Electrostatics

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest, focusing on the electric effects and forces they produce

Electric Charge - Basic Properties

**Definition:** Charge is an intrinsic property of matter that allows it to create and experience electric and magnetic effects [2].

Key Concepts:

Charges are of two types: *positive and negative** [1].

* Like charges repel, and unlike charges attract [1, 3].

The *SI unit is the Coulomb (C)** [1].

Charge is *quantized** (Q = ne, where n is an integer and e is the elementary charge) [3, 4].

Charge is *conserved**; the net charge of an isolated system remains constant [3, 5].

Charge is *additive**, meaning the total charge is the algebraic sum of individual charges [4].

The value of charge *does not depend on its velocity**

Charging by Friction

Charging by friction involves the transfer of electrons between two materials when they are rubbed together, resulting in one becoming positively charged and the other negatively charged [5, 7, 8].

Example: Rubbing a glass rod with silk makes the glass rod positively charged and the silk negatively charged

Charging by Conduction

Charging by conduction involves the transfer of charge when a charged object touches an uncharged conductor, allowing electrons to move directly [7, 9].

Key Concept: If conductors of the same shape and size are connected, the charge distributes equally between them

Charging by Induction

Charging by induction occurs when a charged object is brought near but not touching a neutral conductor, causing a redistribution of charge within the conductor due to electrostatic forces [7, 10].

Key Concept: The induced charge on the side closer to the charging object will be opposite in sign, while the far side acquires the same sign [7, 10].

Coulomb's Law - Force Magnitude

Coulomb's Law quantifies the electrostatic force between two point charges [11, 12].

Formula:

F = k |q₁q₂| / r² [11, 12]

Where:

* F is the magnitude of the force.

* q₁, q₂ are the magnitudes of the charges.

* r is the distance between the charges.

* k is Coulomb's constant (k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C²) [11-13].

* ε₀ is the permittivity of free space (8.85 × 10⁻¹² C²/(N·m²)) [11, 13

Coulomb's Law - Vector Form and Superposition

Concept (Vector Form):** The electrostatic force is a vector quantity, acting along the line joining the two charges [14, 15].

Formula (Vector Form):

F = k (q₁q₂ / r³) r̂ (where r̂ is the unit vector along the line connecting charges) [14, 15]

Principle of Superposition: The net electrostatic force on any charge in a system is the vector sum of all individual forces exerted by other charges [16, 17].

Permittivity and Dielectric Constant

Definition (Relative Permittivity / Dielectric Constant K):** A dimensionless factor (εr or K) that describes how an electric field interacts with a dielectric medium [11, 18].

Formula:

K = ε / ε₀ (where ε is the permittivity of the medium) [11, 18].

Effect on Force: When charges are placed in a medium with dielectric constant K, the force between them is reduced by a factor of K compared to vacuum: F_medium = F_vacuum / K [11, 19].

Electric Field

An electric field (E) is the region around a charge where its electric effects are experienced [20, 21]. It represents the force per unit positive test charge [20, 22].

Formula:

E = F / q₀ (where F is the force on a test charge q₀) [20, 23].

Unit: Newton per Coulomb (N/C) or Volt per meter (V/m) [23].

Direction: Electric field lines point radially outward from positive charges and inward towards negative charges [24-26]. The tangent to an electric field line at any point gives the direction of E at that point [27].

Electric Field Due to a Point Charge

Formula (Magnitude):**

E = k |Q| / r² [20, 28]

Where:

* Q is the point charge.

* r is the distance from the charge.

* k is Coulomb's constant.

Key Concepts:

E is *inversely proportional to the square of the distance** (E ∝ 1/r²) [28].

* E tends to infinity near the charge (r→0) and to zero far from it (r→∞) [28].

Topic:** Electric Field Due to a Uniformly Charged Ring

Formula (on Axis at distance x from center):**

E = kQx / (R² + x²)^(3/2) [29, 30]

Where:

* Q is the total charge.

* R is the radius of the ring.

