10/6/2020 Electromagnetics: Electromagnetic Field Theory The Del Operator & Field Operations Outline • Scalar vs. Vector Fields • The Del Operator ∇ • Gradient of a Scalar Field ∇ • Divergence of a Vector Field ∇ · ⃗ • Curl of a Vector Field ∇⃗ • Laplacian Operation ∇ Slide 2 1 10/6/2020 Scalar vs. Vector Fields Slide 3 Scalar Field Vs. Vector Field (2D) Scalar Field , , Vector Field ⃗ , , • Magnitude only • Magnitude • Direction Slide 4 2 10/6/2020 Scalar Field Vs. Vector Field (3D) Scalar Field , , Vector Field ⃗ , , • Magnitude only • Magnitude • Direction Slide 5 Isocontour Lines Isocontour lines trace the paths of equal value. Closely space isocontours conveys that the function is varying rapidly. Slide 6 3 10/6/2020 The Del Operator Slide 7 The Del Operator ∇ The del operator is the vector differential operator. It is sort of a 3D derivative. Even though is a Coordinate Del Operator vector, it is never System written as ∇. Cartesian aaaˆˆˆ x x yzyz 1 Cylindrical aaaˆˆˆ z z Derived from the Cartesian equation using coordinate 11 Spherical aaˆˆ a ˆtransformation. rrr rsin Slide 8 4 10/6/2020 Summary of Vector Operations Operation Input Output Vector Addition & Subtraction Vectors Vector UV Dot ProductUV Vectors Scalar Cross ProductUV Vectors Vector Gradientf Scalar Function Vector Function DivergenceU Vector Function Scalar Function CurlU Vector Function Vector Function Scalar Laplacian2V Scalar Function Scalar Function Vector Laplacian2U Vector Function Vector Function Slide 9 Gradient of a Scalar Field Slide 10 5 10/6/2020 Gradient of a Scalar Field (1 of 3) Start with a scalar field , . Slide 11 Gradient of a Scalar Field (2 of 3) Plot the gradient ∇ , , on top of , , . The background color is the original scalar field , , . Slide 12 6 10/6/2020 Gradient of a Scalar Field (3 of 3) The gradient will always be perpendicular to the isocontour lines. Slide 13 The Gradient ∇ The gradient calculates how rapidly, and in what direction, a scalar function is increasing. Coordinate Gradient Operator System VVV Cartesian Vaˆˆˆ a a x x yzyz VVV1 Cylindrical Vaˆˆˆ a a z z VV11 V Spherical Vaˆˆ a a ˆ rrr rsin Slide 14 7 10/6/2020 Algebra Rules for the Gradient UV U V Sum/Difference Rule UV U V V U Product Rule UVUUV 2 Quotient Rule VV VnVVnn1 Power Rule Slide 15 Properties of the Gradient 1. The gradient of a scalar function is a vector function. 2. The magnitude of V is the local maximum rate of change in V. 3. V points in the direction of maximum rate of change in V. 4. V at any point is perpendicular to the constant V surface that passes through that point. 5. V points toward increasing numbers in V. Slide 16 8 10/6/2020 Divergence of a Vector Field Slide 17 Divergence of a Vector Field (1 of 2) Start with the following vector field ⃗ , . Observe how the field seems to be converging to point in the upper left and diverging from a point in the low right. Slide 18 9 10/6/2020 Divergence of a Vector Field (2 of 2) The divergence is ∇ · ⃗ , . Divergence measures the tendency of a vector field to diverge from a point or converge to a point. Divergence is a scalar quantity and is shown as the colored regions. Blue indicates negative divergence (convergence) and red indicates positive divergence. Slide 19 Divergence of a Vector Field (2D) Vector Field ⃗ , Divergence ∇ · ⃗ , Slide 20 10 10/6/2020 Divergence ∇ · ⃗ The divergence of a vector field is a scalar field that measures the tendency of a vector field to diverge from a point or converge to a point. Coordinate Gradient Operation System A Ay A Cartesian A x z xyz 11 A A A Cylindrical A z z 2 11 rAr A sin 1A Spherical A rrr2 sin r sin Slide 21 Algebra Rules for Divergence A BAB Distributive Rule Slide 22 11 10/6/2020 Properties of Divergence ∇ · ⃗ 1. The divergence of a vector function is a scalar function. 