Optomagnetismo associado ao spin eletrônico em semicondutores

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❯◆■❱❊❘❙■❉❆❉❊ ❉❊ ❙➹❖ P❆❯▲❖ ■◆❙❚■❚❯❚❖ ❉❊ ❋❮❙■❈❆

❖♣t♦♠❛❣♥❡t✐s♠♦ ❆ss♦❝✐❛❞♦ ❛♦ ❙♣✐♥

❊❧❡trô♥✐❝♦ ❡♠ ❙❡♠✐❝♦♥❞✉t♦r❡s

❘❡♥❛♥ ❈❛r❧♦s ❈♦r❞❡✐r♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳❉r✳❆♥❞ré ❇♦❤♦♠♦❧❡t③ ❍❡♥r✐q✉❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s ❡ ♣r♦♣r✐❡❞❛✲ ❞❡s ♠❛❣♥ét✐❝❛s

❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❆♥❞ré ❇♦❤♦♠♦❧❡t③ ❍❡♥r✐q✉❡s ✭■❋❯❙P✮ Pr♦❢✳ ❉r✳ ❙②❧✈✐♦ ❘♦❜❡rt♦ ❆❝❝✐♦❧② ❈❛♥✉t♦ ✭■❋❯❙P✮ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❙ér❣✐♦ ❙♦❛r❡s ●✉✐♠❛rã❡s ✭❯❋▼●✮

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❊♥tã♦ ❛ r❛♣♦s❛ ❛♣❛r❡❝❡✉✳ ✧❇♦♠ ❞✐❛✧✱ ❞✐ss❡ ❛ r❛♣♦s❛✳

✧❇♦♠ ❞✐❛✧✱ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡ r❡s♣♦♥❞❡✉ ❡❞✉❝❛❞❛♠❡♥t❡✳ ✧◗✉❡♠ é ✈♦❝ê❄ ❱♦❝ê é tã♦ ❜♦♥✐t❛ ❞❡ s❡ ♦❧❤❛r✳✧

✧❊✉ s♦✉ ✉♠❛ r❛♣♦s❛✧✱ ❞✐ss❡ ❛ r❛♣♦s❛✳

✧❱❡♥❤❛ ❜r✐♥❝❛r ❝♦♠✐❣♦✧✱ ♣r♦♣ôs ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡✳ ✧❊✉ ❡st♦✉ tã♦ tr✐st❡✧✳ ✧❊✉ ♥ã♦ ♣♦ss♦ ❜r✐♥❝❛r ❝♦♠ ✈♦❝ê✧✱ ❛ r❛♣♦s❛ ❞✐ss❡✳ ✧❊✉ ❡st♦✉ ❝❛t✐✈❛❞❛✧✳ ✧❖ q✉❡ s✐❣♥✐✜❝❛ ✐ss♦ ✕ ❝❛t✐✈❛r❄✧

✧➱ ✉♠❛ ❝♦✐s❛ q✉❡ ❛s ♣❡ss♦❛s ❢r❡q✉❡♥t❡♠❡♥t❡ ♥❡❣❧✐❣❡♥❝✐❛♠✧✱ ❞✐ss❡ ❛ r❛♣♦s❛✳ ✧❙✐❣♥✐✜❝❛ ❡st❛❜❡❧❡❝❡r ❧❛ç♦s✧✳

✧❙✐♠✧❞✐ss❡ ❛ r❛♣♦s❛✳ ✧P❛r❛ ♠✐♠ ✈♦❝ê é ❛♣❡♥❛s ✉♠ ♠❡♥✐♥✐♥❤♦ ❡ ❡✉ ♥ã♦ t❡♥❤♦ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✈♦❝ê✳ ❊ ✈♦❝ê ♣♦r s✉❛ ✈❡③✱ ♥ã♦ t❡♠ ♥❡♥❤✉♠❛

