1. Vector Calculus - Introduction
Electrodynamics deals with electric and magnetic fields, which are vector fields – quantities that have both magnitude and direction at every point in space. To describe how these fields change in space, we use three important vector differential operators: Gradient, Divergence, and Curl.
Del Operator (∇)
The del operator (nabla) is a vector differential operator used in vector calculus. In Cartesian coordinates:
∇ = î(∂/∂x) + ĵ(∂/∂y) + k̂(∂/∂z)
The del operator can act on scalar fields (giving gradient) or vector fields (giving divergence or curl), depending on how it's applied.
2. Gradient
Gradient of a Scalar Field
The gradient of a scalar field φ(x, y, z) is a vector that points in the direction of the maximum rate of increase of φ, with magnitude equal to that rate of increase.
∇φ = grad φ = (∂φ/∂x)î + (∂φ/∂y)ĵ + (∂φ/∂z)k̂
Gradient in Cartesian Coordinates
Gradient Visualization (Temperature Field Example)
Gradient vectors are perpendicular to contour lines, pointing toward increasing values
Physical Significance of Gradient
- Direction: Points in the direction of maximum increase of the scalar field
- Magnitude: Equals the maximum rate of change per unit distance
- Perpendicular to level surfaces: ∇φ is always perpendicular to surfaces of constant φ
- Electric field relation: E = -∇V (Electric field is negative gradient of potential)
Example: Gradient Calculation
Find the gradient of the scalar field φ = x²y + yz³ at the point (1, 2, 1).
Step 1: Calculate partial derivatives
∂φ/∂x = 2xy = 2(1)(2) = 4
∂φ/∂y = x² + z³ = 1 + 1 = 2
∂φ/∂z = 3yz² = 3(2)(1) = 6
Step 2: Write the gradient vector
∇φ = 4î + 2ĵ + 6k̂
∇φ at (1, 2, 1) = 4î + 2ĵ + 6k̂
3. Divergence
Divergence of a Vector Field
The divergence of a vector field F is a scalar that measures the "outward flux" or "spreading out" of the field from a point. It indicates whether a point is a source (positive divergence) or sink (negative divergence) of the field.
If F = F_x î + F_y ĵ + F_z k̂, then:
∇ · F = div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
Divergence in Cartesian Coordinates
Physical Meaning of Divergence
Physical Significance of Divergence
- ∇ · F > 0: Field lines are spreading out – there's a source at that point
- ∇ · F < 0: Field lines are converging – there's a sink at that point
- ∇ · F = 0: No net flux through closed surface – solenoidal field (incompressible flow)
- Example: For electric field, ∇ · E = ρ/ε₀ (positive charge is a source)
Example: Divergence Calculation
Find the divergence of F = x²yî + y²zĵ + z²xk̂ at point (1, 1, 1).
Step 1: Identify components
F_x = x²y, F_y = y²z, F_z = z²x
Step 2: Calculate partial derivatives
∂F_x/∂x = 2xy = 2(1)(1) = 2
∂F_y/∂y = 2yz = 2(1)(1) = 2
∂F_z/∂z = 2zx = 2(1)(1) = 2
Step 3: Sum for divergence
∇ · F = 2 + 2 + 2 = 6
∇ · F at (1, 1, 1) = 6
4. Curl
Curl of a Vector Field
The curl of a vector field F is a vector that measures the "rotation" or "circulation" of the field around a point. Its direction gives the axis of rotation, and its magnitude gives the intensity of rotation.
∇ × F = | î ĵ k̂ |
| ∂/∂x ∂/∂y ∂/∂z |
| F_x F_y F_z |
| ∂/∂x ∂/∂y ∂/∂z |
| F_x F_y F_z |
Curl as Determinant
Expanding the determinant:
∇ × F = (∂F_z/∂y - ∂F_y/∂z)î + (∂F_x/∂z - ∂F_z/∂x)ĵ + (∂F_y/∂x - ∂F_x/∂y)k̂
Curl in Cartesian Coordinates
Physical Meaning of Curl
Physical Significance of Curl
- Non-zero curl: Field has rotation/circulation – like a whirlpool
- Zero curl: Irrotational field – no circulation (conservative field)
- Direction: Right-hand rule – curl thumb along rotation axis
- Example: For magnetic field around wire, ∇ × B = μ₀J (current causes curl)
Example: Curl Calculation
Find the curl of F = y²î + 2xyzĵ + z²k̂.
