Revision: 12th Std >> Electrostatics MAH-MHT CET (PCM/PCB) Electrostatics

The study of electricity/electric charges at rest is called electrostatics.

The closed surface over which the surface integral of the electric field intensity (i.e. total electric flux) is considered in Gauss' Law is called a Gaussian surface.

A Gaussian surface is an imaginary, closed mathematical surface chosen to apply Gauss's Law conveniently.

The work done against the electrostatic forces to achieve a certain configuration of charges in a given system is called electrostatic potential energy.

"Potential difference is the work done to move a unit charge from one point to another in an electric field."

OR

The difference in electric potential between two points B and A, given by ΔV = V_B − V_A = \[\frac {W_AB}{q_0}\]​​, is called potential difference.

The work done by an external force in bringing a unit positive charge from infinity to that point is called electric potential at that point.

The work done by an external agent in bringing a unit positive test charge slowly from infinity to a point in an electric field, against the electrostatic force, is called the electric potential at that point.

The surface at which electric potential is the same at each point is called an equipotential surface.

Those substances in which electric charge cannot flow are called ‘insulators' (or dielectrics). Glass, hard-rubber, plastics and dry wood are insulators. Insulators have practically no free electrons.

OR

The material in which electrons are tightly bound to the nucleus and thus not available for conductance is called an insulator.

OR

Substances which offer high resistance to the passage of electricity and do not allow electricity to pass through them easily, are called insulators.

Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.

Conductors are those through which electric charge can easily flow. Metals, human body, earth, mercury and electrolytes are conductors of electricity.

OR

The material through which electric charge can flow easily is called a conductor.

The molecule in which the centres of positive and negative charges coincide and which has no permanent dipole moment in its normal state is called a non-polar molecule. (e.g. O₂, H₂, N₂, CO₂, benzene, methane)

The molecule in which the centres of positive and negative charges are separated even when there is no external field, and which has a permanent dipole moment, is called a polar molecule. (e.g. HCl, H₂O, alcohol, NH₃)

Non-conducting substances which cannot transmit electric charge through them are called dielectrics.

A dielectric that has a permanent electric dipole moment even if the external electric field is absent is called a polar dielectric.

A dielectric in which every molecule has zero dipole moment in its normal state is called a non-polar dielectric.

Alignment of dipole moments (permanent or induced) in the direction of an applied electric field is called polarisation.

The maximum electric field that a dielectric medium can withstand without breakdown (of its insulating property) is called its dielectric strength.

The ability of a conductor to store charge is called the capacity of conductor.

The ratio of the charge Q given to one of the conductors of a capacitor to the potential difference V between the conductors is called its capacitance, given by C = Q/V.

A system consisting of two conductors having equal and opposite charges separated by an insulator or dielectric is called a capacitor.

The capacitance of a single capacitor that stores the same charge at the same voltage as the entire combination is called the equivalent capacitance of the combination.

The work done per unit charge in moving a charge from one plate of a capacitor to the other is called the potential difference between the plates.

A parallel plate capacitor in which a dielectric slab is inserted between the plates to increase its capacitance by reducing the electric field between the plates is called a capacitor with a dielectric medium.

The current that exists at any point in space where a time-varying electric field (E) exists, i.e., \[\frac {dE}{dt}\] ≠ 0, is called displacement current (iₐ).

A device used to develop very high potentials of the order of 10⁷ volts is called a Van de Graaff generator.

Potential difference (V) between two points = Work done (W)/Charge (Q)
V = \[\frac {W}{Q}\]

The SI unit of electric potential difference is volt (V)

1 volt = \[\frac{1\mathrm{~joule}}{1\mathrm{~coulomb}}\] = 1 J C^-1

U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]

\[V=\frac{Q}{4\pi\varepsilon_0r}\]

Potential due to System of Charges:

\[U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]

V = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r}\]

Varies on spherical shell carrying charge q and radius R:

• Inside shell (r < R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• On surface (r = R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• Outside shell (r > R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]

\[V(r)=\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]

• V(r) = electric potential at distance rr from the charge
• q = source charge
• ε₀ = permittivity of free space
• K = dielectric constant of medium
• Reference is taken such that V(∞) = 0.

E = \[\frac{1}{4\pi\varepsilon_{0}}\frac{p}{r^{3}}\]

In vector notation:
\[\overrightarrow{\mathbf{E}}=-\frac{1}{4\pi\varepsilon_{0}}\frac{\overrightarrow{\mathbf{p}}}{r^{3}}\]

For a system of n point charges q₁, q₂, q₃,…, qn, the total electric field at point P is:

E(r) = \[{\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_{iP}^2}\hat{\mathbf{r}}_{iP}}\]

Symbol Reference

When a charge q₀​ is moved from point A to point B on the same equipotential surface:

Since V_A = V_B​ on the surface:

Defined as dipole moment per unit volume:

\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]

C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]

C = \[\frac {2πkε₀ l}{2.303 log(b/a)}\]

C = Q/V

\[{C_P=C_1+C_2+C_3+\cdots}\]

For n identical capacitors of capacitance C each: C_P = nC

Physical Insight: Adding capacitors in parallel is like adding more storage tanks — the total storage capacity simply increases.

