Question

Question 1: Stress tensor for a parallel plate capacitor Consider an infinite parallel-plate capacitor, with the lower plate (at = = -d/2) carrying the charge density - and the upper plate (at z =d/2) carrying the charge density o. (a) Determine all nine elements of the Maxwell stress tensor (i) for -1 <d/2 and (ii) for : > d/2. (b) Use the stress tensor to find the force per unit area acting on the top plate. (C) Find the electromagnetic momentum stored in the fields between the plates. (d) Find the momentum per unit area crossing the x-y plane. (d) This momentum will be absorbed by the plates: find the force this produces per unit area on the top plate. What relationship does this have to your answer to (b)?

Answer 1

Stress Tensor for a parallel plate capacitor According to question. She given situation is 10 our 2=-212 first . We have to calculate Formular for h is Tij - Es tit; - 1 Bij t] + 1 Maxwell stress Tensor, та: Tex Tчу | Tyx Tyy T२५ Tyz % Bi Bj. 1 Sij B) How calculate E (Electric Field) and Electric field B (Magnetic Inside field) capacitor is 6 . Ex:o, Eyne inside) E Coutsidel nEr .o Exta Ey= + E- Ezzo N

daangdagan (25 Now, Magnetic field B because there is Current or Electric inside and outside is pero no moving charge present field is also constant. i.e. Therefore B - (0'd Tid 0,1214/2 ↑ Sij Ba] Where Sij is Kronecker delta. Sij (1,{j in. $24Syge 520-1, Swy. Sya : 524 = o Now Try - How to Exo -ty, E co 2 UUUUUUUUUUUUUUUUUUUJIJDWUWW

Similarly (672 672 62}au (Ryan Baez] Test (52të ty] + . (62-6. ] [Buby - Swy ( Twy - to (Ezty - Say (62+63'+ex]]+ + By+ B)] - Eo [Ex Ey - o] + 0 = EoExky so - Similiarly Trex - Tyn - Tyz = Tzre stay=0 { Eros So, Stress Tensor to 121<ols is a for Risala All components are of E and B ris pero All Nine are components zero. UUUUWWWUUUUUUUUWWWUWW

1 (6) Fod f'da Eello dls.de at ! da - Area element, s de Poynting Vector volume Element As we know. 5 EX and Bio sio Fida da is - dx dy - ve sign indicates direction of reo outward work and opposite to F = Stez daz that element Electric Pd. fie Filte dx dy, ale - A 3 & Force per unit Area. ( Electromagnetic Momentum per unit Area per unit Time P fidit af - - o Eod i sa At Jv → Momentum + I T da Per unit Time per unit Area Momentum stored in EM field

your Shinoorten anananan no SEM momentum is e Pems noko sat AS S=0 . Bro I ch Momentum per unit area is 다t force per unit Area on the top plak is - It is equal to expression ikle got in (6) da As he knows (b) Foo Fida - Fida da Area