Step 1: Identify components
F_x = y², F_y = 2xyz, F_z = z²
Step 2: Calculate î component
(∂F_z/∂y - ∂F_y/∂z) = 0 - 2xy = -2xy
Step 3: Calculate ĵ component
(∂F_x/∂z - ∂F_z/∂x) = 0 - 0 = 0
Step 4: Calculate k̂ component
(∂F_y/∂x - ∂F_x/∂y) = 2yz - 2y = 2y(z - 1)
∇ × F = -2xyî + 0ĵ + 2y(z-1)k̂
5. Gauss's Law
Gauss's Law for Electricity
The total electric flux through any closed surface is equal to the total charge enclosed divided by the permittivity of free space.
∮ E · dA = Q_enc/ε₀
Integral Form
∇ · E = ρ/ε₀
Point (Differential) Form
Gauss's Law Illustration
Significance of Gauss's Law
- Relates electric field to its source (charge distribution)
- ∇ · E gives the charge density at any point (scaled by ε₀)
- Positive charge acts as a source (positive divergence)
- Negative charge acts as a sink (negative divergence)
- Useful for calculating E-field with symmetric charge distributions
Example: Gauss's Law
A sphere of radius R has uniform charge density ρ. Using Gauss's law in differential form, verify the electric field inside the sphere at distance r from center is E = ρr/(3ε₀).
Step 1: By symmetry, E is radial
E = E(r)r̂ (only radial component)
Step 2: Apply Gauss's law in integral form
∮ E · dA = E(4πr²) = Q_enc/ε₀
Step 3: Find enclosed charge
Q_enc = ρ × (4/3)πr³
Step 4: Solve for E
E(4πr²) = ρ(4/3)πr³/ε₀
E = ρr/(3ε₀)
E = ρr/(3ε₀) (verified)
6. Ampere's Circuital Law
Ampere's Circuital Law
The line integral of magnetic field around any closed loop equals μ₀ times the total current passing through the loop.
∮ B · dl = μ₀I_enc
Integral Form
∇ × B = μ₀J
Point (Differential) Form
Where J is the current density vector (current per unit area).
Ampere's Law - Current Carrying Wire
Significance of Ampere's Law
- Relates magnetic field to its source (current)
- ∇ × B gives the current density at any point (scaled by μ₀)
- Current creates rotational (curl) magnetic field
- Useful for calculating B-field with symmetric current distributions
- Modified by Maxwell to include displacement current
7. Faraday's Law of Electromagnetic Induction
Faraday's Law
A changing magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced EMF equals the negative rate of change of magnetic flux.
∮ E · dl = -dΦ_B/dt
Integral Form
∇ × E = -∂B/∂t
Point (Differential) Form
Faraday's Law - Changing Magnetic Field
Significance of Faraday's Law
- Changing magnetic field creates electric field with curl
- This induced E-field is non-conservative (closed loop integral ≠ 0)
- Basis of electric generators, transformers, inductors
- Negative sign represents Lenz's law (opposes change)
- Links electricity and magnetism as aspects of same phenomenon
8. Divergence Theorem (Gauss's Theorem)
Divergence Theorem
The volume integral of the divergence of a vector field over a region equals the surface integral of the field over the closed surface bounding that region.
∫∫∫_V (∇ · F) dV = ∮_S F · dA
Divergence Theorem
Physical Interpretation
The total "source strength" inside a volume (left side) equals the total flux flowing out through the surface (right side).
This theorem converts a volume integral to a surface integral, often simplifying calculations.
