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Exercise:
Show that for an electromagnetic wave propagating along the positive z direction with E_xz t E_cosomega t-kz B_yz t B_cosomega t-kz The curl s vecnablatimesvec E -fracpartialvec Bpartial t vecnablatimesvec B -fracpartialvec Epartial t are fulfilled.
Solution:
The field vectors are given by vec E pmatrixE_ pmatrix cosomega t-kz vec B pmatrix B_ pmatrix cosomega t-kz The curl of the electric field is therefore vecnablatimesvec E pmatrixpartial/partial x partial/partial y partial/partial z pmatrixtimes pmatrixE_ cosomega t-kz pmatrix pmatrix fracpartialpartial zE_ cosomega t-kz -fracpartialpartial yE_ cosomega t-kzpmatrix pmatrix -E_sinomega t-kz-k pmatrix pmatrix k E_ pmatrixsinomega t-kz and the time derivative of the magnetic field is fracpartialpartial tpmatrix B_cosomega t-kz pmatrix pmatrix -B_sinomega t-kzomega pmatrix pmatrix -omega B_ pmatrix sinomega t-kz Faraday's law yields vecnablatimesvec E -fracpartialvec Bpartial t Longrightarrow pmatrix k E_ pmatrixsinomega t-kz -pmatrix -omega B_ pmatrix sinomega t-kz The two vectors are equal with fracE_B_ fracomegak c which is the correct relation between the electric and magnetic field vector in an electromagnetic wave. vspace.cm The second relation can be shown in the same way.
Show that for an electromagnetic wave propagating along the positive z direction with E_xz t E_cosomega t-kz B_yz t B_cosomega t-kz The curl s vecnablatimesvec E -fracpartialvec Bpartial t vecnablatimesvec B -fracpartialvec Epartial t are fulfilled.
Solution:
The field vectors are given by vec E pmatrixE_ pmatrix cosomega t-kz vec B pmatrix B_ pmatrix cosomega t-kz The curl of the electric field is therefore vecnablatimesvec E pmatrixpartial/partial x partial/partial y partial/partial z pmatrixtimes pmatrixE_ cosomega t-kz pmatrix pmatrix fracpartialpartial zE_ cosomega t-kz -fracpartialpartial yE_ cosomega t-kzpmatrix pmatrix -E_sinomega t-kz-k pmatrix pmatrix k E_ pmatrixsinomega t-kz and the time derivative of the magnetic field is fracpartialpartial tpmatrix B_cosomega t-kz pmatrix pmatrix -B_sinomega t-kzomega pmatrix pmatrix -omega B_ pmatrix sinomega t-kz Faraday's law yields vecnablatimesvec E -fracpartialvec Bpartial t Longrightarrow pmatrix k E_ pmatrixsinomega t-kz -pmatrix -omega B_ pmatrix sinomega t-kz The two vectors are equal with fracE_B_ fracomegak c which is the correct relation between the electric and magnetic field vector in an electromagnetic wave. vspace.cm The second relation can be shown in the same way.