Field Vectors in Electromagnetic Waves
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Exercise:
The amplitudes of the electric and magnetic field vectors in an electromagnetic wave propagating along the z direction are vectors in the xy plane. abcliste abc For a magnetic field vector vecB pmatrix B_x B_y pmatrix find the components of the corresponding electric field vector. abc For an electric field vector vecE pmatrix E_x E_y pmatrix find the components of the corresponding magnetic field vector. abcliste
Solution:
abcliste abc The electric field vector is given by the relation vec E vec B cross vec c. It follows vec E pmatrix B_x B_y pmatrix cross pmatrix c pmatrix pmatrix B_y c - - B_x c B_x - B_y pmatrix pmatrix B_y c -B_x c pmatrix c pmatrix B_y -B_x pmatrix We can easily verify that vec E is perpicular to vec B and that E B c: vec E vec B B_x B_y + B_y -B_x B_x B_y - B_x B_y E absvec E sqrtE_x^ + E_y^ sqrtc B_y^+-c B_x^ sqrtc^ B_y^ + B_x^ c sqrtB_x^+B_y^ c absvec B c B abc In the xy plane the slope for the vector vec E is m_E fracE_yE_x We know that vec B is perpicular to vec E so its slope is m_B -fracE_xE_y The components of vec B can therefore be written as vec B pmatrixB_x B_y pmatrix k pmatrixE_y -E_x pmatrix The paramter k can be determined using the relation vec E vec B cross vec c Longrightarrow pmatrix E_x E_y pmatrix k pmatrixE_y -E_x pmatrix cross pmatrix c pmatrix k pmatrix - E_x c - - E_y c - pmatrix -k c pmatrix E_x E_y pmatrix Longrightarrow k -fracc The magnetic field vector is therefore vec B pmatrix -E_y/c E_x/c pmatrix vspacecm Alternatively we can use the vector identity vec u cross left vec v cross vec w right vec u vec w vec v - vec u vec v vec w With vec u vec w vec c and vec v vec B and using the relation vec E vec B cross vec c we find vec c cross vec E vec c cross left vec B cross vec c right vec c vec c vec B - vec c vec B vec c c^ vec B where we have used vec c vec c c^ and vec c vec B since vec c perp vec B. It follows for the components pmatrix c pmatrix cross pmatrix E_x E_y pmatrix pmatrix - c E_y c E_x - - pmatrix pmatrix - c E_y c E_x pmatrix c^ pmatrix B_x B_y pmatrix This leads to the same result as the first approach: vec B pmatrix -E_y/c E_x/c pmatrix abcliste
The amplitudes of the electric and magnetic field vectors in an electromagnetic wave propagating along the z direction are vectors in the xy plane. abcliste abc For a magnetic field vector vecB pmatrix B_x B_y pmatrix find the components of the corresponding electric field vector. abc For an electric field vector vecE pmatrix E_x E_y pmatrix find the components of the corresponding magnetic field vector. abcliste
Solution:
abcliste abc The electric field vector is given by the relation vec E vec B cross vec c. It follows vec E pmatrix B_x B_y pmatrix cross pmatrix c pmatrix pmatrix B_y c - - B_x c B_x - B_y pmatrix pmatrix B_y c -B_x c pmatrix c pmatrix B_y -B_x pmatrix We can easily verify that vec E is perpicular to vec B and that E B c: vec E vec B B_x B_y + B_y -B_x B_x B_y - B_x B_y E absvec E sqrtE_x^ + E_y^ sqrtc B_y^+-c B_x^ sqrtc^ B_y^ + B_x^ c sqrtB_x^+B_y^ c absvec B c B abc In the xy plane the slope for the vector vec E is m_E fracE_yE_x We know that vec B is perpicular to vec E so its slope is m_B -fracE_xE_y The components of vec B can therefore be written as vec B pmatrixB_x B_y pmatrix k pmatrixE_y -E_x pmatrix The paramter k can be determined using the relation vec E vec B cross vec c Longrightarrow pmatrix E_x E_y pmatrix k pmatrixE_y -E_x pmatrix cross pmatrix c pmatrix k pmatrix - E_x c - - E_y c - pmatrix -k c pmatrix E_x E_y pmatrix Longrightarrow k -fracc The magnetic field vector is therefore vec B pmatrix -E_y/c E_x/c pmatrix vspacecm Alternatively we can use the vector identity vec u cross left vec v cross vec w right vec u vec w vec v - vec u vec v vec w With vec u vec w vec c and vec v vec B and using the relation vec E vec B cross vec c we find vec c cross vec E vec c cross left vec B cross vec c right vec c vec c vec B - vec c vec B vec c c^ vec B where we have used vec c vec c c^ and vec c vec B since vec c perp vec B. It follows for the components pmatrix c pmatrix cross pmatrix E_x E_y pmatrix pmatrix - c E_y c E_x - - pmatrix pmatrix - c E_y c E_x pmatrix c^ pmatrix B_x B_y pmatrix This leads to the same result as the first approach: vec B pmatrix -E_y/c E_x/c pmatrix abcliste