Electromagnetic Theory Maxwell equations - Waveguides - Radiation Concise CSIR-UGC NET/JRF/Ph.D. class notes
1. Maxwell’s four equations in free space
| Differential form | Physical meaning |
|---|---|
| ∇·E = ρ/ε₀ | Electric flux emerges from charge |
| ∇·B = 0 | No magnetic monopoles |
| ∇×E = −∂B/∂t | Changing B induces electric field |
| ∇×B = μ₀J + μ₀ε₀∂E/∂t | Currents and changing E create B |
∇²E = μ₀ε₀∂²E/∂t² ⇒ speed c = 1/√(μ₀ε₀).
2. Conductors, skin depth & shielding
Skin depth δ = √(2/μσω).
For 1% transmission (e^{-2z/δ}=0.01) the required thickness is z≈2.30 δ.
Example: σ=1×10⁶ Ω⁻¹ m⁻¹, ω=10⁷ rad s⁻¹ ⇒ δ≈1 mm → shield ≈2.3 mm thick.
3. Waveguides (rectangular, perfect conductor)
Cut-off frequency for TEₘₙ mode:
fₘₙ = (c/2)√[(m/a)²+(n/b)²].
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Dominant mode TE₁₀ (m=1,n=0).
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Guide wavelength λg = λ/√(1−(λ/2a)²).
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Phase velocity v_p = c/√(1−(f_c/f)²) > c; group velocity v_g = c²/v_p < c.
Worked pair: For a=3.33 cm, b=2.50 cm, TE₁₁ cuts on at 7.5 GHz; any higher f propagates.
4. Poynting vector & radiation zones
Instantaneous energy flow S = (1/μ₀)E×B.
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Far-field (radiation) of an accelerating charge varies as 1/r for E and 1/r² for intensity (S).
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For a narrow proton beam (current I, velocity u), at distance r ≫ beam width:
|S| = I²/(4π²ε₀ u r²) directed radially outward along u × (azimuthal ϕ̂)
5. Reflection & transmission at normal incidence
Reflection coefficient (E || interface):
R = (n₂−n₁)/(n₂+n₁); T = 1−R² (power).
Phase shift for reflection from metal with refractive index n₂ = n(1+iρ):
φ = –tan⁻¹(ρ/n) (parallel polarisation).
6. Vector and scalar gauge transforms
Gauge change with scalar χ: A′ = A + ∇χ, V′ = V − ∂χ/∂t keeps E,B unchanged.
Example χ = a t x gives A′ = A − a t î, V′ = V + a x (valid gauge).
7. Infinite solenoid field (outside)
For flux Φ along ẑ:
Vector potential in Coulomb gauge (r⊥ > 0): A = Φ/(2πr²)(−y î + x ĵ).
8. Typical CSIR exam problem types
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Standing-wave ratios: maxima/minima ratio 5 ⇒ |R| = 3/4, reflected/incident power = 9/16
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Method of images: point charge q at distance d from grounded plane feels attractive image force F = −q²/(16π ε₀ d²) toward plane.youtube
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Charge distributions: given ρ(r) ∝ e^{−r/r₀}, evaluate ∇²V to find ρ at specific r
9. Quick-reference formulas
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Plane wave: |B₀| = |E₀|/c.
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Intrinsic impedance of free space: η₀ ≈ 377 Ω.
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Radiation from non-relativistic charge: P = (q²a²)/(6π ε₀c³).
Scaling: doubling q and a multiplies intensity by 16.
10-Day Crash Plan
Day 1–2 : Maxwell derivations & boundary conditions.
Day 3–4 : Waveguide modes; solve 15 cut-off problems.
Day 5 : Skin depth, shielding calculations.
Day 6 : Poynting vector & radiation patterns (dipole, charge in circular motion).
Day 7–8 : Reflection/refraction; Brewster & critical angle drills.
Day 9 : Method of images practice (plane, sphere).
Day 10 : Attempt two full CSIR past-year electromagnetic sections (timed).
Master these core pieces and electromagnetic theory yields a reliable 10–14 marks on the CSIR Physical Sciences paper.
