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Mechanics
Gauss's Theorem: Flux, Divergence and Gauss's Law
Gauss's theorem relates the total flux of a vector field through a closed surface to the total source enclosed by that surface. It simplifies the evaluation of electric and gravitational fields for systems possessing high symmetry.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of electric field using Gauss's law for infinite line charges, infinite plane sheets, and spherical shells, along with basic flux concepts.
PGT extension
Differential form of Gauss's theorem (divergence theorem), gravitational analogues, Gauss's law for dielectric media, and field energy calculations.
Prerequisites
Vector calculus and divergence theorem
Solid angle subtended by a surface
Inverse-square central force fields
Statement of Gauss's Law in Electrostatics
The total electric flux through any closed hypothetical surface (Gaussian surface) is equal to 1/ε0 times the net charge enclosed by the surface.
Only charges enclosed within the Gaussian surface contribute to the net flux; charges outside contribute zero net flux.
The Gaussian surface must be real or imaginary, continuous, and closed, chosen cleverly to exploit the symmetry of the charge distribution.
Gravitational Analogue of Gauss's Law
The surface integral of gravitational field intensity over a closed surface equals -4πG times the total mass enclosed.
Expressed mathematically as ∫ g · d$A = -4\piGM_{enclosed}$, reflecting the attractive nature of gravitational forces.
Useful for calculating gravitational acceleration inside and outside solid spheres and spherical shells.
Differential Form of Gauss's Law
Relates the divergence of the field directly to the local source density at any point in space.
For electrostatics, ∇ · $E = \rho / \epsilon_{0}$, where ρ is the volume charge density.
For gravitation, ∇ · $g = -4\piG\rho_{m}$, where ρm is the mass density.
Symmetry Requirements for Application
Spherical symmetry allows the use of a concentric spherical Gaussian surface where the field magnitude E is constant and normal to the surface.
Cylindrical symmetry requires a coaxial cylindrical Gaussian surface for infinite lines or cylinders of charge.
Planar symmetry permits a pillbox Gaussian surface oriented perpendicular to an infinite sheet of charge.
Derivations
Derivation of Gauss's Law from Coulomb's Law
Conditions: Point charge in a vacuum
Consider a point charge q placed at the origin of a coordinate system.
The electric field at a distance r is given by Coulomb's law as $E = (1 / 4\pi\epsilon_{0})(q / r^{2})r̂$.
Calculate the electric flux through a concentric spherical surface of radius r: ΦE = ∫ E · dA.
Since E and dA are parallel everywhere on the sphere and E is constant in magnitude, ΦE = E × (4πr2).
Substitute E to obtain ΦE = (1 / 4πε0)(q / r2) × 4πr2 = q / ε0.
Result: Ph$i_E = q / \epsilon_0$
Differential Form of Gauss's Law via Divergence Theorem
Conditions: Continuous charge distribution and Gauss's divergence theorem
Start with the integral form of Gauss's law: ∫S E · d$A = Q_{enclosed} / \epsilon_{0}$.
Express the enclosed charge in terms of volume charge density ρ: Qenclosed = ∫V ρ dV.
Apply the divergence theorem to convert the surface integral into a volume integral: ∫S E · d$A = \int _{V} (\nabla \cdot E) dV$.
Equate the two volume integrals: ∫V (∇ · E) d$V = \int _{V} (\rho / \epsilon_{0}) dV$.
Since this holds for any arbitrary volume, the integrands must be equal pointwise.