For two capacitors in series, the voltage across each is:

\[V_1=\frac{C_2}{C_1+C_2}\cdot V\]

\[V_2=\frac{C_1}{C_1+C_2}\cdot V\]

Physical Insight: The smaller the capacitor, the larger the voltage drop across it in a series combination. This is why identical series capacitors share voltage equally.

\[{\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots}\]

For n identical capacitors of capacitance C each: C_S = \[\frac {C}{n}\]

\[\frac {dE}{dt}\] ≠ 0 ⇒ i_d​ exists

W = \[\frac {1}{2}\]qV

OR

U = \[\frac {Q^2}{2C}\] ​= \[\frac {1​}{2}\]QV = \[\frac {1}{2}\]​CV²

SI unit: Joule (J)

Gauss' Law states that the net electric flux through any closed surface is equal to `1/epsilon_0` times the net electric charge within that closed surface.

`oint  vec" E".d vec" s" = (q_(enclosed))/epsilon_o`

In the diagram, we have taken a  cylindrical gaussian surface of radius = r and length = l.
The net charge enclosed inside the gaussian surface `q_(enclosed) = lambdal`
By symmetry, we can say that the Electric field will be in radially outward direction.

According to gauss' law,

`oint  vec"E".d  vec"s" = q_(enclosed)/epsilon_o`

`int_1 vec"E" .d  vec"s" + int_2  vec"E" .d  vec"s" + int_3  vec"E". d  vec"s" = (lambdal)/epsilon_o`

`int_1  vec"E". d  vec"s"  &  int_3  vec"E". d  vec"s"  "are zero", "Since"  vec"E"  "is perpendicular to"  d  vec"s"`

`int_2  vec"E" . d  vec"s" = (lambdal)/epsilon_o`

`"at"  2,  vec"E" and d  vec"s"  "are in the same direction, we can write"`

`E.2pirl = (lambdal)/epsilon_o`

`E = lambda/(2piepsilon_o r)`

"The total electric flux through any closed surface is equal to \[\frac {1}{ε_0}\] times the net charge enclosed by that surface."

Three Forms of the Law

1. Verbal Form:
The net outward electric flux through a closed surface equals the net enclosed charge divided by ε₀.

2. Algebraic Form:

Φ_E = \[\frac {Q_enc}{ε_0}\]

3. Integral Form:

\[\oint\vec{E}\cdot d\vec{S}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}\]

Variable Legend

"The electric field at any point due to a group of charges is the vector sum of the electric fields at that point due to each individual charge, calculated as if the other charges were not present."

• Each charge in the system contributes its own independent electric field at the point of interest.
• These individual fields are then added vectorially to give the total (resultant) field.

\[\oint_c\vec{B}\cdot d\vec{l}=\mu_0I_c+\varepsilon_0\mu_0\frac{d\Phi_E}{dt}\]

This equation states that not only the current but also the changing electric field produces a changing magnetic field. This equation is known as the Ampere-Maxwell Circuital Law.

Works on:

• Corona discharge
• Charge distribution on a hollow conductor (outer surface)
• A continuous supply of charge increases potential
• Can generate potentials of order 10⁷ volts.

• Applicable to any closed surface, regardless of shape or size — sphere, cube, irregular shape
• Only enclosed charges contribute to the net flux; external charges do not
• The electric field E at the Gaussian surface is due to all charges (inside and outside), but the net flux depends only on enclosed charge​
• Gauss's Law is valid for both stationary and moving charges​
• It is one of Maxwell's four equations of electromagnetism​
• Gauss's Law can be derived from Coulomb's Law for static charges, and vice versa — both are equivalent​
• If net enclosed charge = 0, net flux = 0 (but E ≠ 0 necessarily)

• Electric potential at a point is the work done per unit positive test charge in bringing it slowly from infinity to that point, against the electric field.
• For a point charge q in air/vacuum:
  V(r) = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\]
• In a medium of dielectric constant K:
  V(r) = \[\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]
• Positive charge produces positive potential; negative charge produces negative potential.
• Potential due to a point charge is spherically symmetric and depends only on distance r.
• Distance dependence:
  F ∝ 1/r², E ∝ 1/r², V ∝ 1/r.
• The potential at infinity is taken as zero; only potential differences are physically significant.
• The electrostatic field is conservative, so the work done in moving a charge between two points is path independent.

• The resultant field E is the vector sum of all individual fields.
• Each individual field E_i is calculated independently, as if no other charges exist.
• The unit vector \[\hat r_i\]P points from each charge q_i toward point P.
• The principle holds for any number of charges in any configuration.
• This is a direct application of the Superposition Principle to electric fields.

Capacitors in Series:

Equivalent capacitance: \[\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor

Capacitors in Parallel:

\[C_p=C_1+C_2+C_3+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor