- Electric potential is a scalar quantity, and it is positive near a positive charge and negative near a negative charge.
- Electric potential is taken as zero at infinity because the force between charges becomes zero at infinite separation.
- The potential difference between two points is measured using a voltmeter, which is connected in parallel with the circuit, with its positive terminal at the higher-potential point.
Definitions [42]
Definition: Electric Charge
Electric charge is an intrinsic property of certain fundamental particles (like electrons and protons) that gives rise to electric and magnetic forces, causing them to experience a force when placed in an electromagnetic field.
OR
Electric charge is the physical property of matter that causes it to experience a force when placed in an electric field.
Key facts:
- SI Unit: Coulomb (C)
- Dimensional Formula: [A T] (Ampere × Time)
- Two types exist: positive and negative
Define a unit charge.
One coulomb is the amount of charge which, when placed at a distance of one metre from another charge of the same magnitude in vacuum, experiences a force of 9.0 × 109 N.
Definition: Principle of Superposition
The total electrostatic force on any charge in a system of multiple charges is the vector sum of the forces exerted on it individually by each of the other charges, with each pairwise force being unaffected by the presence of the remaining charges.
Definition: Volume Charge Density
The charge per unit volume in a region of space, is called volume charge density.
OR
When charge is distributed over the volume of an object, it is called volume charge distribution.
Definition: Surface Charge Distribution
When charge is distributed over a surface, the charge distribution is called surface charge distribution.
OR
The surface charge density σσ is the charge per unit area at any point on the surface.
Definition: Volume Charge Distribution
When charge is distributed over the volume of an object, it is called volume charge distribution.
OR
The volume charge density ρρ is the charge per unit volume at any point inside the body.
Definition: Continuous Charge Distribution
A charge distribution in which charge is treated as continuously spread over a line, surface, or volume (ignoring microscopic discreteness), is called continuous charge distribution.
Definition: Linear Charge Distribution
When charge is distributed along a line, the charge distribution is called a linear charge distribution.
OR
The linear charge density λ is the charge per unit length at any point on the line.
Definition: Surface Charge Density
The charge per unit area on a surface, is called surface charge density.
Definition: Linear Charge Density
The charge per unit length along a line (such as a wire), is called linear charge density.
OR
When charge is distributed along a line, the charge distribution is called linear charge distribution.
Definition: Electric Field
The space surrounding an electric charge q in which another charge q0 experiences a (electrostatic) force of attraction or repulsion, is called the electric field of the charge q.
OR
Electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point.
OR
The region surrounding an electric charge or a group of charges in which another charge experiences a force is called an electric field.
Definition: Source Charge
The charge Q that produces the electric field is called the source charge.
Definition: Electric Field Intensity (E)
The electric field intensity at any point is the strength of the electric field at that point.
- It is defined as the force experienced by a unit positive charge placed at that point.
\[\vec{E}=\frac{\vec{F}}{q_0}=\frac{kq}{r^2}\hat{r}=\frac{kq}{r^3}\vec{r}\]
- The SI unit of E is NC−1 (newtons per coulomb).
Define electric field.
The region in which the charge experiences an electric force is the electric field around the charge.
Definition: Test Charge
The charge q that tests the effect of the source charge is called the test charge.
Definition: Electric Lines of Force
Electric line of force is an imaginary curve drawn in an electric field such that the tangent at any point on it gives the direction of the electric field at that point.
- It represents the path along which a unit positive test charge would tend to move if free to do so.
- Lines are imaginary — they have no physical existence; they are a visualization tool only.
Definition: Centre of Dipole
The midpoint of the line joining the two charges is called the centre of the dipole.
Definition: Equatorial Line
The line passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial line.
OR
The plane passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial plane; the line along which the equatorial field is evaluated is the equatorial line (perpendicular bisector).
Definition: Electric Dipole Moment
Electric dipole moment \[\vec p\] is a vector quantity defined as the product of the magnitude of either charge and the separation between them.
Mathematical definition: \[\vec p\] = q × 2a
| Symbol | \[\vec p\] |
| Magnitude | p = q × 2a |
| Direction | From −q to +q (along the dipole axis) |
| SI Unit | Coulomb-metre (C·m) |
| Dimensional Formula | [M0L1T1A1] |
Definition: Electric Dipole
An electric dipole is a pair of equal and opposite point charges placed at a short distance apart.