Result: \nabla cdot $E = \frac{\rho}{\epsilon_0}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Integral Form of Electrostatic Gauss's Law
$$oint E \cdot dA = \frac{Q_{enclosed}}{\epsilon_0}$$
E = Electric field, dA = Area vector, $Q_enclosed = Net enclosed charge$, epsilo$n_0 = Permittivity of free space$ N m² / C
Any closed surface enclosing stationary charges
Differential Form of Electrostatic Gauss's Law
$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$
\nabla cdot E = Divergence of electric field, \rho = Volume charge density, epsilo$n_0 = Permittivity$ C / m³ (or V / m²)
Valid at every point in space where fields are differentiable
Gauss's Law in Gravitation (Integral Form)
$$oint g \cdot dA = -4pi G M_{enclosed}$$
g = Gravitational field, G = Universal gravitational constant, $M_enclosed = Enclosed mass$ m³ / (kg s²)
Closed surface in a gravitational field
Gravitational Field of an Infinite Solid Cylinder
$$g = \frac{2 G \lambda}{r} \text{ (for r geq R)}$$
lambda = Mass per unit length, r = Radial distance, R = Radius of cylinder m / s²
Outside an infinitely long solid cylinder
Electric Field due to Infinite Plane Sheet
$$E = \frac{\sigma}{2 \epsilon_0}$$
sigma = Surface charge density, epsilo$n_0 = Permittivity$ V / m
Near an infinite, uniformly charged flat sheet
Electric Field inside a Uniformly Charged Solid Sphere
$$E = \frac{q r}{4 \pi \epsilon_0 R^3}$$
q = Total charge, R = Radius of sphere, r = Distance from center (r < R) V / m
Internal points of a solid insulating sphere
Electric Field due to Thin Spherical Shell (Outside)
$$E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2}$$
q = Total charge on shell, r = Distance from center, R = Radius of shell V / m
For r > R
Electric Field due to Thin Spherical Shell (Inside)
$$E = 0$$
E = Electric field V / m
For r < R (inside a hollow charged conductor or shell)
Graphs and experiments
Electric Field vs. Distance for a Solid Insulating Sphere
Draw: A graph with E on the y-axis and r on the x-axis. For r < R, a straight line passing through the origin (E proportional to r). For r > R, a hyperbolic decay curve (E proportional to 1/r²).
Interpret: Shows continuous behavior at the boundary r = R, peaking at the surface and dropping off inversely with distance squared externally.
Exam use: Frequently asked in multiple-choice questions regarding discontinuity or continuity of field magnitude at boundaries.
Electric Field vs. Distance for a Charged Conducting Shell
Draw: A graph with E on the y-axis and r on the x-axis. E = 0 from r = 0 up to r = R. A sharp vertical discontinuity at r = R jumping to a maximum value, followed by an inverse-square decay for r > R.
Interpret: Demonstrates that the electric field inside a closed conductor is zero, with all excess charge residing on the outer surface.
Exam use: Tested via conceptual questions comparing insulating and conducting spherical distributions.
Verification of Inverse Square Law conceptually via Flux
Draw: Concentric spheres of radii r and 2r surrounding a point charge, showing identical total flux lines passing through both spherical areas.
Interpret: Proves that total flux is independent of the size of the enclosing surface, depending only on enclosed charge.
Exam use: Used to explain the solid angle approach and independence of surface geometry.
Solved numericals
TGT core
A point charge of 8.85 μC is placed at the center of a cube of side 0.1 m. What is the total electric flux passing through the cube?
Identify the total charge enclosed: $Q = 8.85 \muC = 8.85 \times 10^{-6} C$.
Recall Gauss's law: Total flux $\Phi = Q / \epsilon_{0}$.
A solid non-conducting sphere of radius R has a volume charge density that varies with distance r from the center as ρ(r) = ρ0(1 - r/R), for r ≤ R. Find the total charge of the sphere.
Set up the volume integral for total charge: $Q = \int _{0}^{R} \rho(r) (4\pir^{2}) dr$.
Substitute the given density function: $Q = 4\pi\rho_{0} \int _{0}^{R} (r^{2} - r^{3}/R) dr$.
Evaluate the integral: [r3/3 - r4/(4R)] evaluated from 0 to R.
Gauss's law is fundamentally a consequence of the inverse-square nature of forces (Coulomb's and Newton's laws); it fails for force fields that do not obey the inverse-square law.
While Gauss's law is always true for any closed surface, it is only useful for calculating fields when the physical system possesses high symmetry (spherical, cylindrical, or planar).
Common misconceptions
Common error: The term Qenclosed in Gauss's law includes charges located outside the Gaussian surface.
