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Здравствуйте! Прошу помощи в следующем вопросе: У меня возник вопрос при разборе темы "Электромагнитные волны". Как известно, векторы k, H, E образуют правую тройку. К такому выводу мы приходим из уравнений Максвелла. Векторное произведение [k, Е] даёт положительное число, а [k, H] - отрицательное. Почему на основе этого мы полагаем, что это и есть правая тройка?
Если придерживаться рисунка, который я привёл, то правой будет тройка, в которой первым является орт оси абсцисс, вторым - вектор E, третьим - вектор H/ Ведь поворот орта оси Ox на кратчайший угол до совмещения с вектором E наблюдается с конца вектора H как поворот против направления часовой стрелки. Поворот вектора E на кратчайший угол до совмещения с вектором H наблюдается с конца орта оси абсцисс тоже как поворот против направления часовой стрелки.
Относительно перечисленной Вами тройки векторов рисунок делает её левой. Поэтому фраза
условна. Недаром векторное умножение даёт в результате псевдовектор. Какой рисунок дан в Вашем учебнике?
Векторное произведение никак не может давать положительное или отрицетельное число, т.к. это вектор (точнее, псевдовектор). Чтобы разобраться, какая тройка векторов правая, какая левая, можно воспользоваться свойствами смешанного произведения векторов. Тройка a1, a2, a3 правая, если смешанное произведение a1*(a2 x a3) > 0. Известно, что смешанное произведение не меняется при циклической перестановке сомножителей, а при перестановке соседних аргументов меняет знак на противоположный. Для плоской волны из уравнений Максвелла можно получить: k[$215$]E=[$956$][$956$]0[$969$]H, k[$215$]H=-ee0[$969$]E. Умножая скалярно первое уравнение на H, второе на Е, находим: H*(k[$215$]E) > 0 (H,k,E - правая тройка) E*(k[$215$]H) < 0 (E,k,H - левая тройка) Перестановкой аргументов, как описано выше, из одного из этих неравенств получается другое. Также легко установить, что тройка k, E, H - правая.
Векторное произведение - псевдовектор, смешанное - псевдоскаляр. Это значит что они меняют знак при переходе от правой системы координат к левой, как при отражении в зеркале.
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