- Equipotential surfaces for a point charge are concentric spheres, and for a line charge, they are cylindrical in shape.
- Electric field is always perpendicular (normal) to an equipotential surface at every point.
- No work is done in moving a charge along an equipotential surface, and such surfaces never intersect each other.
Definitions [28]
Definition: Gaussian Surface
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.
Definition: Electrostatics
The study of electricity/electric charges at rest is called electrostatics.
Definition: Electric Flux
The surface integral of the electric field intensity over a closed surface S is called the electric flux through that surface.
Definition: Surface Charge Density
The charge per unit area of a uniformly charged infinite plane sheet or spherical shell is called surface charge density (σ).
Definition: Radial Unit Vector
The unit vector directed along the perpendicular (radial) direction from the charge configuration is called the radial unit vector (\[\hat n\]).
Definition: Linear Charge Density
The charge per unit length of an infinitely long line charge is called linear charge density (λ).
Definition: Potential Difference
"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 = VB − VA = \[\frac {W_AB}{q_0}\], is called potential difference.
Definition: Electric Potential
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.
Definition: Electric Potential Energy
The work done against the electrostatic forces to achieve a certain configuration of charges in a given system is called electrostatic potential energy.
Definition: Equipotential Surface
The surface at which electric potential is the same at each point is called an equipotential surface.
Definition: Displacement Current
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).
Definition: Energy Stored in a Capacitor
The work done in the transfer of charge q between the two plates of a capacitor, which gets stored in the form of potential energy of the system, is called the energy stored in a capacitor.
Obtain the expression for the energy stored in a capacitor connected across a dc battery. Hence define energy density of the capacitor
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.
Definition: Van de Graaff Generator
A device used to develop very high potentials of the order of 107 volts is called a Van de Graaff generator.
Definition: One Farad
A capacitor has a capacitance of 1 farad if a charge of 1 coulomb produces a potential difference of 1 volt across it.
1 F = 1 C/V
Definition: Displacement Current
The current due to the time rate of change of electric field in a dielectric (or in space), even in the absence of free charge flow.
Definition: Energy Stored in a Capacitor
The work done in charging a capacitor is stored as electrostatic potential energy in the electric field between its plates.
Definition: Van de Graaff Generator
A Van de Graaff generator is a device that produces very high electric potentials (on the order of 107 volts) by accumulating charge on a hollow metallic conductor.
Definition: Potential Difference
Potential difference between two points is the work done per unit charge in moving a charge between them.
Definition: Electrostatic Energy of Point Charges
Electrostatic potential energy of a system of point charges is defined as the total amount of work done to assemble the system of charges by bringing them from infinity to their present locations.
Definition: Dielectric
A dielectric is an insulating material that can be polarised when placed in an external electric field.
Definition: Polarization
Polarization is the process in which positive and negative charges inside a dielectric are slightly displaced in opposite directions under the influence of an external electric field, producing a dipole moment.
Definition: Non-Polar Molecule
A non-polar molecule is a molecule in which the centre of positive charge coincides with the centre of negative charge, resulting in zero dipole moment in the normal state.
Definition: Non-Polar Dielectric
A non-polar dielectric is a dielectric material made up of non-polar molecules that do not possess permanent dipole moments.
Definition: Polar Molecule
A polar molecule is a molecule in which the centre of positive charge does not coincide with the centre of negative charge, resulting in a permanent dipole moment.
Definition: Polar Dielectric
A polar dielectric is a dielectric material made up of polar molecules having permanent dipole moments.
Definition: Equipotential Surfaces
An equipotential surface is a surface on which the electric potential is the same at every point.
No work is done in moving a charge along an equipotential surface.
Definition: Capacitance
Capacitance is defined as the ratio of charge to potential difference.
