- 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 [45]
Definition: Electric Charge
The basic property of matter due to which it experiences electric force and shows attraction or repulsion, is called electric charge.
OR
The fundamental property of subatomic particles that gives rise to the phenomenon of experiencing force in the presence of electric and magnetic fields is called electric charge.
- Positive charge: Deficiency of electrons
- Negative charge: Excess of electrons
- SI unit: Coulomb (C)
- Dimension: [M0L0T1A1]
Definition: Test Charge
A unit positive charge used to test the strength of electric fields is called a test charge.
Definition: Point Charge
An electric charge which can be considered to exist at a single point is called a point charge.
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: Linear Charge Distribution
When charge is distributed along a line, the charge distribution is called linear charge distribution.
Definition: Surface Charge Distribution
When charge is distributed over a surface, the charge distribution is called surface charge distribution.
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: Volume Charge Distribution
When charge is distributed over the volume of an object, it is called volume charge distribution.
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: 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: Source Charge
The charge Q that produces the electric field is called the source charge.
Define electric field.
The region in which the charge experiences an electric force is the electric field around the 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).
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: Non-Uniform Electric Field
A field whose magnitude and direction are not the same at all points is called a non-uniform electric field.
Definition: Uniform Electric Field
A field whose magnitude and direction is the same at all points is called a uniform electric field.
Definition: Test Charge
The charge q that tests the effect of the source charge is called the test charge.
Definition: Electric Lines of Force
“An electric line of force is an imaginary smooth curve drawn in an electric field along which a free, isolated positive charge moves. The tangent drawn at any point on the electric line of force gives the direction of the force acting on a positive charge placed at that point.”
OR
An imaginary curve drawn in such a way that the tangent at any given point on this curve gives the direction of the electric field is called an electric line of force.
Definition: Equatorial Line
The line passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial line.
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: 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’.”
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.
Definition: Centre of Dipole
The midpoint of the line joining the two charges is called the centre of the dipole.
Definition: Electric Potential
Electric potential at a point is the work done in bringing a unit positive test charge from infinity to that point in an electric field.
The SI unit of electric potential is the volt.
Definition: Potential Difference
The difference in electrical potentials between two points is known as potential difference or voltage.
The unit of potential difference or potential is joule/coulomb, called a volt (V).
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 potential difference between those two points.
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.
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: Potential at a Point
The potential at a point is defined as the amount of work done per unit charge in bringing a positive test charge from infinity to that point.
Definition: Equipotential Surface
The surface at which electric potential is the same at each point is called an equipotential surface.
Definition: Potential Difference
The difference in electrical potentials between two points is known as potential difference or voltage.
The unit of potential difference or potential is joule/coulomb, called a volt (V).
Definition: Electric Potential
Electric potential at a point is the work done in bringing a unit positive test charge from infinity to that point in an electric field.
The SI unit of electric potential is the volt.
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.
Definition: Semiconductors
Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.
Definition: Conductors
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.
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: Capacitor
A system consisting of two conductors having equal and opposite charges separated by an insulator or dielectric is called a capacitor.
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: Capacity of Conductor
The ability of a conductor to store charge is called the capacity of conductor.
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 [22]
Formula: Linear Charge Density
λ = \[\frac {ΔQ}{Δl}\] C/m
where ΔQ is the charge distributed over a small length Δl of the wire.
Formula: Surface Charge Density
σ = \[\frac {ΔQ}{ΔS}\] C/m2
where ΔQ is the charge distributed over a small surface area ΔS.
Formula: Volume Charge Density
ρ = \[\frac {ΔQ}{ΔV}\] C/m3
where ΔQ is the charge distributed over a small volume ΔV of the material.
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: 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: Electric Field at a Point
E = \[\frac{1}{4\pi\varepsilon_{0}}\frac{q}{r^{2}}\] newton / coulomb
where \[\frac{1}{4\pi\varepsilon_{0}}\] = 9.0 × 109 newton meter2 / coulomb2.
Formula: Electric Dipole Moment
p = q × 2a
It is a vector quantity; its direction is from −q to +q.
