9/28/2009. t kz H a x. in free space. find the value(s) of k such that E satisfies both of Maxwell s curl equations.

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1 9//9 3- E3.1 For E E cos 6 1 tkz a in free space,, J=, find the value(s) of k such that E satisfies both of Mawell s curl equations. Noting that E E (z,t)a,we have from B E, t 3-1 a a a z B E t z E B E t z E cos 6 1 tkz z ke sin 6 1 t kz ke 6 1 B cos 6 1 tkz Thus, ke B 6 1 B B H cos 6 1 t 7 ke cos 6 1 tkz 4 Then, noting that D H, t kz H H (z,t)a, we have from a a 1

2 9//9 3-3 D H t a a a z z H D t H z ke sin 6 1 t 4 kz 3-4 ke cos t kz D ke cos t D E ke cos 6 1 t 4 D kz a kz a 3-5 Comparing with the original given E, we have E ke 4 k E E cos 6 1 t z Sinusoidal traveling waves in free space, propagating in the z directions with velocit, 3 1 ( c) ms. a

3 9//9 3-6 E Review Questions Discuss the applicabilit of integral forms of Mawell s equations versus that of the differential forms for obtaining the solutions for the fields. 3.. State Farada s law in differential form for the special case of E = E (z, t)a and H = H (z, t)a. How is it derived from Farada s law in integral form? How would ou derive Farada s law in differential form from its integral form for the general case of an arbitrar electric field? 3.4. What is meant b the net right-lateral differential of the - and - components of a vector normal to the z- direction? Give an eample in which the net right-lateral differential of E and E normal to the z-direction is zero, although the individual derivatives are nonzero What is the determinant epansion for the curl of a vector in Cartesian coordinates? 3

4 9//9 3-9 Review Questions (Continued) 3.6. State Ampere s circuital law in differential form for the general case of an arbitrar magnetic field. How is it obtained from its integral form? 3.7. State Ampere s circuital law in differential form for the special case of H = H (z, t)a. How is it derived from Ampere s circuital law for the general case in differential form? 3.. If a pair of E and B at a point satisfies Farada s law in differential form, does it necessaril follow that it also satisfies Ampere s circuital form and vice versa? 3.9. Discuss the determination of magnetic field for one dimensional current distributions, in the static case, using Ampere s circuital law in differential form, without the displacement current densit term Gauss Laws and the Continuit Equation 3-11 GAUSS LAW FOR THE ELECTRIC FIELD D ds dv S V Δz z (,, z) Δ Δ D z D z D z D z D D z zz z z z 4

5 9//9 Lim z z 3-1 D D z D D ΔzΔ +Δ D D z Lim z z zz z z z 3-13 D Dz z Longitudinal derivatives of the components of D The quantit on the left side is the net longitudinal differential of the components of D, that t is, the algebraic sum of the derivatives of components of D along their respective directions. It can be written as D, which is known as the divergence of D. Thus, the equation becomes D 3-14 The figure below illustrates the case of (a) zero value, and (b) nonzero value for. D z D D z D z D D D z D z D (a) (b) 5

6 9//9 E3.3 Given that Find D everwhere for a a otherwise = a = =a Noting that = () and hence D = D(), we set and, so that z D D D Dz z 3-16 D Thus, D= gives ( ) which also means that D has onl an - component. Proceeding further, we have D dc where C is the constant tof fintegration. ti Evaluating the integral graphicall, we have the following: () d a a a a a 3-17 From smmetr considerations, the fields on the two sides of the charge distribution must be equal in magnitude and opposite in direction. Hence, C = a aa for a D a for a a aa for a a D a a a 6

7 9//9 3-1 GAUSS LAW FOR THE MAGNETIC FIELD D ds = dv S V D = From analog B ds == dv S V B = B Solenoidal propert of magnetic field lines. Provides test for phsical realizabilit of a given vector field as a magnetic field LAW OF CONSERVATION OF CHARGE J ds + dt d S V dv = J + ( ) = t J t Continuit Equation 3- SUMMARY B E (1) t D H J () t (3) D B (4) J (5) t (4) is, however, not independent of (1), and (3) can be derived from () with the aid of (5). 7

8 9//9 3-1 The interdependence of fields and sources through Mawell s equations J Law of Conservation of Charge (5) + + Ampere s Circuital Law () H,B Farada s Law (1) Gauss Law for E (3) D,E Review Questions State Gauss law for the electric field in differential form. How is it derived from its integral form? What is meant b the net longitudinal differential of the components of a vector field? Give an eample in which the net longitudinal differential of the components of a vector field is zero, although h the individual derivatives are nonzero What is the epression for the divergence of a vector in Cartesian coordinates? Discuss the determination of electric field for one dimensional charge distributions, in the static case, using Gauss law for the electric field in differential form. 3-3 Review Questions (Continued) State Gauss law for the magnetic field in differential form. How is it obtained from its integral form? How can ou determine if a given vector field can be realized as a magnetic field? State the continuit equation Summarize Mawell s equations in differential form and the continuit equation, stating which of the equations are independent Discuss the interdependence of fields and sources through Mawell s equations.