C = \[\frac {Q}{V}\]
Formulae [22]
Formula: Electric Field due to Infinitely Long Line Charge
\[\vec{E}=\frac{\lambda}{2\pi\varepsilon_0r}\hat{n}=\frac{2k\lambda}{r}\hat{n}\]
- λ = linear charge density
- \[\hat n\] = radial unit vector
- r = perpendicular distance from the line charge
- Variation: E ∝ \[\frac {1}{r}\] (decreases as rr increases)
Formula: Electric Field due to Uniformly Charged Infinite Plane Sheet
\[\vec{E}=\frac{\sigma}{2\varepsilon_0}\hat{n}\]
For a sheet of uniform thickness (both surfaces contribute):
E = \[\frac{\sigma}{\varepsilon_{0}}\]
- σ = surface charge density
- \[\hat n\] = unit vector normal to the plane, going away from it
- E is directed away from plate if σ is positive; towards plate if σ is negative
- Variation: E is constant (independent of distance)
Formula: Potential Difference
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
Formula: Electric Potential Energy of Two Point Charges
U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]
Formula: Electric Potential due to a Point Charge
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 = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]
Formula: Displacement Current Condition
\[\frac {dE}{dt}\] ≠ 0 ⇒ id exists
Formula: Energy Stored / Work Done in a Capacitor
W = \[\frac {1}{2}\]qV
Formula: Capacitance with Partial Dielectric Filling
\[C=\frac{\varepsilon_0A}{d-t+\frac{t}{k}}\]
Formula: Displacement Curren
In a Dielectric:
\[i_d=Ak\varepsilon_0\frac{dE}{dt}\]
In Vacuum/Air:
\[i_d=A\varepsilon_0\frac{dE}{dt}\]
Formula: Electrostatic Energy Stored in a Capacitor
U = \[\frac {Q^2}{2C}\]
Using Q = CV:
U = \[\frac {1}{2}\]CV2
U = \[\frac {1}{2}\]QV
Formula: Potential Energy
\[U\left(r\right)=\left(\frac{1}{4\pi\epsilon_{0}}\right)\left(\frac{q_{1}q_{2}}{r}\right)\]
SI unit = joule (J)
l eV = 1.6 × 10-19 J
1 meV = 1.6 × 10-22 J
1 kev = 1.6 × 10-16 J
Formula: Electric Potential due to a System of Charges
V = \[\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_i}\]
For continuous distribution:
V = \[\frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r}\]
Formula: Potential Difference
\[V_2-V_1=\frac{U_2-U_1}{q}=\frac{W}{q}\]
Formula: Electric Potential due to a Point Charge
v = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\]
Potential Energy of Two Point Charges:
U = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}\]
Formula: Electric Potential due to an Electric Dipole
General Expression:
V = \[\frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}\]
Vector Form of Dipole Potential:
V = \[\frac{1}{4\pi\varepsilon_0}\frac{\vec{p}\cdot\vec{r}}{r^3}\]
Formula: Electric Field due to an Infinite Plane Sheet
\[E=\frac{\sigma}{2\varepsilon_0}\]
Where:
- σ = surface charge density
- ε0 = permittivity of free space
Formula: Electric Field of a Charged Spherical Shell
\[E=\frac{\sigma R^2}{\varepsilon_0r^2}\]
Case (i): Electric Field on the Surface of the Shell
\[E=\frac{q}{4\pi\varepsilon_0R^2}=\frac{\sigma}{\varepsilon_0}\]
Case (ii): Electric Field Inside a Uniformly Charged Spherical Shell
E = 0
Formula: Electric Field of a Charged Wire
\[E=\frac{\lambda}{2\pi\varepsilon_0r}\]
Where:
- λ = linear charge density
- r = distance from the wire
- ε0 = permittivity of free space
Formula: Potential Energy of a Dipole in Uniform Field
Formula: Electric Field–Potential Relation
E = -\[\frac {dV}{dx}\]
Formula: Capacitance of a Parallel Plate Capacitor
C = \[=\frac{Q}{V}=\frac{Q}{\left(\frac{Qd}{A\varepsilon_{0}}\right)}=\frac{A\varepsilon_{0}}{d}\]
Formula: Electric Field in a Dielectric-Filled Capacitor
E = \[\frac {Q}{Aε_{0}k}\] or Q = Akε0E
Theorems and Laws [2]
Law: Van de Graaff Generator
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.
