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

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

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

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

OR

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

OR

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

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.

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

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

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

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)

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)

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

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

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

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

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 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 capacitor is connected across the terminals of a d.c. battery.

The energy stored on a capacitor is equal to the work done by the battery.

The work required to transport a small amount of charge (dQ) from the negative to positive plates of a capacitor is equal to V dQ, where V represents the voltage across the capacitor.

dU = V dQ

= `Q/C dQ`

∴ Energy stored (U) = ∫V dQ

= `1/C int Q dQ`

= `1/2 Q^2/C`

= `1/2 CV^2`    ...(i)

Energy density is defined as the total energy per unit volume of the capacitor.

For a parallel plate capacitor,

C = `(A epsilon_0)/d`

Putting in eqn. (i),

U = `1/2 (A epsilon_0)/d V^2`

= `epsilon_0/2 Ad(V/d)^2`

= `epsilon_0/2 Ad E^2`    ...[Putting `V/d` = E]

A × d = Volume of space between plates

So, energy is stored per unit volume.

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

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 = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]

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

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

C = Q/V

C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]

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

\[\frac {dE}{dt}\] ≠ 0 ⇒ i_d​ exists

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

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 flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε_0​.

Formula - Gauss's Law:

Key Points of Gauss's Law:

• Applicable to any closed surface of regular or irregular shape.
• Only the enclosed charge contributes to the electric flux.
• The electric field at a point depends on the total charge distribution, both inside and outside the Gaussian surface.

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.

1. Infinite Line Charge
\[E=\frac{\lambda}{2\pi\varepsilon_0r}\]

2. Infinite Plane Sheet
\[E=\frac{\sigma}{2\varepsilon_0}\]

3. Charged Conducting Plate
\[E=\frac{\sigma}{\varepsilon_0}\]

4. Spherical Shell

Inside (r < R):
E = 0

Outside (r > R):
\[E=\frac{Q}{4\pi\varepsilon_0r^2}\]

On surface (r = R):
\[E=\frac{Q}{4\pi\varepsilon_0R^2}\]

5. Charged Solid Sphere (Non-conducting)

Inside:
\[E\propto r\quad\left(E=\frac{Qr}{4\pi\varepsilon_0R^3}\right)\]

Outside:
\[E=\frac{Q}{4\pi\varepsilon_0r^2}\]

• The electric field due to many charges is the force on a unit test charge at that point.
• The total electric field is the vector sum of fields due to each charge (superposition principle).
• The electric field depends on the positions of the charges and changes from point to point in space.

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