* x is the distance along the axis from the center.

Key Concepts:

At the *center (x=0)**, the electric field is zero [29-31].

The electric field is *maximum** at x = ±R/√2 [32, 33].

Electric Field Due to an Infinite Non-Conducting Sheet

Formula:

E = σ / (2ε₀) [34, 35]

Where:

* σ is the uniform surface charge density.

* ε₀ is the permittivity of free space.

Key Concept: The electric field is uniform and independent of distance from the sheet [34, 35].

Electric Field Due to an Infinitely Long Wire

Formula:**

E = λ / (2πε₀r) = 2kλ / r [36-38]

Where:

* λ is the uniform linear charge density.

* r is the perpendicular distance from the wire.

* k is Coulomb's constant.

Direction: The field points radially outward for positive λ [36].

Electric Field Due to a Uniformly Charged Hollow Sphere (Shell)

Formula:

* Inside (r < R): E = 0 [39, 40]

* Outside (r > R): E = kQ / r² [40, 41]

Where:

* Q is the total charge.

* R is the radius of the shell.

* r is the distance from the center.

Key Concept: For external points, a uniformly charged shell behaves like a point charge at its center [41, 42].

Electric Field Due to a Uniformly Charged Solid Sphere

Formula:**

* Inside (r ≤ R): E = (kQr) / R³ = (ρr) / (3ε₀) [43, 44]

* Outside (r > R): E = kQ / r² [43, 44]

Where:

* Q is the total charge.

* R is the radius of the sphere.

* r is the distance from the center.

* ρ is the volume charge density.

Key Concept: E increases linearly inside and decreases as 1/r² outside [43, 45, 46].

Electric Dipole

An electric dipole is a system of two equal and opposite charges (+q and -q) separated by a small distance (2a) [47, 48].

Dipole Moment (p): A vector quantity that characterizes the strength and orientation of an electric dipole [47, 49].

Formula:

p = q(2a)l (where l is the vector from -q to +q) [47, 49].

Direction: From negative to positive charge [47, 49].

Unit: Coulomb-meter (C·m) [49].

Torque on an Electric Dipole in a Uniform Electric Field

Concept: A dipole in a uniform electric field experiences a torque that tends to align its dipole moment with the field [50-52].

Formula:

τ = pE sinθ or τ = p x E [50-53]

Where:

* p is the dipole moment.

* E is the electric field.

* θ is the angle between p and E.

Equilibrium:

* **Stable Equilibrium:** θ = 0° (p aligned with E), torque is zero [54].

* **Unstable Equilibrium:** θ = 180° (p anti-aligned with E), torque is zero [54].

Potential Energy of an Electric Dipole in a Uniform Electric Field

Concept: The potential energy (U) of an electric dipole in a uniform electric field depends on its orientation [51, 55].

Formula:

U = -pE cosθ or U = -p ⋅ E [51, 55, 56]

Key Values:

* **Minimum Energy (Stable):** U = -pE (at θ = 0°) [56].

* **Maximum Energy (Unstable):** U = +pE (at θ = 180°) [56].

Electric Potential

**Definition:** Electric potential (V) at a point is the work done by an external agent to bring a unit positive test charge slowly from infinity to that point [57-59].

Formula (for a point charge Q at distance r):

V = kQ / r [57, 60]

Key Concepts:

V is a *scalar quantity** [57, 61].

* **Unit:** Volt (V) = Joule/Coulomb (J/C) [57, 61].

The electric field always points from *high potential to low potential** [62].

Electric Potential Due to a Uniformly Charged Hollow Sphere (Shell)

Formula:

* Inside or on the surface (r ≤ R): V = kQ / R [63, 64]

* Outside the sphere (r > R): V = kQ / r [63, 64]

Where:

* Q is the total charge.

* R is the radius.

* r is the distance from the center.

Key Concept: The potential inside a uniformly charged shell is constant and equals its surface potential [63, 64].

