Question

A plane EM wave travelling in vacuum along z direction is given by 

E = E_o sin(kz –ωt )i and B=B_osin (kz ωt)j .

(i) Evaluate ∮ E.dl over the rectangular loop 1234 shown in Fig 8.2.

(ii) Evaluate ∮B.ds over the surface bounded by loop 1234.

(iii) Use equation ∮E.dl=-dϕ_B/dt to prove E_o/B_o=c.

(iv) By using similar process and the equation

∮B.dl= μ_oI+ɛ_o dϕ_E/dt, prove that c=1/√μ_oɛ_o 

Answer 1

(ii) For evaluating ∫B.ds let us consider the rectangle 1234 to be made of strips of area ds = h dz each.

Using the relations obtained in Equations (1) and (2) and simplifying, we get

(iv) For evaluating ∫NB.dl , let us consider the loop 1234 in yz plane as shown in Fig.

Now to evaluate ϕ_E = ∫E.ds , let us consider the rectangle 1234 to be made of strips of area hdz each.

Using relations obtained in Equations (3) and (4) and simplifying, we get