--- title: "Verify Stokes' theorem, conical surface" subject: "Electromagnetism" chapter: 1 key: "semester_3/electromagnetism:verify-stokes-conical-surface" source: https://www.bctengineer.com/q/semester_3/electromagnetism/verify-stokes-conical-surface site: BCT Engineering --- # Verify Stokes' theorem, conical surface Electromagnetism · Chapter 1: Introduction - Asked in: 2074 Asoj, 8 marks - Video: [Stokes' Theorem Example // Verifying both Sides // Vector Calculus](https://www.youtube.com/watch?v=ms4JjH0BANU) — Dr. Trefor Bazett - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/verify-stokes-conical-surface - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1 ## Question Verify Stokes' theorem for the field H = (3r²/sinθ) a_θ + 54r cosθ a_φ A/m in free space for the conical surface defined by θ = 20°, 0 ≤ φ ≤ 2π, 0 ≤ r ≤ 5. Let the positive direction of dS be a_θ. ## Short answer The surface integral of ∇×H over the cone equals the line integral of H around the rim r=5, θ=20°. Both evaluate to −2726 A, verifying Stokes' theorem. ## Answer **Given:** H = (3r²/sinθ) a_θ + 54 r cosθ a_φ A/m on the cone θ = 20°, 0 ≤ φ ≤ 2π, 0 ≤ r ≤ 5, with dS along a_θ. **Stokes' theorem:** ∮ H · dl = ∫ (∇ × H) · dS. We evaluate both sides. ```text z │ cone θ = 20° │ ╱│╲ │ ╱ │ ╲ edge: circle r = 5 at θ = 20° │ ╱ │ ╲ radius = 5 sin20° = 1.710 ● ───────── ``` #### Right side: surface integral With H_r = 0, H_θ = 3r²/sinθ, H_φ = 54 r cosθ, the a_θ component of the curl is ```text (∇ × H)_θ = (1/r) [ (1/sinθ) ∂H_r/∂φ − ∂(r H_φ)/∂r ] = −(1/r) ∂(54 r² cosθ)/∂r = −108 cosθ ``` On the cone dS = r sinθ dr dφ a_θ: ```text ∫ (∇ × H) · dS = ∫₀^(2π) ∫₀⁵ (−108 cosθ)(r sinθ) dr dφ = −108 cos20° sin20° (25/2)(2π) = −108 (0.32140)(12.5)(6.2832) = −2726.2 A ``` (Exact form: −1350π sin40° = −2726.16 A.) #### Left side: line integral The surface is bounded by the circle r = 5, θ = 20°. The edge r = 0 is a single point and contributes nothing. By the right-hand rule, with dS along a_θ, the direction of travel along the rim (where the outward in-surface direction is a_r) is a_θ × a_r = −a_φ. So on the rim dl = −r sinθ dφ a_φ, and H_φ = 54 r cosθ: ```text ∮ H · dl = −∫₀^(2π) 54 (5) cos20° · (5 sin20°) dφ = −54 (25) cos20° sin20° (2π) = −54 (25)(0.32140)(6.2832) = −2726.2 A ``` **Answer:** Both sides = **−2726 A** (= −1350π sin40°). Stokes' theorem is verified. (The negative sign arises because, with dS along a_θ, the rim must be traversed in the −φ sense.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With dS along a_θ, which curl component is needed and what is it for this H? A. (curl H)_θ = -108 cosθ B. (curl H)_r = 108 cosθ C. (curl H)_φ = -108 cosθ D. (curl H)_θ = 54 cosθ Answer: A. (curl H)_θ = -(1/r) ∂(r H_φ)/∂r = -(1/r) ∂(54 r^2 cosθ)/∂r = -108 cosθ. (Key point: Curl component) 2. What is dS on the cone θ = 20°? A. r^2 sinθ dθ dφ a_r B. r sinθ dr dφ a_θ C. r dr dθ a_φ D. dr dφ a_z Answer: B. On a surface of constant θ, dS = r sinθ dr dφ a_θ. (Key point: Surface element on cone) 3. What is the boundary of the surface, and what about the apex r = 0? A. The circle r = 5, θ = 20° of radius 1.710; the apex is a point and contributes nothing B. The apex only C. Two circles at r = 0 and r = 5 D. The z-axis Answer: A. The open cone is bounded by the rim circle at r = 5, radius 5 sin20°; the apex is a single point with zero length. (Key point: Boundary of the cone) 4. Why is dl = -r sinθ dφ a_φ on the rim? A. Because H_φ is negative B. Because by the right-hand rule, dS along a_θ requires travelling in the -φ sense C. Because phi decreases with r D. Because the cone is upside down Answer: B. The rim direction is a_θ × a_r = -a_φ, so the path is traversed in the -φ direction, which gives the negative sign. (Key point: Direction around the rim) 5. What common value do both sides of Stokes' theorem give? A. +2726 A B. -2726 A C. 0 D. -1350 A Answer: B. Both sides equal -1350π sin40° = -2726 A, verifying the theorem. 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