♥❡❝❡ss✐❞❛❞❡ ❞❡ ♠✐♠✳ P❛r❛ ✈♦❝ê ❡✉ ♥ã♦ s♦✉ ♥❛❞❛ ♠❛✐s ❞♦ q✉❡ ✉♠❛ r❛♣♦s❛✱ ♠❛s s❡ ✈♦❝ê ♠❡ ❝❛t✐✈❛r ❡♥tã♦ ♥ós ♣r❡❝✐s❛r❡♠♦s ✉♠ ❞♦ ♦✉tr♦✧✳

❆ r❛♣♦s❛ ♦❧❤♦✉ ✜①❛♠❡♥t❡ ♣❛r❛ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡ ❞✉r❛♥t❡ ♠✉✐t♦ t❡♠♣♦ ❡ ❞✐ss❡✿ ✧P♦r ❢❛✈♦r ❝❛t✐✈❛✲♠❡✳✧

✧❖ q✉❡ ❡✉ ❞❡✈♦ ❢❛③❡r ♣❛r❛ ❝❛t✐✈❛r ✈♦❝ê❄✧♣❡r❣✉♥t♦✉ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡✳ ✧❱♦❝ê ❞❡✈❡ s❡r ♠✉✐t♦ ♣❛❝✐❡♥t❡✧✳ ❉✐ss❡ ❛ r❛♣♦s❛✳ ✧Pr✐♠❡✐r♦ ✈♦❝ê ✈❛✐ s❡♥t❛r ❛ ✉♠❛ ♣❡q✉❡♥❛ ❞✐stâ♥❝✐❛ ❞❡ ♠✐♠ ❡ ♥ã♦ ✈❛✐ ❞✐③❡r ♥❛❞❛✳ P❛❧❛✈r❛s sã♦ ❛s ❢♦♥t❡s ❞❡ ❞❡s❡♥t❡♥❞✐♠❡♥t♦✳ ▼❛s ✈♦❝ê s❡ s❡♥t❛rá ✉♠ ♣♦✉❝♦ ♠❛✐s ♣❡rt♦ ❞❡ ♠✐♠ t♦❞♦ ❞✐❛✳✧ ❊♥tã♦ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡ ❝❛t✐✈♦✉ ❛ r❛♣♦s❛ ❡ ❞❡♣♦✐s ❝❤❡❣♦✉ ❛ ❤♦r❛ ❞❛

♣❛rt✐❞❛ ❞❡❧❡ ✕ ✧❖❤✦✧❞✐ss❡ ❛ r❛♣♦s❛✳ ✧❊✉ ✈♦✉ ❝❤♦r❛r✧✳ ✧❆ ❝✉❧♣❛ é s✉❛✧✱ ❞✐ss❡ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡✱

✧♠❛s ✈♦❝ê ♠❡s♠❛ q✉✐s q✉❡ ❡✉ ❛ ❝❛t✐✈❛ss❡✧✳ ✧❆❞❡✉s✧✱ ❞✐ss❡ ♦ P❡q✉❡♥♦ Prí♥❝✐♣❡✳

✧❆❞❡✉s✧✱ ❞✐ss❡ ❛ r❛♣♦s❛✳

✧❊ ❛❣♦r❛ ❡✉ ✈♦✉ ❝♦♥t❛r ❛ ✈♦❝ê ✉♠ s❡❣r❡❞♦✿ ♥ós só ♣♦❞❡♠♦s ✈❡r ♣❡r❢❡✐t❛♠❡♥t❡ ❝♦♠ ♦ ❝♦r❛çã♦❀ ♦ q✉❡ é ❡ss❡♥❝✐❛❧ é ✐♥✈✐sí✈❡❧ ❛♦s ♦❧❤♦s✳ ❖s ❤♦♠❡♥s tê♠