Law: Gauss' Law
The total electric flux through a closed surface is equal to \[\frac {1}{ε_0}\] times the total charge enclosed within the surface.
Mathematical Form:
It is one of Maxwell’s equations and is widely used to calculate electric fields for symmetrical charge distributions (spherical, cylindrical, planar symmetry).
Key Points
Key Points: Von de Graaff Generator
- It produces very high voltage (about 107 V) by collecting charge on a hollow metal dome.
- It works on corona discharge and the property that the charge stays on the outer surface of a conductor.
- A moving insulating belt carries charge to the dome, thereby continuously increasing its potential.
- It is used to accelerate charged particles for nuclear experiments and other applications.
Key Points: Potential Energy of Charges and Dipoles
- For two charges, only the second charge requires work to assemble the system.
- For many charges, total energy is the sum of all pairwise interaction energies.
- In an external field, a charge has potential energy depending on its position.
- For charges in an external field, the total energy includes mutual energy and external-field energy.
- A dipole in a uniform field has minimum energy when aligned with the field and maximum energy when opposite to it.
Key Points: Conductors, Insulators and Charges
- Conductors have free electrons; insulators do not.
- Inside a conductor, the electric field is zero, and the potential is constant.
- An excess charge on a conductor remains on its surface.
- The electric field outside a conductor is perpendicular to the surface.
- Free charges can move; bound charges remain fixed to atoms.
Key Points: Polarization of Polar Dielectrics
- In a non-polar dielectric, an external electric field induces dipoles by slightly shifting the charges.
- In a polar dielectric, permanent dipoles align with the applied electric field.
- Polarisation is the dipole moment per unit volume and increases with the applied field.
- Polarisation produces induced surface charges that create an opposing internal field.
- The net electric field inside a dielectric is reduced, and very strong fields can cause dielectric breakdown.
Key Points: Equipotential Surfaces
Key Points: Capacitors: Principle and Combinations
- A capacitor stores electric charge and electrical energy.
- Capacitance is given by C = \[\frac {Q}{V}\] and depends on plate size, distance, and dielectric.
- In series: the same charge; the voltage drops.
\[\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots\] - In parallel: the voltage is the same; the charge divides.
Ceq = C1 + C2 + ⋯ - A series is used for high voltage; a parallel is used for large capacitance.
Important Questions [6]
- A spherical metal ball of radius 15 cm carries a charge of 2μC. Calculate the electric field at a distance of 20 cm from the center of the sphere.
- State the formula giving the relation between electric field intensity and potential gradient.
- The work done in bringing a unit positive charge from infinity to a given point against the direction of electric field is known as ______.
- The angle at which maximum torque is exerted by the external uniform electric field on the electric dipole is ______.
- An electric dipole consists of two opposite charges each of magnitude 1 μC, separated by 2 cm. The dipole is placed in an external electric field of 105 N/C. Calculate the: maximum torque experienced
- Derive an expression for energy stored in a capacitor.
Concepts [13]
- Concept of Electrostatics
- Application of Gauss' Law
- Electric Potential and Potential Difference
- Electric Potential Due to a Point Charge
- Equipotential Surfaces
- Electrical Energy of Two Point Charges and of a Dipole in an Electrostatic Field
- Free Charges and Bound Charges Inside a Conductor
- Combination of Capacitors
- Displacement Current
- Energy Stored in a Charged Capacitor
- Van De Graaff Generator
- Uniformly Charged Infinite Plane Sheet and Uniformly Charged Thin Spherical Shell (Field Inside and Outside)
- Overview: Electrostatics
