5)Gradient We use Gradient to describe the change of a scalar quantity along an infinitesimal distance of dl, which is df=V·dl the operator is defined by 00:0 V=i-+ Cx ay Xk OZ 6)Divergence We use Divergence to describe flux of a vector quantity for an element volume dv. which is dop=v. adv
5) Gradient We use Gradient to describe the change of a scalar quantity along an infinitesimal distance of , which is, dl df f dl = • the operator is defined by , z k y j x i + + = 6) Divergence We use Divergence to describe flux of a vector quantity for an element volume dv, which is, d Adv = •
cUrl We use curl describe line integral along an infinitesimal small path of a vector which is Aodl=(V×A)·dl verysmalldosepath Where i j k 000 V×A= ax ay ox 8)Laplace operator We use the Laplace operator to describe the divergent of gradient of A scalar, which is V·Vf=V2f=△ 02f.02f,02f ax av 02
7)Curl We use curl describe line integral along an infinitesimal small path of a vector which is, A dl A da verysmallclosepath • = • ( ) Ax Ay Az x y x i j k A = Where, 8) Laplace operator We use the Laplace operator to describe the divergent of gradient of A scalar, which is , 2 2 2 2 2 2 2 z f y f x f f f f + + • = = =
9)Stoke's formula Stoke's theorem gives relationship between the line integral of a vector along any size of closed loop and the curl of the vector, which is Adl=|(V×A)odl 1. 3 Electric Potential 1.3. 1 The proof of the existence of the electric potential The electric field is a special case of the stoke's formula El=(V×E)odl From Section 1.2, For A Stationary charge, E is purely function of r Therefore V×E=0
9) Stoke’s formula Stoke’s theorem gives relationship between the line integral of a vector along any size of closed loop and the curl of the vector, which is • = • C s E dl E da ( ) 1.3 Electric Potential 1.3.1 The proof of the existence of the electric potential The electric field is a special case of the Stoke’s formula • = • C s A dl A da ( ) From Section 1.2, For A Stationary charge, E is purely function of r . Therefore, E = 0
Discussion: 1)The curl of electric field is null.(understand this point 2)It is only true for stationary field. ( illustrate this point) Prove your self that E=Vv Discussion: boundary condition and uniqueness theorem 1.3.2 The great convenience of introduction of V Superposition of scalar is much easier than that of vector! The new method to study the electric field Step 1: superposition of potential Step 2: Differential V to obtain the solution of field 1.2.2 V produced by different shape of charges ∑ Q 1)Point shape charge 2)Linear shape charge a(r 4兀
Discussion: 1)The curl of electric field is null. (understand this point) 2) It is only true for stationary field. (illustrate this point) Prove your self that E = −V 1.3.2 The great convenience of introduction of V Superposition of scalar is much easier than that of vector!! The new method to study the electric field: Step 1: superposition of potential V Step 2: Differential V to obtain the solution of field 1.2.2 V produced by different shape of charges dl r r V l = ( ) 4 1 0 2) Linear shape charge: 1)Point shape charge: = = n i i i r Q V 1 Discussion: boundary condition and uniqueness theorem
3. Surface shape as 4兀Ea 4. Volume shape D(r 4丌 1. 4 EXamples of electric field 1.4. 1 Electric dipole P(X, y, Z) The potential of dipole at far distance 0 Q Q X
3. Surface shape: ds r V o s r ( ) 4 1 = 4. Volume shape: dv r V o v r ( ) 4 1 = 1.4 Examples of electric field 1.4.1 Electric dipole The potential of dipole at far distance Q -Q P(x,y,z) x y z r r1 r2 l/2 θ