Electrostatics: An electrostatics is a science related to the electric charge which is statics i.e., are at rest. Charge in terms of various charge distributions
where, ρL, ρS and ρV are line charge, surface charge and volume charge density respectively.
Charge Distribution: In electrostatics, we deal with point charges and different types of charge distributions like volume charge distribution, line charge distribution and surface charge distribution.
Volume Charge Distribution: It is defined as charge per unit volume.
where ΔQ is a small amount of charge in small volume ΔQ.
The total charge within a defined volume is obtained by taking the volume integral throughout the volume.
where, ρV = Volume charge density, and dV = Differential volume
Line Charge Distribution: In line charge distribution, while the charge is linearly distributed along the length of the line and it is defined by linear charge density. It is the charge per unit length.
where, ρL = Linear charge density, and dL = Differential length
Surface (Sheet) Charge Distribution: In surface charge distribution, the charge is uniformly distributed over the surface of a sheet and it is defined by surface charge density.
It is charge per unit surface area.
Here, ρS = Surface charge density, and dS = Differential area
Coulomb's Law of Force:
Experiment has shown that a force exists between any two bodies that have a net electric charge, that is an excess of positive over negative charge or vice versa. When these bodies are called electrostatic. Such forces were studied by coulomb who experimentally determined a law about them.
If two charges of magnitude Q1 and Q2 respectively are situated on bodies whose dimensions are small compared to the distance, R, between them:
- Is proportional to Q1 Q2
- Is inversely proportional to R2
- Acts along the line joining the bodies, and
- Is attractive or repulsive according to whether Q1 and Q2 are of unlike or like signs respectively. Thus, the force experienced by Q2 due to the field of Q1 can be written as
Note: The force is repulsive if Q1 and Q2 are of the same sign and attractive if they are of the opposite sign.
Electric Field Intensity: Force per unit charge is called electric field intensity (E).
where ρL is line charge density in C/m and r is radial distance.
where ρS surface charge density in C/m2 and an 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 ρL 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 ρL C/m.
Electric Field due to Infinite Sheet of Charge: Consider an infinite sheet of charge having uniform surface charge density ρS C/m2, placed in XY plane. We want to find E at point P present at Z-axis.
Gauss’s Law – Maxwell Equations
The total outward electric flux Ψ through any closed surface is equal to the total charge enclosed by the surface.
In equation from, gauss’s law is written as
Comparing the two volume integrals
which is the first of the four Maxwell’s equations to be derived.
It states that the volume charge density is the same as the divergence of the electric flux density.
Applications of Gauss’s Law
Let us now see how we many apply gauss’s law to find electric fields or electric flux density if the charge distributions is known. The solutions is easy if we choose surface known as gauss’s surface which satisfies the following conditions.
Electric Potential
The voltage defined across a circuit element (such as a resistor, a capacitor or an inductor) is related to the electric field within the element. Gwen that the electric field defines the force per unit charge acting on a positive lest charge, any attempt to move the test charge against the electric field requires that work be performed, The potential difference or voltage between two points In an electric field is defined as the work per unit charge performed when moving a positive lest charge from one point to the other
From Coulomb's law, the vector force on a Positive point charge in an electric field is given by 
The amount of work (J) performed in moving this point charge In the electric field is the product of the force (N) and the distance moved (m). When the positive point charge is moved against the force (against the electric field) the work done is positive. When the point charge Is moved in the direction of the force, the work done Is negative. If me point charge is moved in a direction perpendicular to the force, the amount of work done Is zero,
Work = Force component along path × Distance
The incremental amount of work (dW) done when moving the positive test charge a distance of dl along the path is
where is a vector defining the direction of the path. The minus sign in the previous equation is necessary to obtain the proper sign on the work done (positive when moving the test charge against the electric field). When the point charge is moved along a path from point A to B. the total amount of work performed (W) is found by integrating dW along the path.
The potential difference between A and B is then
Note: The potential difference is written as a line integral of the electric field he>d
The potential difference equation may be written as
where VA and VB are the absolute potentials at points A and B. respectively The absolute potential at a point is defined as the potential difference between the point and a reference point an infinite distance away.
Note: The definition of the potential difference in terms of the absolute potentials at the starting and ending points of the path shows that the potential difference between any two points is Independent of the path taken between the points (example multiple parallel paths in a circuit).
For a closed path (point A = pent B), the line Integral of the electric field yields the potential difference between a point and itself yielding a value of zero.
(example: Kirchhoff’s voltage law applied to a closed loop).
Vector fields having zero-valued closed path line integrals are designated as conservative fields All electrostatic fields are conservative fields.
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