Anotação Rotacional-2021 1

∇⃗ x 𝐻⃗ = Rotacional Cartesians ∇ x 𝐻 = (∂𝐻_z / ∂y - ∂𝐻_y / ∂z) 𝑎_x + (∂𝐻_x / ∂z - ∂𝐻_z / ∂x) 𝑎_y + (∂𝐻_y / ∂x - ∂𝐻_x / ∂y) 𝑎_z Cilíndricas ∇ x 𝐻 = (1/ρ ∂𝐻_z / ∂φ - ∂𝐻_φ / ∂z) 𝑎_ρ + (∂𝐻_ρ / ∂z - ∂𝐻_z / ∂ρ) 𝑎_φ + 1/ρ (∂/∂ρ (ρ𝐻_φ) - ∂𝐻_ρ / ∂φ) 𝑎_z Esféricas ∇ x 𝐻 = 1/r sin θ (∂/∂θ (𝐻_φ sin θ) - ∂𝐻_θ / ∂φ) 𝑎_r + 1/r (1/sin θ ∂𝐻_r / ∂φ - ∂/∂r (r𝐻_φ)) 𝑎_θ + 1/r (∂(r𝐻_θ) / ∂r - ∂𝐻_r / ∂θ) 𝑎_φ ∇⃗ x 𝐻⃗ = 1/r ∂(r Π r^2 / 3) / ∂r 𝑎_z = 1/r Π r^2 / 3 𝑎_z ∇⃗ x 𝐻⃗ = r𝑎_z ∫_sup ∇⃗ x 𝐻⃗ ⋅ dś = ∫_𝑆ν𝐹𝜋 r𝐚_z ⋅ r dφ dr 𝑎_z = = ∫ from r=0 to φ=0 r^2 dr dφ = (2Π r^3 / 3) Rotacional Cartesianas \nabla \times \vec{H} = \left(\frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z} \right)\hat{x} + \left(\frac{\partial H_x}{\partial z} - \frac{\partial H_z}{\partial x} \right)\hat{y} + \left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y} \right)\hat{z} Cilindricas \nabla \times \vec{H} = \left(\frac{1}{\rho} \frac{\partial (\rho H_z)}{\partial \phi} - \frac{\partial H_\phi}{\partial z} \right)\hat{\rho} + \left(\frac{\partial H_\rho}{\partial z} - \frac{\partial H_z}{\partial \rho} \right)\hat{\phi} + \frac{1}{\rho} \left(\frac{\partial (\rho H_\phi)}{\partial \rho} - \frac{\partial H_\rho}{\partial \phi} \right)\hat{z} \rho = l Esfericas \nabla \times \vec{H} = \frac{1}{r \sin \theta} \left(\frac{\partial (H_\phi \sin \theta)}{\partial \theta} - \frac{\partial H_\theta}{\partial \phi} \right)\hat{r} + \frac{1}{r} \left(\frac{1}{\sin \theta} \frac{\partial H_r}{\partial \phi} - \frac{\partial (r H_\phi)}{\partial r} \right)\hat{\theta} + \frac{1}{r} \left(\frac{\partial (r H_\theta)}{\partial r} - \frac{\partial H_r}{\partial \theta} \right)\hat{\phi} EXEMPLOS \mu_\phi a) \vec{H} = \frac{I r}{2 \pi a^2} \hat{\phi} \quad 0 \le r \le A \nabla \times \vec{H} = \vec{J} = \frac{1}{r} \frac{\partial (Ir)}{2 \pi a^2} \hat{a_z} \nabla \times \vec{H} = \vec{J} = \frac{2AI}{r\cdot 2 \pi a^2} \hat{a_z} = \frac{I}{\pi A^2} \hat{a_z} \nabla \times \vec{H} = \vec{J} \hat{a_z} \nabla \times \vec{H} b) \vec{H} = \frac{I}{2 \pi r} \hat{\phi} \quad A \le r \le B \nabla \times \vec{H} = \frac{1}{r} \frac{\partial }{\partial r} \left(\frac{I r}{2 \pi a^2} \right) \hat{a_z} = 0 Rotacional Cartesianas \nabla \times \vec{H} = \left(\frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z} \right)\hat{x} + \left(\frac{\partial H_x}{\partial z} - \frac{\partial H_z}{\partial x} \right)\hat{y} + \left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y} \right)\hat{z} Cilindricas \nabla \times \vec{H} = \left(\frac{1}{\rho} \frac{\partial (\rho H_z)}{\partial \phi} - \frac{\partial H_\phi}{\partial z} \right)\hat{\rho} + \left(\frac{\partial H_\rho}{\partial z} - \frac{\partial H_z}{\partial \rho} \right)\hat{\phi} + \frac{1}{\rho} \left(\frac{\partial (\rho H_\phi)}{\partial \rho} - \frac{\partial H_\rho}{\partial \phi} \right)\hat{z} Esfericas \nabla \times \vec{H} = \frac{1}{r \sin \theta} \left(\frac{\partial (H_\phi \sin \theta)}{\partial \theta} - \frac{\partial H_\theta}{\partial \phi} \right)\hat{r} + \frac{1}{r} \left(\frac{1}{\sin \theta} \frac{\partial H_r}{\partial \phi} - \frac{\partial (r H_\phi)}{\partial r} \right)\hat{\theta} + \frac{1}{r} \left(\frac{\partial (r H_\theta)}{\partial r} - \frac{\partial H_r}{\partial \theta} \right)\hat{\phi} c) \vec{H} = \frac{I (c^2 - r^2)}{2 \pi r (c^2 - B^2)} \hat{\phi} \quad B \le r \le C \nabla \times \vec{H} = \frac{1}{r} \frac{\partial }{\partial r} \left(\frac{I (c^2 - r^2)}{2 \pi r (c^2 - B^2)} \right) \hat{a_z} = 0 \nabla \times \vec{H} = \left(-\frac{1}{r} \frac{I r^2}{2 \pi (c^2 - B^2)} \right) \hat{a_z} \nabla \times \vec{H} = -\frac{I}{\pi (c^2 - B^2)} \hat{a_z} \quad B \le r \le C "VIDE CABO COAXIAL" Φ = ∫ B⃗ ⋅ dS⃗ = ∫ (μ₀I / 2πr) âϕ ⋅ drdz aϕ̂ sup aϕ̂ sur Φ = ∫∫ (μ₀I / 2πr) dr dz L = {RA z = 0 Φ = (μ₀I / 2π) h ln (RB / RA) [wb]