Correct: Only charges inside the closed Gaussian surface determine the net flux through that surface.
External charges contribute equally to entering and exiting flux lines, resulting in zero net flux contribution.
Common error: The electric field E in Gauss's law ∫ E · d$A = Q/\epsilon_{0}$ is produced only by the enclosed charges.
Correct: The field E is the total electric field at the surface due to all charges, both inside and outside.
Although external charges cancel out in the surface integral, they still contribute to the local value of the total electric field vector.
Common error: Gauss's law is valid only for symmetric charge distributions.
Correct: Gauss's law is universally valid for any distribution of charges, but analytical evaluation of E using it is only tractable under symmetry.
The law is a mathematical identity derived from Maxwell-Faraday equations; symmetry only makes the surface integral solvable.
Quick revision
Total electric flux through a closed surface depends solely on the net enclosed charge and is completely independent of the shape or size of the Gaussian surface.
Integral Form of Electrostatic Gauss's Law: oint E cdot d$A = \frac{Q_{enclosed}}{\epsilon_0}$
Differential Form of Electrostatic Gauss's Law: \nabla cdot $E = \frac{\rho}{\epsilon_0}$
Must remember
The total electric flux through any closed hypothetical surface is equal to $1/\epsilon_0$ times the net charge enclosed.
Only charges enclosed within the Gaussian surface contribute to the net electric flux; external charges contribute zero net flux.
The differential form of Gauss's law in electrostatics is given by $\nabla \cdot E = \rho / \epsilon_0$.
The gravitational analogue of Gauss's law is expressed as $\int g \cdot dA = -4\pi G M_{enclosed}$.
The electric field inside a hollow uniformly charged thin spherical shell is identically zero ($E = 0$).
For points outside a thin spherical shell, the electric field behaves as if the entire charge is concentrated at the center.
Gauss's law is universally valid for any distribution of charges, though analytical evaluation using symmetry is practical only for specific geometries.
Spherical symmetry allows the use of a concentric spherical Gaussian surface where field magnitude $E$ is constant and normal to the surface.
Glossary
Gaussian SurfaceA hypothetical closed surface chosen to calculate the electric or gravitational flux using symmetry properties.
Electric FluxThe measure of flow of the electric field through a given surface area, denoted by $\Phi_E = \oint E \cdot dA$.
Permittivity of Free SpaceA physical constant denoted by $\epsilon_0$ that measures the ability of a vacuum to permit electric field lines.
Volume Charge DensityCharge per unit volume, denoted by $\rho$, acting as the source term in the differential form of Gauss's law.
DivergenceA vector operator that measures the magnitude of a vector field's source or sink at a given point, represented as $\nabla \cdot$.
Electrostatic vs. Gravitational Gauss's Law
Property
Electrostatic Gauss's Law
Gravitational Gauss's Law
Integral Form
$\oint E \cdot dA = \frac{Q_{enclosed}}{\epsilon_0}$
$\oint g \cdot dA = -4\pi G M_{enclosed}$
Differential Form
$\nabla \cdot E = \frac{\rho}{\epsilon_0}$
$\nabla \cdot g = -4\pi G \rho_m$
Nature of Force
Attractive or repulsive
Always attractive
Coupling Constant
$1 / \epsilon_0$
$-4\pi G$
Practice Quiz
Choose an option to reveal the explanation.
1. What does the term $Q_{enclosed}$ represent in Gauss's law $\oint E \cdot dA = \frac{Q_{enclosed}}{\epsilon_0}$?
Explanation: By definition, only charges enclosed within the closed Gaussian surface contribute to the net flux and are included in $Q_{enclosed}$.
2. What is the electric field inside a thin spherical shell of radius $R$ carrying a uniform charge $q$?
Explanation: For any point inside a thin spherical shell ($r < R$), a concentric spherical Gaussian surface encloses zero net charge, making $E = 0$.
3. What is the differential form of Gauss's law in electrostatics?
Explanation: The differential form relates the divergence of the electric field directly to the local volume charge density via $\nabla \cdot E = \frac{\rho}{\epsilon_0}$.
