Gauss' Law and Divergence Theorem PDF

Gauss’ Law and Divergence Theorem Course: Electromagnetic Fields Academic Year: 2023/2024 Lecturer: Dr. Mohamed A. El-Shimy Textbook: William Hayt, ‘Engineering Electromagnetics’, Ed. 8. 1. Electric Flux 1.1. Electric Flux Lines (𝝍) It is a scalar field where 𝑄 coulombs of charges produce 𝜓(=...

Gauss’ Law and Divergence Theorem Course: Electromagnetic Fields Academic Year: 2023/2024 Lecturer: Dr. Mohamed A. El-Shimy Textbook: William Hayt, ‘Engineering Electromagnetics’, Ed. 8. 1. Electric Flux 1.1. Electric Flux Lines (𝝍) It is a scalar field where 𝑄 coulombs of charges produce 𝜓(= 𝑄) lines of electric flux. Since there is a direct proportionality between the electric flux and the produced charge with a unity constant, then ψ=𝑄 and the electric flux is measure in coulombs. 1.2. Electric Flux Density (𝐃) It is a vector field where the direction of 𝐃 at a point is the direction of the flux lines at that point, and the magnitude is given by the number of flux lines crossing a surface normal to the lines divided by the surface area. It measured in ‘coulombs per square meter’ or ‘lines per square meter’. The flux density for an amount of flux 𝑑ψ crosses a differential area 𝑑𝑆 normal to its direction: 𝑑ψ 𝐃= 𝒂𝛙 , [C/m2 ] 𝑑𝑆 If, at the surface element 𝑑𝑆, 𝐃 makes an angle 𝜃 with the normal, then the differential flux crossing 𝑑𝑆 is 𝑑ψ = 𝐷 𝑑𝑆 cos 𝜃 = 𝐃 ∙ 𝑑𝐒 1.3. Relation Between Electric Flux Density (𝐃) and Field Intensity (𝐄) A point charge 𝑄 produces 𝜓 = 𝑄 lines of flux directed outward from the point. The electric flux density at a point 𝑟 for the flux passes symmetrically through an imaginary spherical surface of area 4𝜋𝑟 2 : 𝑄 𝐃= 𝐚 4𝜋𝑟 2 𝐫 Recall that, the radial electric field intensity of a point charge in free space, 𝑄 𝐄= 𝐚 4𝜋𝜖𝑜 𝑟 2 𝐫 Therefore, in free-space, 𝐃 = 𝜖𝑜 𝐄 2. Gauss’ Law The total electric flux passing out through any imaginary closed surface is equal to the total charge enclosed by that surface (often called a gaussian surface). ∮ 𝐃 ∙ 𝑑𝐒 = 𝑄𝑒𝑛𝑐 Special Gaussian Surfaces To simplify the integration in the law, a closed surface is chosen to satisfy two conditions: o 𝐃 is either normal or tangential to surface. o 𝐃 is constant over the portion of the surface for 𝐃 ∙ 𝑑𝐒 is not zero. Enclosed Charge Charge Distribution Total Charge Point Charge 𝑄 Multiple Point