Formula: Torque on a Dipole in a Uniform Electric Field
\[\vec τ\] = \[\vec p\] × \[\vec E\]
Magnitude: τ = pE sin θ
Formula: Potential Due to a Dipole
\[V=\frac{q(2a\cos\theta)}{4\pi\varepsilon_0(r^2-a^2\cos^2\theta)}\]
if r ≫ a:
\[V=\frac{p\cos\theta}{4\pi\varepsilon_0r^2}\]
Formula: Potential at a Point Due to a System of Charges
\[V=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_i}\]
Formula: Electric Potential at a Point
V = \[\frac {W}{Q}\]
or
W = QV
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: Potential Due to a Point Charge
\[V=\frac{Q}{4\pi\varepsilon_0r}\]
Potential due to System of Charges:
\[U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]
Formula: Potential Due to an Electric Dipole
\[V=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{p\cos\theta}{r^{2}}=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{\vec{p}\cdot\vec{r}}{r^{3}}(r>>a)\]
Formula: Potential at a Point Due to a System of Charges
\[V=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_i}\]
Formula: Potential Due to a Dipole
\[V=\frac{q(2a\cos\theta)}{4\pi\varepsilon_0(r^2-a^2\cos^2\theta)}\]
if r ≫ a:
\[V=\frac{p\cos\theta}{4\pi\varepsilon_0r^2}\]
Formula: Cylindrical Capacitor
C = \[\frac {2πkε₀ l}{2.303 log(b/a)}\]
Formula: Spherical Capacitor
C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]
Formula: Basic Capacitance
C = Q/V
Formula: Energy Stored / Work Done in a Capacitor
W = \[\frac {1}{2}\]qV
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
Statement
Coulomb’s law states that the electrostatic force 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 line joining the two charges and is repulsive for like charges and attractive for unlike charges.
Explanation/Mathematical Form
Let two point charges q1 and q2 be placed at a distance r apart in vacuum (or air).
According to Coulomb’s law:
F ∝ q1q2
Combining the above relations:
F = k\[\frac {q_1q_1}{r^2}\]
where
F = electrostatic force between the charges,
r = distance between the charges,
k = proportionality constant.
In vacuum (or air),
k = 9.0 × 109 N m2C−2
Hence,
F = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\]
where ε0 is the permittivity of free space, given by
ε0 = 8.85 × 10−12 C2N−1m−2
If the charges are placed in a dielectric medium of permittivity ε,
F = \[\frac{1}{4\pi\varepsilon}\frac{q_1q_2}{r^2}\]
and since ε = Kε0,
F = \[\frac{1}{4\pi K\varepsilon_0}\frac{q_1q_2}{r^2}\]
where K is the dielectric constant of the medium.
Conclusion
Coulomb’s law quantitatively describes the force of attraction or repulsion between two point charges.
The force:
- depends on the magnitudes of charges,
- varies inversely as the square of the distance,
- acts along the line joining the charges, and
- decreases in a dielectric medium by a factor equal to its dielectric constant.
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)`
Law: Gauss's Law
The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε0.
Formula - Gauss's Law:
\[\oint\vec{E}\cdot d\vec{S}=\frac{Q}{\varepsilon_0}\]
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.
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
Additivity of Charge
- Electric charge is additive — the total charge of a system is the algebraic sum of all charges in it.
- Example: A system with +5 C and −2 C has a net charge of +3 C.
- Electrostatic forces between two point charges obey Newton's Third Law — action and reaction are equal and opposite.
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: Properties of the Electric Lines of Force
- Electric field lines originate from positive charges and terminate on negative charges (or at infinity).
- The tangent to a field line at any point gives the direction of the electric field; in a uniform field, the lines are parallel and straight.
- No two electric field lines intersect, as this would imply more than one direction of the electric field at a point.
- Electric field lines do not pass through a conductor, showing that the electric field inside a conductor is zero.
- The density of field lines indicates field strength—closer lines represent a stronger field, while wider spacing represents a weaker field; the lines are continuous and imaginary, though the field is real.
Key points: Potential and Potential Difference
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 [25]
- Electric Charge
- Scalar Form of Coulomb’s Law
- Principle of Superposition
- 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
- Potential and Potential Difference
- Electric Potential Due to a Point Charge
- Potential Due to an Electric Dipole
- Equipotential Surfaces
- Electric Potential
- 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