4. What is the gravitational analogue of Gauss's law for a closed surface?
Explanation: The surface integral of the gravitational field intensity equals $-4\pi G M_{enclosed}$, reflecting the attractive nature of gravity.
5. Which type of symmetry requires a coaxial cylindrical Gaussian surface for applying Gauss's law?
Explanation: Infinite line charges or cylinders exhibit cylindrical symmetry, requiring a coaxial cylindrical Gaussian surface.
6. What is the electric field due to an infinite plane sheet of charge with surface charge density $\sigma$?
Explanation: Applying a pillbox Gaussian surface across an infinite plane sheet yields $E = \frac{\sigma}{2 \epsilon_0}$.
7. What is the electric field inside a uniformly charged solid insulating sphere of radius $R$ at a distance $r$ ($r < R$)?
Explanation: For internal points of a solid sphere, the enclosed charge is proportional to $r^3$, giving $E = \frac{q r}{4\pi \epsilon_0 R^3}$.
8. Do external charges contribute to the net electric flux through a closed Gaussian surface?
Explanation: External charges contribute equally to entering and exiting flux lines, resulting in a net flux contribution of zero.
9. What is the differential form of Gauss's law in gravitation?
Explanation: The gravitational divergence equation is $\nabla \cdot g = -4\pi G \rho_m$, where $\rho_m$ is the mass density.
10. Is Gauss's law valid only for highly symmetric charge distributions?
Explanation: Gauss's law is a universal mathematical identity derived from Maxwell's equations; symmetry is only required for easy analytical evaluation of $E$.
11. What is the unit of electric flux in SI units?
Explanation: Electric flux is calculated as $E \cdot A$, with units of electric field ($\text{N/C}$ or $\text{V/m}$) multiplied by area ($\text{m}^2$), yielding $\text{N m}^2 / \text{C}$.
12. What kind of Gaussian surface is suitable for planar symmetry?
Explanation: A pillbox Gaussian surface oriented perpendicular to an infinite sheet of charge effectively captures the uniform electric field flux.
13. What is the gravitational field outside an infinitely long solid cylinder of radius $R$ and mass per unit length $\lambda$ at a distance $r \ge R$?
Explanation: Using the gravitational analogue of Gauss's law for cylindrical symmetry, the field outside a solid cylinder is $g = \frac{2 G \lambda}{r}$.
14. What is the electric field outside a thin spherical shell of radius $R$ carrying total charge $q$ at a distance $r$ ($r > R$)?
Explanation: For points outside a thin spherical shell, the shell behaves as a point charge at its center, giving $E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2}$.
15. In the surface integral $\oint E \cdot dA$, what does $E$ represent?
Explanation: The field $E$ in Gauss's law is the total electric field at any point on the surface, produced by all charges present in the universe.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A point charge $q$ is placed at the center of a cube of side $L$. What is the electric flux passing through one face of the cube?
Explanation: The total flux through all 6 faces of the cube is $\frac{q}{\epsilon_0}$. By symmetry, the flux through each identical face is $\frac{q}{6 \epsilon_0}$.
2. If the electric field in a region is given by $E = 3x^2 \hat{i}$, what is the volume charge density at $x = 2\text{ m}$? Given $\epsilon_0$
Explanation: Using the differential form $\nabla \cdot E = \frac{\rho}{\epsilon_0}$, we find $\nabla \cdot E = \frac{\partial}{\partial x}(3x^2) = 6x$. At $x = 2$, the divergence is $12$, hence $\rho = 12 \epsilon_0$. Wait, differentiating $3x^2$ gives $6x$. At $x=2$, $6(2) = 12$. Option 1 is $6 \epsilon_0$ and option 4 is $12 \epsilon_0$. Let's check correct_index: option index 3 corresponds to $12 \epsilon_0$. Wait, let's adjust correct_index to 3 if option 4 is chosen, or change options. Let's make options: A) $3\epsilon_0$, B) $6\epsilon_0$, C) $12\epsilon_0$, D) $24\epsilon_0$. Then correct index is 2 ($12\epsilon_0$). Let's rewrite options.