2. The divergence of a scalar field does not make sense. 3. The original vector field will point form larger to smaller numbers in the scalar field. Slide 23 Curl of a Vector Field Slide 24 12 10/6/2020 Curl of a Vector Field (1 of 2) Start with the following vector field ⃗ , . Observe how the field seems to be rotating around two different axes. Slide 25 Curl of a Vector Field (1 of 2) The curl is ∇⃗ , . Curl measures the tendency of the vector field to circulate around an axis. Curl is a vector quantity and is shown as the blue arrows. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation. Slide 26 13 10/6/2020 Curl of a Vector Field (2D) Vector Field ⃗ , Curl ∇⃗ , y x z y x z Slide 27 Curl The curl of a vector function is a vector function that quantifies the tendency of the vector field to circulate around an axis. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation. Coordinate Curl Operation System AA AAzzyyAAxx A aaaˆˆˆxyz Cartesian yz zx xy 11AAAA A A A zz aaˆˆ a ˆ Cylindrical z zz 1111ArAsin A AA rA A aaaˆˆˆ rr Spherical r r rrrrrsin sin Slide 28 14 10/6/2020 Algebra Rules for Curl A BAB Distributive Rule A B A BB A B AA B Slide 29 Properties of Curl 1. The curl of a vector function is a vector function. 2. The curl of a scalar field does not make sense. 3. Curl follows the right‐hand rule. A 4. The divergence of the curl of a vector field is always zero. ∇ · ∇⃗ 0 5. The curl of the gradient of a scalar field is always zero. ∇ ∇ 0 6. Note that ∇ · ⃗ ⃗ · ∇. ∇ · ⃗ calculates the ⃗ ⃗ derivative of , whereas · ∇ sets up A a derivative operation that is scaled by ⃗. Slide 30 15 10/6/2020 Laplacian Operation Slide 31 Scalar Laplacian ∇ The scalar Laplacian is defined as the divergence of the gradient. It is 2 sort of a measure of the tendency of the scalar function to form bowls. VV Coordinate Scalar Laplacian Operation System 222VVV 2V Cartesian x222yz 22 2 11VVV Cylindrical V 22 2 z 2 2211VVV 1 Spherical Vr22 sin 222 rr r rsin r sin Slide 32 16 10/6/2020 Visualization of Scalar Laplacian ∇ 2 Vx, y Vx , y Slide 33 Vector Laplacian ∇⃗ The vector Laplacian is defined as the 2 AA A gradient of the divergence minus the curl. Coordinate Scalar Laplacian Operation System 22 2 2 Cartesian A Aax ˆˆˆxyyzz Aa Aa 22 2 2 Cylindrical A Aaˆˆˆ Aa Aaz z 22 2 2 Spherical A Aarrˆˆˆ Aa Aa Slide 34 17 10/6/2020 Visualization of Vector Laplacian ∇⃗ Axy, 2 Axy, Slide 35 Properties of the Laplacian 1. The Laplacian of a scalar function is a scalar function. 2. The scalar Laplacian arises when dealing with electric potential. 3. The Laplacian of a vector function is a vector function. 4. The vector Laplacian arises in the wave equation. Slide 36 18 10/6/2020 Classification of Vector Fields Slide 37 Irrotational A vector field is irrotational if A 0 Examples of Irrotational Fields Gravity Electric potential Irrotational Rotational Consequences: 1. Irrotational fields must flow in essentially straight lines. 2. If A 0, then Ad 0 and the field is said to be conservative. L Slide 38 19 10/6/2020 Solenoidal (Divergenceless) A vector field is solenoidal if A 0 Examples of Solenoidal Fields Fluid flows Magnetic fields Consequences: 1. If the vector function does not diverge from any points, it must form loops. 2. If A 0, then Ads0 and the field will have zero net flux. S Slide 39 Laplacian Vector Fields A vector field is Laplacian if it is both irrotational and solenoidal A 0 A 0 Slide 40 20.