OR
A system formed by two equal and opposite point charges placed at a small distance apart is called an electric dipole.
OR
A system of two equal and opposite point charges +q and −q separated by a small fixed distance 2a is called an electric dipole.
- The total charge of an electric dipole is zero
- Zero net charge does not mean zero electric field - the field exists because the charges are spatially separated
- The midpoint of the line joining −q and +q is called the centre of the dipole
Definition: Direction of Dipole Axis
“The line joining the two charges, pointing from the negative charge to the positive charge. This is known as the ‘direction of dipole axis’.”
OR
The line passing through both charges +q and −q is called the dipole axis (also called the axial line or axis of the dipole).
Define electric dipole moment.
The electric dipole moment is defined as the product of the magnitude of one of the charges and the distance between the two equal and opposite charges.
Definition: Gaussian Surface
A Gaussian surface is an imaginary, closed mathematical surface chosen to apply Gauss's Law conveniently.
Define the following:
Potential difference
Potential difference: The potential difference between two points may be defined as the work done in moving a unit positive charge from one point to the other.
Definition: Potential Difference
The potential difference (p.d.) between two points is equal to the work done per unit charge in moving a positive test charge from one point to the other.
OR
The work done per unit positive charge in moving a charge from one point to another in an electric field is called the potential difference between those two points.
OR
The potential difference between any two points in the circuit is the amount of energy needed to move one unit of electric charge from one point to the other.
Define Electric potential.
Electric potential is a measure of work done on the unit's positive charge to bring it to that point against all electrical forces. It is represented as ‘V’.
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: Insulators
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.
Definition: Conductors
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.
Definition: Semiconductors
Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.
Definition: Capacity of Conductor
The ability of a conductor to store charge is called the capacity of conductor.
Definition: Capacitor
A system consisting of two conductors having equal and opposite charges separated by an insulator or dielectric is called a capacitor.
Definition: Dielectric Strength
The maximum electric field that a dielectric medium can withstand without breakdown (of its insulating property) is called its dielectric strength.
Definition: Capacitance
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.
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: 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: Parallel Plate Capacitor with Dielectric Medium
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.
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: Electric Field
Electric Field \[\vec E\] at a point is the electrostatic force \[\vec F\] experienced by a vanishingly small positive test charge q0 placed at that point:
\[\vec E\] = \[\frac {\vec F}{q_0}\]
| Quantity | Symbol | SI Unit |
|---|---|---|
| Electric Field | \[\vec E\] | N C⁻¹ or V m⁻¹ |
| Force | \[\vec F\] | Newton (N) |
| Test Charge | q0 | Coulomb (C) |
Definition: Electromagnetic Field
A time-dependent combination of electric and magnetic fields that propagates through space and can transport energy is called an electromagnetic field.
Formulae [17]
Formula: Linear Charge Distribution
λ = \[\frac {ΔQ}{Δl}\] ⇒ dq = λdl
where ΔQ is the charge distributed over a small length Δl of the wire.
- SI Unit: C m⁻¹ (coulomb per metre)
- Nature: Scalar quantity
Formula: Surface Charge Distridution
σ = \[\frac {ΔQ}{ΔS}\] ⇒ dq = σ dS
where ΔQ is the charge distributed over a small surface area ΔS.
- SI Unit: C m⁻² (coulomb per square metre)
- Nature: Scalar quantity
Formula: Electric Field Due to a Continuous Charge Distribution
\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\sum\frac{\rho\Delta V}{r^{\prime2}}\hat{r}^{\prime}\]
Formula: Volume Charge Distribution
ρ = \[\frac {ΔQ}{ΔV}\] ⇒ dq = ρ dV
where ΔQ is the charge distributed over a small volume ΔV of the material.