Charge 𝑄 = ∑𝑄𝑛 Line Charge 𝑄 = ∫ 𝜌𝐿 𝑑𝐿 Surface Charge 𝑄 = ∫𝜌𝑆 𝑑𝑆 𝑆 Volume Charge 𝑄 = ∫ 𝜌𝑣 𝑑𝑣 𝑣𝑜𝑙 3. Application of Gauss’ Law 3.1. Point Charge A point charge 𝑄 is placed at the origin of a spherical coordinates. The gaussian surface is a spherical surface, centered at the origin of any radius 𝑟 where 𝐷 is normal and constant at all points on the surface. 𝑄 = ∮ 𝐃 ∙ 𝑑𝐒 𝜙=2𝜋 𝜃=𝜋 = 𝐷∫ ∫ 𝑟 2 sin 𝜃 𝑑𝜃 𝑑𝜙 𝜙=0 𝜃=0 2 = 4𝜋𝑟 𝐷 Therefore, 𝑄 𝑄 𝐃= 𝐚, 𝑬= 𝐚 4𝜋𝑟 2 𝐫 4𝜋𝜖𝑜 𝑟 2 𝐫 3.2. Infinite Line Charge Consider a uniform line charge distribution 𝜌𝐿 lying along the 𝑧 axis and extending from −∞ to +∞. Using cylindrical coordinates, the only radial component of 𝐃 is present 𝐃 = 𝐷𝜌 𝐚𝛒 where 𝐷𝜌 is function of 𝜌 only. The chosen gaussian surface is a cylindrical surface along the 𝑧 axis of radius 𝜌 closed by plane surfaces normal to the 𝑧 axis at 𝑧 = 0 and 𝑧 = 𝐿. By applying Gauss’ Law, 𝑄 = ∮ 𝐃 ∙ 𝑑𝐒 = ∫ 𝐷𝜌 𝑑𝑆 + ∫ 0 𝑑𝑆 + ∫ 0 𝑑𝑆 𝑠𝑖𝑑𝑒𝑠 𝑡𝑜𝑝 𝑏𝑜𝑡𝑡𝑜𝑚 𝐿 2𝜋 = 𝐷𝜌 ∫ ∫ 𝜌 𝑑𝜙 𝑑𝑧 𝑧=0 𝜙=0 = 𝐷𝜌 2𝜋 𝜌 𝐿 The total charge enclosed is 𝑄 = 𝜌𝐿 𝐿. Therefore, 𝜌𝐿 𝜌𝐿 𝐃= 𝐚 , 𝑬= 𝐚 2𝜋𝜌 𝛒 2𝜋𝜖𝑜 𝜌 𝛒 3.3. Infinite Plane of Sheet Consider an extent plane of sheet with a constant surface charge density 𝜌𝑠 C/m2. The gaussian surface is chosen as shown in the figure where the flux is normal to the top and the bottom surfaces and tangent to the sides. By applying the Gauss’ law, 𝑄 = ∮ 𝐃 ∙ 𝑑𝐒 = ∫ 0 𝑑𝑆 + ∫ 𝐷𝑧 𝑑𝑆 + ∫ 𝐷𝑧 𝑑𝑆 𝑠𝑖𝑑𝑒𝑠 𝑡𝑜𝑝 𝑏𝑜𝑡𝑡𝑜𝑚 𝜌𝑠 𝐴 = 2 𝐷 𝐴 where 𝐴 is the area. Therefore, 𝜌𝑠 C 𝜌𝑠 𝐃 = 𝐚n [ 2 ] , 𝐄= 𝐚 [V/m] 2 m 2𝜖𝑜 𝐧 Example Two infinite parallel-plate capacitor located at 𝑥𝑧-plane. The plates are uniformly charged where the left plate is on 𝑦 = −1 with a surface charge density 𝜌𝑠 and the right plate is on 𝑦 = 1 with charge density −𝜌𝑠. Determine 𝐄 in the regions: (a) 𝑦 < −1, (b) −1 < 𝑦 < +1, and (c) 𝑦 > +1. Ans. (a) 0, (b) 𝜌𝑠 /𝜖𝑜 𝐚𝐲 , (c) 0 4. Divergence 4.1. Differential Volume For a problem that does not posses any symmetric at all, the small change in 𝐃 at a point can be evaluated by applying the Gauss’ law for a very small gaussian surface chosen such that 𝐃 is almost constant over the surface. Consider the gaussian surface be a small cuboid with edges Δ𝑥, Δ𝑦, and Δ𝑧 as shown in the figure. The flux density 𝐃 at point 𝑃 (at the corner of the cube) is defined as: 𝐃 = 𝐷𝑥 𝐚𝐱 + 𝐷𝑦 𝐚𝐲 + 𝐷𝑧 𝐚𝐳 By applying Gauss’ law for an enclosed charge 𝑄, ∮ 𝐃 ∙ 𝑑𝐒 = 𝑄  The integration