Electromagnetism and Vector Calculus Essential Concepts

1. Vector Calculus

Gradient

The gradient of a scalar field represents the rate of maximum increase of the function and its direction.

Formula: ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Procedure

• Step 1: Differentiate w.r.t x
• Step 2: Differentiate w.r.t y
• Step 3: Differentiate w.r.t z
• Step 4: Substitute the coordinates

Divergence

Divergence measures the outward flow of a vector field.

Formula: ∇·A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z

Curl

Curl measures the rotational nature of a vector field.

Formula: ∇×A (determinant method)

2. Electrostatics

Coulomb's Law

The force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance. The direction is along the line joining the charges.

Formula: F = (1/4πε₀)(q₁q₂/r²)

Electric Field

Electric field is the force experienced per unit positive charge.

Formula: E = F/q (Unit: N/C)

Electric Potential

Work done per unit charge.

Formula: V = (1/4πε₀)(q/r) (Unit: Volt)

3. Gauss Law

The total electric flux through a closed surface equals the enclosed charge divided by ε₀.

Formula: ∮E·dA = Q/ε₀

4. Electric Dipole

Dipole Moment: p = q(2a) (Direction: Negative to positive charge)

Potential at General Point: V = (1/4πε₀)(p cosθ / r²)

Electric Field on Axis: E = (1/4πε₀)(2p/r³)

5. Dielectrics

Polarization: Dipole moment per unit volume (P = Dipole Moment / Volume)

Bound Volume Charge Density: ρb = –∇·P

Bound Surface Charge Density: σb = P·n

6. Capacitors

Parallel Plate Capacitor: C = εA/d (where ε = Kε₀)

Energy Stored: U = ½CV²

7. Magnetostatics

Biot-Savart Law: dB = (μ₀/4π)(Idl sinθ/r²)

Magnetic Field (Long Wire): B = μ₀I/(2πr)

8. Maxwell Equations

1. Gauss Law: ∇·E = ρ/ε₀
2. Gauss Law for Magnetism: ∇·B = 0
3. Faraday Law: ∇×E = –∂B/∂t
4. Ampere-Maxwell Law: ∇×B = μ₀J + μ₀ε₀(∂E/∂t)

9. Electromagnetic Waves

Speed: c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s

Properties: EM waves are transverse; electric and magnetic fields are perpendicular to each other and to the direction of propagation.