- 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 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: 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 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: Electric Potential Due to a Point Charge
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.
Definition: Equipotential Surface
The surface at which electric potential is the same at each point is called an equipotential surface.
OR
Any surface over which the electric potential is the same everywhere is called an equipotential surface.
Definition: Free Charges
Charges that are loosely bound to atoms/molecules and can move freely through a material when an external electric field is applied.
Example: conduction electrons in metals, ions in electrolytes.
Definition: Bound Charges
Charges that are tightly bound to atoms/molecules and cannot migrate through the material. They can only undergo small displacement or realignment, producing polarisation.
Example: electrons bound to atomic nuclei in a dielectric.
Definition: Potential Difference (V)
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.
Definition: Equivalent Capacitance
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.
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.
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 [24]
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 due to a Point Charge
\[V(r)=\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]
- 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.
Formula: Work Done on an Equipotential Surface
When a charge q0 is moved from point A to point B on the same equipotential surface:
W = q0(VA − VB)
Since VA = VB on the surface:
W = 0
Formula: Net Field Inside Dielectric
E = E0 − Einduced
Formula: Series Combination
\[{\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots}\]
For n identical capacitors of capacitance C each: CS = \[\frac {C}{n}\]
Formula: Parallel Combination
\[{C_P=C_1+C_2+C_3+\cdots}\]
For n identical capacitors of capacitance C each: CP = nC
Physical Insight: Adding capacitors in parallel is like adding more storage tanks — the total storage capacity simply increases.
Formula: Voltage Distribution (Special Formula)
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.
Formula: Displacement Current Condition
\[\frac {dE}{dt}\] ≠ 0 ⇒ id exists
Formula: Energy Stored / Work Done in a Capacitor
W = \[\frac {1}{2}\]qV
OR
U = \[\frac {Q^2}{2C}\] = \[\frac {1}{2}\]QV = \[\frac {1}{2}\]CV2
SI unit: Joule (J)
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: 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: 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
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: 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 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 = \[\frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r}\]
Theorems and Laws [3]
Ampere-Maxwell Circuital Law
\[\oint_c\vec{B}\cdot d\vec{l}=\mu_0I_c+\varepsilon_0\mu_0\frac{d\Phi_E}{dt}\]
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.
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: Electric Potential Due to a Point Charge
- 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.
Key Points: Free and Bound Charges
- Free charges → conductors → net internal field = 0.
- Bound charges → dielectrics → polarisation, not conduction.
- Bound charges never leave their parent atom/molecule.
- Polarisation charge is also called induced charge.
- Free charges are the basis of electric current; bound charges are the basis of dielectric behaviour.
Key Points: Combination of Capacitors
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
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.
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.
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 and Bound Charges
- 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
