Surface Elements | Santa Fe College - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

283

Pages:

5
Author:

Jabin F.

Surface Elements: - dx, dy, dz → / Closed Surfaces Those surfaces which Bounds some volume. Like Cube, Cylinder, sphere Open Surfaces Those Surfaces which are don't bounds any wlume. Like- Disc, Square, reetong Cube of Sto make of wbe of side a cornerd at origin. (outgoing hormal) d\vec{S_1} = dy dz \hat{x} |_{x=a} d\vec{S_2} = dy dz (-\hat{x}) |_{x=0} d\vec{S_3} = dx dz (-\hat{y}) |_{y=0} d\vec{S_4} = dx dz (\hat{y}) |_{y=a} d\vec{S_5} = dx dy (\hat{z}) |_{z=a} d\vec{S_6} = dx dy (-\hat{z}) |_{z=0} Dus: Make a Cube of Side 'à Centorcol ∫∫ E.dS = Electric flux ∫∫ B.dS = magnetic flux ∫∫ E.dS = Electric flux ∫∫ B.dS = 0 always Cube - 1 edge a placed in a uniform electric field E = Eo(x+y). Calculate the Electric flux (No. of Electric field lines) associated with the cube. Uniform Independent of position Coordinate Non-Uniform → Dependent of position Coordinate Constant → Independent of time non-Constant → Dependent of time Soln Φtube = ∫E.dS = Φ₁ + Φ₂ + Φ₃ + Φ₄ + Φ₅ + Φ₆ Φ₂ = ∫∫E.dS₂ = ∫∫ Eo(x+y) dy dz (-x̂) |x=0 Φ₂ = -E₀α² Φ₁ = ∫∫∫ E₀(x+y) dy dz (+x̂) |x=+a 0 0 Φ₁ = +E₀α² Similarly Φ₃ = -E₀α² Φ₄ = +E₀α² Net flux Φtube = 0 Net flux or Net No. of field Lines associated with a closed surface in a uniform field is always zero. Surface elements:- d\vec{S_r} = dr d\theta d\phi \hat{r} d\vec{S} = r^2 sin\theta d\theta d\phi \hat{r} Spherical surface Area - S_r = ∫(r^2 sinθ dθ dφ) |_(r=R) Sr = 4πR² dSo = dlr dφ r̂ = r sinθ dr dφ r̂ |θ=90° dSdisc = r dr dφ r̂ area- Sdisc = ∫₀ᴿ rdr ∫₀²π dφ = πR² d⃗Sφ = r dr dφ r̂ dSφdisc = r dr dφ

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