is over each face of the closed surface where for any two parallel faces only one component of 𝐃 will cross normally while the others will be tangent. Thus, ∮ =∫ +∫ +∫ +∫ +∫ +∫ front back left right top bottom o ∫back = − 𝐷𝑥 Δ𝑦 Δ𝑧 𝜕𝐷𝑥 o ∫front ≈ 𝐷𝑥+Δ𝑥/2 Δ𝑦 Δ𝑧 = [𝐷𝑥 + Δ𝑥] Δ𝑦 Δ𝑧 𝜕𝑥 o ∫left = − 𝐷𝑦 Δ𝑥 Δ𝑧 𝜕𝐷𝑦 o ∫right ≈ 𝐷𝑦+Δ𝑦 Δx Δ𝑧 = [𝐷𝑦 + Δ𝑦] Δ𝑥 Δ𝑧 𝜕𝑦 o ∫top = − 𝐷𝑧 Δx Δ𝑦 𝜕𝐷𝑧 o ∫bottom ≈ 𝐷𝑧+Δ𝑧 Δx Δ𝑦 = [𝐷𝑧 + Δ𝑧] Δ𝑥 Δ𝑦 𝜕𝑧 By combining all the results, we get, 𝜕𝐷𝑥 𝜕𝐷𝑦 𝜕𝐷𝑧 ∮ 𝐃 ∙ 𝑑𝐒 = 𝑄 ≈ ( + + ) Δ𝑥 Δ𝑦 Δ𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 4.2. Divergence of 𝐃 The divergence of the vector flux density 𝐃 is defined as the outflow of flux from a small, closed surface per unit volume as the volume shrinks to zero. Dividing the Gauss’ law by the volume element Δ𝑣 and letting Δ𝑣 → 0, we obtain ∮ 𝐃 ∙ 𝑑𝐒 𝑄 div 𝐃 = lim = lim = 𝜌𝑣 Δ𝑣→0 Δ𝑣 Δ𝑣→0 Δ𝑣 where 𝜌𝑣 is the volume charge density. Noted that: o Positive divergence indicates a source at that point. o Negative divergence indicates a sink. o Zero divergence indicates no source or sink exists. Thus, for a differential volume unit Δ𝑣 = Δ𝑥 Δ𝑦 Δ𝑧 in rectangular coordinates 𝜕𝐷𝑥 𝜕𝐷𝑦 𝜕𝐷𝑧 div 𝐃 = ( + + ) (rectangular) 𝜕𝑥 𝜕𝑦 𝜕𝑧 By expressing 𝐃 in cylindrical coordinates of a differential volume 𝜌 𝑑𝜌 𝑑𝜙 𝑑𝑧, it can be shown that 1 𝜕(𝜌𝐷𝜌 ) 1 𝜕𝐷𝜙 𝜕𝐷𝑧 div 𝐃 = + + (cylindrical) 𝜌 𝜕𝑥 𝜌 𝜕𝜙 𝜕𝑧 Also, in spherical coordinates of a differential volume 𝑟 2 sin 𝜃 𝑑𝑟 𝑑𝜃 𝑑𝜙, it can be shown that 1 𝜕(𝑟 2 𝐷𝑟 ) 1 𝜕(sin 𝜃 𝐷𝜃 ) 1 𝜕𝐷𝜙 div 𝐃 = 2 + + (spherical) 𝑟 𝜕𝑟 𝑟 sin 𝜃 𝜕𝜃 𝑟 sin 𝜃 𝜕𝜙 4.3. The Del Operator (𝛁) It is a vector operator defined in Cartesian coordinates by 𝜕( ) 𝜕( ) 𝜕( ) 𝛁= 𝐚𝐱 + 𝐚𝐲 + 𝐚 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝐳 Then 𝜕𝐷𝑥 𝜕𝐷𝑦 𝜕𝐷𝑧 div 𝐃 = + + 𝜕𝑥 𝜕𝑦 𝜕𝑧 = 𝛁 ⋅ (𝐷𝑥 𝐚𝐱 + 𝐷𝑦 𝐚𝐲 + 𝐷𝑧 𝐚𝐳 ) = 𝛁 ⋅ 𝐃 4.4. Divergence Theorem From Gauss’ law, ∮ 𝐃 ∙ 𝑑𝐒 = 𝑄 = ∫ 𝜌𝑣 𝑑𝑣 𝑣𝑜𝑙 And since 𝜌𝑣 = div 𝐃 = 𝛁 ⋅ 𝐃, then ∮ 𝐃 ∙ 𝑑𝐒 = ∫ 𝛁 ⋅ 𝐃 𝑑𝑣 vol The divergence theorem states that the total flux crossing the closed surface is equal to the integral of the divergence of the flux density throughout the enclosed volume. 5. Maxwell’s First Equation Basic concepts to all electromagnetic theory can be defined by Maxwell’s four equations. They apply to electrostatics and steady magnetic fields. The Maxwell’s first equation: Integral Form Point Form ∮ 𝐃 ∙ 𝑑𝐒 = ∫ 𝜌𝑣 𝑑𝑣 𝛁 ⋅ 𝐃 = div 𝐃 = 𝜌𝑣 𝑣𝑜𝑙