Applications
- Deriving Gauss's law in differential form from integral form
- Conservation laws (mass, charge, energy)
- Fluid dynamics (continuity equation)
9. Stokes' Theorem
Stokes' Theorem
The line integral of a vector field around a closed curve equals the surface integral of the curl of the field over any surface bounded by that curve.
∮_C F · dl = ∫∫_S (∇ × F) · dA
Stokes' Theorem
Physical Interpretation
The total circulation around a curve (left side) equals the total "rotation" passing through any surface bounded by that curve (right side).
This theorem converts a line integral to a surface integral.
Applications
- Deriving Ampere's and Faraday's laws in differential form
- Understanding conservative fields (if ∇ × F = 0, then ∮ F · dl = 0)
- Circulation and vorticity in fluid mechanics
10. Maxwell's Equations
Maxwell's equations are the four fundamental equations that describe all classical electromagnetic phenomena. They unify electricity and magnetism into a single theory and predict the existence of electromagnetic waves.
| Name | Point Form | Integral Form | Significance |
|---|---|---|---|
| Gauss's Law (E) | ∇ · E = ρ/ε₀ | ∮ E · dA = Q/ε₀ | Charges are sources of E-field |
| Gauss's Law (B) | ∇ · B = 0 | ∮ B · dA = 0 | No magnetic monopoles exist |
| Faraday's Law | ∇ × E = -∂B/∂t | ∮ E · dl = -dΦ_B/dt | Changing B creates E |
| Ampere-Maxwell Law | ∇ × B = μ₀J + μ₀ε₀∂E/∂t | ∮ B · dl = μ₀I + μ₀ε₀dΦ_E/dt | Current and changing E create B |
Significance of Each Equation
- Gauss's Law (E): Electric field diverges from positive charges and converges to negative charges. Charges are sources/sinks of electric field lines.
- Gauss's Law (B): Magnetic field lines always form closed loops – they have no beginning or end. There are no magnetic monopoles.
- Faraday's Law: A time-varying magnetic field produces a circulating electric field. This is the basis of electromagnetic induction.
- Ampere-Maxwell Law: Magnetic field circulates around currents AND around changing electric fields. Maxwell's addition of the displacement current term (ε₀∂E/∂t) made electromagnetic waves possible.
Maxwell's Key Insight
Maxwell added the "displacement current" term μ₀ε₀∂E/∂t to Ampere's law. This showed that:
- Changing E-field creates B-field (just as changing B creates E)
- E and B fields can sustain each other, propagating as electromagnetic waves
- Speed of EM waves: c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s (speed of light!)
This unified light, electricity, and magnetism as manifestations of the same phenomenon.
Example: Interpreting Maxwell's Equations
In a region of space, the electric field is E = E₀ sin(kz - ωt) î. Using Faraday's law, find the associated magnetic field.
Step 1: Apply Faraday's Law
∇ × E = -∂B/∂t
Step 2: Calculate ∇ × E
E = E₀ sin(kz - ωt) î, so E_x = E₀ sin(kz - ωt), E_y = E_z = 0
∇ × E = (∂E_x/∂z)(-ĵ) = -kE₀ cos(kz - ωt) ĵ
Step 3: Integrate to find B
-∂B/∂t = -kE₀ cos(kz - ωt) ĵ
B = ∫ kE₀ cos(kz - ωt) dt ĵ = (kE₀/ω) sin(kz - ωt) ĵ
B = (E₀/c) sin(kz - ωt) ĵ (since ω/k = c)
B = (E₀/c) sin(kz - ωt) ĵ (B perpendicular to E, in phase, ratio E/B = c)
Summary: Key Formulas
∇φ = (∂φ/∂x)î + (∂φ/∂y)ĵ + (∂φ/∂z)k̂
Gradient
∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
Divergence
∇ × F = determinant form
Curl
∇ · E = ρ/ε₀
Gauss's Law
∇ × B = μ₀J + μ₀ε₀∂E/∂t
Ampere-Maxwell Law
∇ × E = -∂B/∂t
Faraday's Law