Electric Potential Due to a Uniformly Charged Solid Sphere

Formula:

* **Inside the sphere (r ≤ R): V = (kQ / (2R³)) (3R² - r²)* [63, 65]

* Outside the sphere (r > R): V = kQ / r [63, 65]

Where:

* Q is the total charge.

* R is the radius.

* r is the distance from the center.

**At center (r=0):** V_center = (3kQ) / (2R) [65].

Relation Between Electric Field and Electric Potential

Relationship: The electric field is the negative gradient of the electric potential [66-68].

Formula (1D):

E = -dV/dr [62, 67]

Formula (3D):

E = - (∂V/∂x î + ∂V/∂y ĵ + ∂V/∂z k̂) [68, 69]

Key Concept: The electric field points in the direction where the potential decreases most rapidly [62].

Electric Potential Energy (EPE) of a System of Charges

Definition: The EPE of a system of charges is the work required by an external agent to assemble that system by bringing charges from infinity to their final positions [70, 71].

Formula (for point charges):

U_system = Σ (k q_i q_j / r_ij) for all unique pairs (i < j) [70-72].

Work Done:

* Work done by external agent (W_ext) = U_final - U_initial [58, 73].

* Work done by electric force (W_elect) = U_initial - U_final [73, 74].

Equipotential Surfaces

Definition: An equipotential surface is a surface where the electric potential is constant at every point [75, 76].

Key Properties:

* No work is done by the electric field when a charge moves along an equipotential surface [75-77].

Electric field lines are *always perpendicular** to equipotential surfaces [75, 78].

Equipotential surfaces *never intersect** [75, 78].

* Conductors in electrostatic equilibrium are equipotential volumes/surfaces [77].

Electric Flux

**Definition:** Electric flux (Φ) is a measure of the number of electric field lines passing through a given surface [79, 80]. It is a scalar quantity [80, 81].

Formula:

Φ = ∫ E ⋅ dS (general form) [80, 81]

For uniform E and a flat surface: Φ = E A cosθ [81]

Where:

* E is the electric field.

* dS is the differential area vector.

* θ is the angle between E and the area's normal.

Unit: N·m²/C or V·m [82].

Gauss's Law

Gauss's Law states that the total electric flux (Φ) through any closed surface (Gaussian surface) is directly proportional to the net electric charge enclosed (Q_enclosed) within that surface [41, 82].

Formula:

Φ = Q_enclosed / ε₀ [41, 82]

Key Concept: This law is powerful for calculating electric fields for symmetrical charge distributions [41, 83].

Conductors in Electrostatic Equilibrium

Key Properties:

The *net electric field inside the bulk of a conductor is zero** [77, 84, 85].

Any *excess charge resides entirely on the outer surface** [84, 86, 87].

The *electric potential is constant** throughout the conductor's volume and on its surface [77, 84].

Electric field lines are *perpendicular to the conductor's surface** [75, 88].

Surface charge density is *higher at sharper points** (smaller radius of curvature) [89, 90].

Topic: Electrostatic Shielding

**Definition:** Electrostatic shielding is the phenomenon where the electric field inside a conductor or a cavity within a conductor is zero, protecting the interior from external electric fields [91, 92].

Key Concept: This principle is used to safeguard sensitive equipment from external electric influences [91].

Electric Pressure

**Definition:** Electric pressure is the outward force per unit area on the surface of a charged conductor, caused by the mutual repulsion of the surface charges [93, 94].

Formula:

P = σ² / (2ε₀) [94, 95]

Where:

* σ is the local surface charge density.

* ε₀ is the permittivity of free space.

Direction: Always directed outward normal to the surface [94, 96].

Self-Energy of a Charged Body

Definition: The self-energy of a charged body is the work required to assemble its charge distribution by bringing infinitesimal charge elements from infinity [97, 98].

Formula (Uniformly charged solid sphere Q, radius R):

U_self = (3kQ²) / (5R) [97, 99]

Formula (Uniformly charged hollow sphere Q, radius R):

U_self = (kQ²) / (2R) [100]