Electromagnetic theory Required boundary condition from (a) is E(parallel left) = E(parallel right) 2020 Q2 b
Posted: Mon May 02, 2022 4:17 pm
Electromagnetic theory
Required boundary condition from (a) is E(parallel left) =
E(parallel right)
2020 Q2 b
(b) Electromagnetic waves with amplitude E, propagating at angle o with respect to the z direction and with vacuum wavenumber, k, are confined between a pair of perfectly conducting, infinite sheets separated by a distance, b, as shown in the diagram below. The vertical component of the electric field and horizontal component of magnetic induction field between the plates are Ex = Re{2E, i sin(ky sin 8)et(wt – kgz)} By , 2wE. Ref: kg i sin(ky sin ) e'(wt – kg2)} XAz
(i) Impose the boundary condition stated in part (a) on the E field at the conducting sheet surfaces. Hence obtain expressions for allowed envelope functions, sin(ky sin o), for the E and B fields. [3 marks] = = (ii) The propagation wavenumber inside the sheets is kg = k cos 0. Using this, show that kg2 = k2 – (na/b)2 and that the group velocity of the wave is do/dkg = c kg/k. [4 marks] = (iii) The value of b2 for a waveguide is 9/76 m2. Calculate the power transmitted by the waveguide in the n = 1 mode if E. = 1 Vm-1 and = 210 x109 rad s-1. Hint: use the Poynting vector, E x H. At what frequency does the guide begin to conduct in the n = 2 mode? [8 marks]
- Electromagnetic Theory Required Boundary Condition From A Is E Parallel Left E Parallel Right 2020 Q2 B 1 (79.18 KiB) Viewed 47 times
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E(parallel right)
2020 Q2 b
(b) Electromagnetic waves with amplitude E, propagating at angle o with respect to the z direction and with vacuum wavenumber, k, are confined between a pair of perfectly conducting, infinite sheets separated by a distance, b, as shown in the diagram below. The vertical component of the electric field and horizontal component of magnetic induction field between the plates are Ex = Re{2E, i sin(ky sin 8)et(wt – kgz)} By , 2wE. Ref: kg i sin(ky sin ) e'(wt – kg2)} XAz
(i) Impose the boundary condition stated in part (a) on the E field at the conducting sheet surfaces. Hence obtain expressions for allowed envelope functions, sin(ky sin o), for the E and B fields. [3 marks] = = (ii) The propagation wavenumber inside the sheets is kg = k cos 0. Using this, show that kg2 = k2 – (na/b)2 and that the group velocity of the wave is do/dkg = c kg/k. [4 marks] = (iii) The value of b2 for a waveguide is 9/76 m2. Calculate the power transmitted by the waveguide in the n = 1 mode if E. = 1 Vm-1 and = 210 x109 rad s-1. Hint: use the Poynting vector, E x H. At what frequency does the guide begin to conduct in the n = 2 mode? [8 marks]