- SI Unit: C m⁻³ (coulomb per cubic metre)
- Nature: Scalar quantity
Formula: Electric Field Due to a Point Charge
\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]
The dimensional formula of the electric field E is:
E = \[\frac {F}{q_0}\] = \[\frac{[LMT^{-2}]}{[IT]}=[MLT^{-3}I^{-1}]\]
Formula: Torque on a Dipole in a Uniform Electric Field
| Expression | Formula | Condition |
|---|---|---|
| Magnitude of Torque | τ = pE sin θ | θ = angle between \[\vec p\] and \[\vec E\] |
| Vector form | \[\vec τ\] = \[\vec p\] × \[\vec E\] | Cross product |
| Maximum Torque | τmax = pE | When θ = 90° |
| Minimum Torque | τmin = 0 | When θ = 0° or 180° |
Formula: Potential Difference between A and B
\[V_A-V_B=\frac{W}{Q}\]
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: Cylindrical Capacitor
C = \[\frac {2πkε₀ l}{2.303 log(b/a)}\]
Formula: Basic Capacitance
C = Q/V
Formula: Spherical Capacitor
C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]
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: 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: 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: 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 Due to a System of Charges
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
| Symbol | Meaning |
|---|---|
| E(r) | Resultant electric field at point P |
| qi | The i-th source charge in the system |
| riP | Distance from charge qi to point P |
| \[\hat r_i\]P | Unit vector directed from qi toward point P |
| ε0 | Permittivity of free space |
| \[\frac {1}{4πε_0}\] | Coulomb's constant ≈ 9 × 109 Nm²C⁻² |
Theorems and Laws [6]
Law: Coulomb's Law (Scalar Form)
Coulomb's Law: The electrostatic force of interaction between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force acts along the straight line joining the two charges.
Derivation (Step-by-Step)
Step 1: Force is directly proportional to the product of charge magnitudes:
F ∝ q1q2
Step 2: Force is inversely proportional to the square of the separation distance:
F ∝ \[\frac {1}{r^2}\]
Step 3: Combining Steps 1 and 2:
F ∝ \[\frac {q_1q_2}{r^2}\]
Step 4: Introducing the proportionality constant k:
F = k ⋅ \[\frac {q_1q_2}{r^2}\]
where k = 1/(4πε0) ≈ 9 × 109 N·m2/C2 (in vacuum/air)
Law: Principle of Superposition of Electric Forces
Statement
The principle of superposition states that the net electric force acting on a given charge due to a number of other charges is equal to the vector sum of the individual forces exerted on it by each charge taken separately, assuming the other charges are absent.
Explanation / Mathematical Form
Consider a system of nnn point charges q1,q2,q3,…,qn.
The force acting on charge q1 due to the other charges is:
where
\[\vec F_{12}\] is the force on q1 due to q2,
\[\vec F_{13}\] is the force due to q3, and so on.
According to Coulomb’s law, the force on q1 due to q2 is:
\[\vec F_{12}\] = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{12}\]
Similarly, forces due to other charges can be written, and their vector sum gives the resultant force on q1.
Thus, the force between any two charges is independent of the presence of other charges.
Conclusion
The principle of superposition shows that:
- Electric forces obey vector addition.
- Each pair of charges interacts independently.
- The net force on a charge in a multi-charge system is found by adding all individual Coulomb forces vectorially.
State Gauss’s law on electrostatics and drive expression for the electric field due to a long straight thin uniformly charged wire (linear charge density λ) at a point lying at a distance r from the wire.
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)`
Statement of Gauss's Law
"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:
ΦE = \[\frac {Q_enc}{ε_0}\]
3. Integral Form:
\[\oint\vec{E}\cdot d\vec{S}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}\]
Variable Legend
| Symbol | Meaning | SI Unit |
|---|---|---|
| ∮ | Closed surface integral | — |
| E | Electric field at the surface | N C⁻¹ |
| dS | Area element vector (outward normal) | m² |
| Qenc | Net charge enclosed by the surface | Coulomb (C) |
| ε0 | Permittivity of free space = 8.85 × 10⁻¹² C² N⁻¹ m⁻² | C² N⁻¹ m⁻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: Principle of Superposition
"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.
Key Points
Conservation and Additivity of Charge
- Law of Conservation of Charge: Charge can neither be created nor destroyed; it can only be transferred from one body to another.
- Additivity: If a system has charges q1, q2, q3, ... qn, the total charge is:
Q = q1 + q2 + q3 + ... +qn
Analogy: Think of charge like money in a closed economy — it only moves between accounts (bodies); no new "charge currency" is printed or destroyed.
Key Points: Electric Field
- A charge creates an electric field around it, and the field exists even if the charge is removed because the space has already been modified.
- The electric field exists at every point in three-dimensional space and does not depend on the test charge used to measure it (if the test charge is very small).
- For a positive source charge, the electric field is directed radially outward, while for a negative source charge, it is directed radially inward.
- The strength of the electric field decreases as the distance from the charge increases, and at equal distances from a point charge, the field has the same magnitude.
- The force on a charge in an electric field is given by \[\vec F\](r) = q\[\vec E\](r), and the SI unit of electric field is N/C.
Key Points: Electric Field Intensity Due to a Point-Charge
- Electric field intensity: \[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{r}\].
- It is a vector quantity, directed outward for +q and inward for −q.
- Follows an inverse-square law with distance.
- Independent of the test charge used to measure it.
- For multiple charges, use vector superposition.
- In a medium, divide by dielectric constant K.
Key Points: Gauss's Law
- 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)
Key Points: Electric Potential: A Quantitative Approach
- Electric potential (V) = work done per unit positive test charge, brought from infinity to a point without acceleration: V = \[\frac {W}{q_0}\].
- SI unit: volt (V) = 1 J/C.
- Dimensional formula: [ML2T−3A−1]
- Positive charge flows from higher to lower potential — analogous to liquid flowing downhill or heat flowing from hot to cold.
- Infinity is the standard reference in theoretical electrostatics; Earth is the standard reference in practical circuits and grounding — each valid in its own context.
Key points: Potential and Potential Difference
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: Conductors and Insulators
- Conductors allow charge to flow easily; insulators resist it strongly.
- In metals, free electrons carry charge; in electrolytes, ions carry it.
- Charge spreads across a conductor's surface but stays localized on an insulator.
- Semiconductors lie between the two and are sensitive to heat/doping.
- Insulators can still polarize under an electric field without conducting current.
Key Points: Capacitors
- Capacitance depends on the geometry (shape, size, separation) of the conductors and on the dielectric between them.
- In a series, the charge on each capacitor is the same, but the voltage across each is different.
- A series combination divides high voltage — the capacitor with the smallest capacitance gets the largest P.D., and it cannot store much charge.
- In parallel, the voltage across each capacitor is the same, but the charge on each is different, and it handles only low voltage.
- A parallel combination is used when a large capacitance at low potential is needed, as it can store a large amount of charge.
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: Physical Significance of Electric Field
- \[\vec E\] = \[\vec F\]/q0 — force per unit positive test charge
- Static case → Coulomb's Law is sufficient; field is a descriptive tool
- Accelerated charges → field becomes a real physical entity (EM waves)
- Time delay = d/c — information travels at the speed of light, not instantaneously
- An electric field carries and transports energy
- Field exists independently of whether any test charge is present
- Gravity is negligible for charged particles in typical electric fields
Key Points: Electric Field Due to a System of Charges
- The resultant field E is the vector sum of all individual fields.
- Each individual field Ei is calculated independently, as if no other charges exist.
- The unit vector \[\hat r_i\]P points from each charge qi 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.
Concepts [24]
- Electric Charge
- Coulomb's Law (Scalar Form): Force Between Two Point-Charges
- Forces Between Multiple Charges: Superposition Principle
- Continuous Charge Distribution
- Electric Field
- Electric Field Intensity Due to a Point-Charge
- Electric Lines of Force
- Electric Dipole
- Dipole in a Uniform External Field
- Gauss’s Law
- Uniformly Charged Infinite Plane Sheet and Uniformly Charged Thin Spherical Shell (Field Inside and Outside)
- Electric Potential: A Quantitative Approach
- Potential Difference
- Electric Potential Due to a Point Charge
- Potential Due to an Electric Dipole
- Equipotential Surfaces
- Conductors and Insulators
- Capacitors and Capacitance
- Combination of Capacitors
- Capacitance of a Parallel Plate Capacitor with and Without Dielectric Medium Between the Plates
- Energy Stored in a Charged Capacitor
- Van De Graaff Generator
- Physical Significance of Electric Field
- Electric Field Due to a System of Charges
