Physics — JES

Electrostatics: Coulomb's Law, Field & Gauss's Law

Apply concepts from Electrostatics: Coulomb's Law, Field & Gauss's Law to problem-solving. Focus on numerical practice, shortcuts, and real-world applications.

6%55 minPhase 1 · APPLICATIONMCQ + Numerical

Concept Core

Electrostatics deals with stationary electric charges and the forces, fields, and potentials they produce. The fundamental quantity is electric charge, measured in coulombs (C), with dimensional formula [A T]. Charge is quantized (q = ne, where e = 1.6 x 10^(-19) C, n is an integer) and conserved in all processes.

Coulomb's Law: The electrostatic force between two point charges q1 and q2 separated by distance r in vacuum is:

F = ( $\frac{1}{4}$ $\pi$ $\epsilon_{0}$ ) * |q1 * q2| / $r^{2}$

where $\epsilon_{0}$ = 8.854 x 10^(-12) $C^{2}$ N^(-1) m^(-2) is the permittivity of free space, and 1/(4* $\pi$ $\epsilon_{0}$ ) = k = 9 x $10^{9}$ N $m^{2}$ C^(-2). Dimensional formula of $\epsilon_{0}$ : [M^(-1) L^(-3) $T^{4} \; A^{2}$ ]. In a medium with dielectric constant K (or kappa), replace $\epsilon_{0}$ with K $\epsilon_{0}$ , so force reduces by factor K. The force is attractive for unlike charges, repulsive for like charges. Coulomb's law obeys the principle of superposition: net force on a charge equals the vector sum of individual forces.

Electric Field (E): The electric field at a point is the force per unit positive test charge: E = F/ $q_{0}$ (SI unit: N/C or V/m, dimensional formula: [M L T^(-3) A^(-1)]). For a point charge Q: E = kQ/ $r^{2}$ (radially outward for +Q, inward for -Q).

Electric Field Lines: Imaginary lines tangent to the field direction at every point. They originate on positive charges and terminate on negative charges. Density of field lines indicates field strength. Lines never cross. For a uniform field, lines are parallel and equispaced.

Continuous Charge Distributions: For a linear charge density $\lambda$ (C/m), surface charge density $\sigma$ (C/ $m^{2}$ ), or volume charge density $\rho$ (C/ $m^{3}$ ), the field is found by integration: dE = k * dq / $r^{2}$ , then integrate over the distribution.

Key results:

Infinite line charge: E = $\lambda$ / (2* $\pi$ * $\epsilon_{0}$ *r), directed radially outward (r = perpendicular distance).
Infinite plane sheet: E = $\sigma$ / (2* $\epsilon_{0}$ ), uniform and independent of distance, directed away from the sheet.
Uniformly charged sphere (outside, r >= R): E = kQ/ $r^{2}$ (behaves like point charge). Inside (r < R): E = kQr/ $R^{3}$ for solid sphere; E = 0 for hollow sphere.

Electric Dipole: Two equal and opposite charges +q and -q separated by distance 2a.
Dipole moment p = q * 2a (SI unit: C m, dimensional formula: [A T L]). Direction: from -q to +q.
Axial field: $E_{axial}$ = 2kp/ $r^{3}$ (for r >> a).
Equatorial field: $E_{eq}$ = kp/ $r^{3}$ (opposite to p).
Torque on dipole in uniform field: $\tau$ = p x E, | $\tau$ | = pE*sin( $\theta$ ).

Gauss's Law: The total electric flux through any closed surface equals the net charge enclosed divided by $\epsilon_{0}$ :

$\Phi_{E}$ = integral(E . dA) = $Q_{enclosed}$ / $\epsilon_{0}$

$\Phi_{E}$ has SI unit: N $m^{2}$ C^(-1) (or V m), dimensional formula: [M $L^{3}$ T^(-3) A^(-1)]. Gauss's law is most useful when the charge distribution has high symmetry (spherical, cylindrical, or planar). The Gaussian surface is chosen to exploit this symmetry so that E is constant on the surface and either parallel or perpendicular to dA.

Key Testable Concept

Comparison Tables

A) Electric Field for Standard Configurations

Configuration	Electric Field Expression	Direction	Condition
Point charge Q	E = kQ/ $r^{2}$	Radially outward (+Q)	r > 0
Infinite line ( $\lambda$ )	E = $\lambda$ /(2* $\pi$ * $\epsilon_{0}$ *r)	Radially outward	r = perp. distance
Infinite plane ( $\sigma$ )	E = $\sigma$ /(2* $\epsilon_{0}$ )	Normal to surface	Uniform, independent of r
Two parallel sheets (+ $\sigma$ , - $\sigma$ )	E = $\frac{\sigma}{\epsilon_{0}}$ (between), 0 (outside)	From + to - sheet	Between the sheets
Solid sphere (uniform $\rho$ ) outside	E = kQ/ $r^{2}$	Radial	r >= R
Solid sphere inside	E = kQr/ $R^{3}$ = $\rho$ r/(3 $\epsilon_{0}$ )	Radial outward	r < R
Hollow sphere outside	E = kQ/ $r^{2}$	Radial	r >= R
Hollow sphere inside	E = 0	—	r < R
Ring on axis	E = kQx/( $x^{2}$ + $R^{2}$ )^( $\frac{3}{2}$ )	Along axis	x = axial distance
Disk on axis	E = ( $\frac{\sigma}{2}$ * $\epsilon_{0}$ )[1 - x/ $\sqrt{x^{2}+R^{2}}$ ]	Along axis	—

B) Dimensional Formulae in Electrostatics

Quantity	Symbol	SI Unit	Dimensional Formula
Charge	q	C	[A T]
Electric field	E	N/C or V/m	[M L T^(-3) A^(-1)]
Electric flux	$\Phi_{E}$	N $m^{2}$ /C	[M $L^{3}$ T^(-3) A^(-1)]
Permittivity	$\epsilon_{0}$	$C^{2}$ N^(-1) m^(-2)	[M^(-1) L^(-3) $T^{4} \; A^{2}$ ]
Coulomb constant	k	N $m^{2}$ C^(-2)	[M $L^{3}$ T^(-4) A^(-2)]
Dipole moment	p	C m	[A T L]
Linear charge density	$\lambda$	C/m	[A T L^(-1)]
Surface charge density	$\sigma$	C/ $m^{2}$	[A T L^(-2)]

C) Comparison: Coulomb's Law vs Gravitational Law

Feature	Coulomb's Law	Gravitational Law
Force formula	F = kq1q2/ $r^{2}$	F = Gm₁m2/ $r^{2}$
Nature	Attractive or repulsive	Always attractive
Constant	k = 9 x $10^{9}$ N $\frac{m^{2}}{C^{2}}$	G = 6.67 x 10^(-11) N $\frac{m^{2}}{kg^{2}}$
Relative strength	~ $10^{36}$ times stronger	Much weaker
Medium dependence	Yes (factor K)	No
Superposition	Yes	Yes

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