Lecture Notes on The Divergence - Electricity and Magnetism I | PHY 481, Study notes of Physics

Download Lecture Notes on The Divergence - Electricity and Magnetism I | PHY 481 and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 3 Sections 2.2-2.5 of Pollack and Stump (PS) The Divergence The divergence is the dot product of the gradient operator and a vector function, ~F = (Fx, Fy, Fz), so that ~∇ · ~F = ∂Fx ∂x + ∂Fy ∂y + ∂Fz ∂z (1) We want to find a co-ordinate independent representation of the divergence, which we achieve by considering a small cube of dimension ǫ. Now consider an integral of the flux through this surface, that is, ∮ dS ~F · n̂dA = 3 ∑ i=1 [Fi(~x + ǫêi/2) − Fi(~x − ǫêi/2)]ǫ 2 (2) where (ê1, ê2, ê3) = (̂i, ĵ, k̂) are introduced to simplify the expression. Eq. (22) reduces to, ∮ dS ~F · n̂dA = 3 ∑ i=1 ∂Fi(~x) ∂xi ǫ3 = (~∇ · ~F )ǫ3 (3) The co-ordinate independent representation of the divergence of a vector function is then, ~∇ · ~F = limV→0 1 V ∮ dS ~F · d ~A (4) The divergence is then proportional to the flux of the function, ~F though the surface of the volume V . The Laplacian The Laplacian ∇2 is a scalar operator found by taking the dot product of the gradient operator with itself, ie., ∇ 2 = ~∇ · ~∇; in cartesion co − ordinates ∇2 = ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 (5) The Curl The curl of a vector function, ~∇ ∧ ~F is defined in the same way as the cross product ~∇∧ ~F = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ î ĵ k̂ ∂x ∂y ∂z Fx Fy Fz ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ( ∂Fz ∂y − ∂Fy ∂z , ∂Fx ∂z − ∂Fz ∂x , ∂Fy ∂x − ∂Fx ∂y ) (6) 1 Using Levi-Civita and suffix notation, this is, (~∇∧ ~F )i = εijk ∂Fk ∂xj (7) To find a co-ordinate independent representation, consider a square loop placed in the x-y plane, with edge length ǫ. Consider a path integral around the loop, the circulation, ∮ loop ~F · d~l = ǫFi(~x − ǫêj/2) + ǫFj(~x + ǫêi/2) − ǫFi(~x + ǫêj/2) − ǫFj(~x − ǫêi/2) (8) This reduces to, ∮ loop ~F · d~l = ( ∂Fj ∂xi − ∂Fi ∂xj ) = (~∇∧ ~F )kǫ 2 (9) The general co-ordinate independent form of ~∇∧ ~F is then n̂ · (~∇∧ ~F ) = limA→0 1 A ∮ C ~F · d~l (10) F. Derivative operator identities Table 2.2 of PS gives a list of indentities for derivative operators. We will look at a couple of interesting examples and in the homework you will need to use both these identities and the vector identities in Table 2.1 of the text. Identity. ~∇∧ (~∇f) = 0. Proof: (~∇∧ ~∇f)i = εijk ∂2f ∂xj∂xk . Because there is a sum over j, k pairs of terms appear with opposite signs, due to the assymmetry of the Levi-Civita tensor. Summing these pairs of terms gives zero. Identity. ~∇ · (~∇ ∧ ~F ) = 0. Proof: Using Levi-Civita notation, the LHS is εijk ∂2Fk ∂xi∂xj . Again the terms on the RHS can be collected into pairs with opposite signs due to the antisymmetric nature of the Levi-Civita tensor. The identity is then proved. G. Summary of key geometrical concepts This has been a busy lecture with many mathematical details. If you have not seen these mathematical tools before, it will take some time for you to become familiar with them. However the most important lessons to take from these tools is the key concepts that enable insight into the physics behind the math. Here are four key concepts. 1. The cross product ~A ∧ ~B is perpendicular to both ~A and ~B. 2. The definition of the gradient through df = d~x · ~∇f demonstrates that ~∇f is perpen- diclar to equipotentials of the function f . 2