{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

309-2008-Solutions9

309-2008-Solutions9 - ECE 309 Electromagnetic Fields...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 309 — Electromagnetic Fields University of Virginia Fall 2008 Homework # 9 Solutions 1. From either Gauss’s Law or Laplace’s equation, the field between the inner and outer con- ductors of a cylindrical capacitor is, ~ E = V r ln( b/a ) ˆ r where V is the voltage across the capacitor, a is the inner radius and b is the outer radius. The displacement current density is thus, ~ J d = ~ E ∂t ~ J d = V r ln( b/a ) where the second relation above holds for phasor representations of ~ J d and V . The total current flowing across a cylindrical surface of radius r is thus, I d = V ln( b/a ) Z 2 π 0 Z 0 rdφdz r = 2 π‘ V ln( b/a ) which is obviously independent of r . The capacitance of the coaxial capacitor is given by, C = Q V = λ‘ V = 2 π ‘ ln( b/a ) where λ is the charge per unit length on the coaxial capacitor. Thus, the charging current for the capacitor is given (from circuit theory) by, I c = C dV dt I c = jωCV = 2 π ‘ V ln( b/a ) which is the same as the displacement current found above. 2. Given the electric field, ~ E = ˆ xA ( x + y ) + ˆ yB ( x - y ) cos ωt
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
is in a source-free medium, we know that it is divergenceless, ~ ∇ · ~ E = 0 . Thus, ~ ∇ · ~ E = ∂E x ∂x + ∂E y ∂y + ∂E z ∂z = A - B = 0 Thus, the relation between the coefficients is, A = B 3. To find the ~ H -field, let’s first express the electric field in phasor form, ~ E = ˆ yE 0 sin πx a e - z z ( * ) From Faraday’s Law, ~ ∇ × ~ E = - jωμ ~ H, ~ H = j ωμ ~ ∇ × ~ E so that ~ H = j ωμ h - ˆ x ∂E y ∂z + ˆ z ∂E y ∂x i = j ωμ h z E 0 sin πx a e - z z ˆ x + π a E 0 cos πx a e - z z i ˆ z = - β z ωμ E 0 sin πx a e - z z ˆ x + j π ωμa E 0 cos πx a e - z z ˆ z To find the instantaneous field, we multiply by exp( jωt )
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern