Flux Density

  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)

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