3. A hollow conducting sphere of inner radius $a$ and outer radius $b$ carries a total charge $+Q$. If a point charge $-q$ is placed at the center of the cavity ($q < Q$), what is the charge on the inner surface?
Explanation: To maintain zero electric field inside the conducting material of the shell, a Gaussian surface inside the conductor must enclose zero net charge, inducing a charge of $+q$ on the inner surface.
4. The electric flux through a closed surface enclosing a dipole is:
Explanation: An electric dipole consists of equal and opposite charges ($+q$ and $-q$). The net enclosed charge is zero, so by Gauss's law, the net electric flux is zero.
5. What is the gravitational field intensity at a distance $r$ inside a uniform solid sphere of mass $M$ and radius $R$ ($r < R$)?
Explanation: Using Gauss's law for gravitation, the enclosed mass inside radius $r$ is $M_{enclosed} = M (r^3/R^3)$, leading to $g = \frac{G M r}{R^3}$.
6. If a charge $q$ is placed at one corner of a cube, what is the total electric flux passing through the entire cube?
Explanation: A corner of a cube is shared by 8 adjacent cubes in a symmetric lattice. Thus, only $\frac{1}{8}$th of the electric field lines pass through the given cube, making the flux $\frac{q}{8 \epsilon_0}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the electric field is given by $E = 2x \hat{i} + 3y^2 \hat{j}$, find the volume charge density at point $(1, 2)$.
2. A spherical volume contains a uniform charge density $\rho$. What is the electric field at a distance $r$ from the center inside the sphere ($r < R$)?
Explanation: Using Gauss's law: $4\pi r^2 E = \frac{1}{\epsilon_0} (\frac{4}{3}\pi r^3 \rho)$, which simplifies to $E = \frac{\rho r}{3 \epsilon_0}$.
3. Which of the following statements is correct regarding Gauss's law?
Explanation: Gauss's law is mathematically equivalent to Coulomb's law for static charges and can be derived directly from it.
4. What is the gravitational potential or field behavior inside a thin spherical mass shell?
Explanation: By the gravitational analogue of Gauss's law, no mass is enclosed inside a thin shell, hence the gravitational field is zero.
5. If the electric flux through a closed surface is zero, does it mean that the electric field is necessarily zero everywhere on the surface?
Explanation: Net flux being zero indicates zero enclosed charge, but individual components of the electric field can still be non-zero across parts of the surface.
6. An infinite line charge produces a field $E$ at a distance $r$ proportional to:
Explanation: Applying Gauss's law with a cylindrical Gaussian surface around an infinite line charge yields $E = \frac{\lambda}{2\pi \epsilon_0 r}$, which is proportional to $1/r$.
7. What happens to the total electric flux through a Gaussian surface if the volume enclosed by the surface is doubled while the enclosed charge remains the same?
Explanation: By Gauss's law, total flux depends solely on the net enclosed charge ($Q_{enclosed} / \epsilon_0$), not on the size or shape of the Gaussian surface.
8. In the gravitational differential form $\nabla \cdot g = -4\pi G \rho_m$, why is there a negative sign?
Explanation: The negative sign reflects the fact that gravitational fields point inward toward masses (attractive nature), unlike electric fields which can diverge outward.
9. A charge $q$ is situated at a distance $a/2$ directly above the center of a square of side $a$. What is the total electric flux through the square?
Explanation: Imagine the square as one face of a hypothetical cube of side $a$ with the charge at its center. The total flux through the cube is $\frac{q}{\epsilon_0}$, so the flux through one face is $\frac{q}{6 \epsilon_0}$.
10. Which physical law or theorem is fundamentally used to convert the surface integral of a vector field into a volume integral in the differential form of Gauss's law?
Explanation: The Divergence Theorem relates the surface integral of a vector field to the volume integral of its divergence, forming the bridge between integral and differential forms of Gauss's law.