❡sq✉❡❝✐❞♦ ❡st❛ ✈❡r❞❛❞❡✳ ▼❛s ✈♦❝ê ♥ã♦ ❞❡✈❡ ❡sq✉❡❝ê✲❧❛✳

❱♦❝ê s❡ t♦r♥❛ ❡t❡r♥❛♠❡♥t❡ r❡s♣♦♥sá✈❡❧ ♣♦r ❛q✉✐❧♦ q✉❡ ❝❛t✐✈❛✳✧

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✐

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ Pr♦❢✳ ❉r✳ ❆♥❞ré ❇♦❤♦♠♦❧❡t③ ❍❡♥r✐q✉❡s✱ ♦r✐❡♥t❛❞♦r✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦ ❡ ♣❡❧♦ ♠❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣r♦✜ss✐♦♥❛❧✴♣❡ss♦❛❧❀ ❆♦s ♠❡✉s ♣❛✐s✱ ❊❧✐s❛❜❡t❡ ❆♣❛r❡❝✐❞❛ ❇❛r❜❡❞♦ ❈♦r❞❡✐r♦ ❡ ▲✉✐③ ❈❛r❧♦s ❈♦r✲ ❞❡✐r♦✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝❛r✐♥❤♦✱ ❞❡❞✐❝❛çã♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❡♥s✐♥❛♠❡♥t♦s q✉❡ ❢♦r♠❛✲ r❛♠ ♦ q✉❡ s♦✉ ❤♦❥❡✳ ▼❡✉s ♣✐❧❛r❡s ❞❡ ✈✐❞❛❀

❆ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ s❡♠♣r❡ s❡ ❞❡♠♦♥str♦✉ ❞✐s♣♦st❛ ❛ ♠❡ ✐♥❝❡♥t✐✈❛r ❡♠ ♠✐♥❤❛s ❞❡❝✐sõ❡s❀

❆♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❙P✱ ♣❡❧♦ ❢♦r♥❡❝✐♠❡♥t♦ ❞❛ ✐♥❢r❛❡str✉t✉r❛ ♥❡❝❡s✲ sár✐❛ à r❡❛❧✐③❛çã♦ ❞♦ ♠❡str❛❞♦❀

❆♦ ❈◆Pq ✭❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦✮ ♣❡❧♦ ❛♣♦✐♦ ❝♦♥❝❡❞✐❞♦ ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❞❡ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦❀ ❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞♦s ▼❛t❡r✐❛✐s ❡ ▼❡❝â♥✐❝❛✱ ♣❡❧♦s s❡r✈✐ç♦s ❡ ❛t❡♥❞✐♠❡♥t♦s ♣r❡st❛❞♦s❀

❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✱ ❊❞✉❛r❞♦ ❙❡❧❧ ●♦♥ç❛❧✈❡s✱ ●❛❜r✐❡❧ ▼❛r✐♥❡❧❧♦✱ ❋r❛♥❝✐❡❧❧❡ ❆✳ ❈✳ ❙❡③♦t③❦✐✱ ◆❛t❛❝❤❛ ❊♥♦❦✐✱ ❘❛❢❛❡❧ ▼❛rt✐♥❡③ ❞❡ ❆r❛✉❥♦✱ ❚✐❜ér✐♦ ❋❡rr❛r✐✱ ❋❧á✈✐♦ ❈✳ ❞❡ ▼♦r❛❡s✱ ❡♥tr❡ ♠✉✐t♦s ♦✉tr♦s✱ ♣♦r ♠❡ ✐♥❝❡♥t✐✈❛r❡♠ ♥❛ ❝♦♥st✐t✉✐çã♦ ❞❡ss❡ tr❛❜❛❧❤♦❀

➚s ♠✐♥❤❛s ❛♠✐❣❛s ❞❡s❞❡ t❡♠♣♦ ❞❡ ❝♦❧é❣✐♦✱ ❈❛r♦❧✐♥❛ ❨♦s✐♥♦✱ ▲❛ís ❊♠② ■s❤✐❜❛s❤✐✱ P❛trí❝✐❛ ❙❝❤❧✐t❤❧❡r ❡ ❇❡❛tr✐③ ❇❡r❡s ▼❡✐r❛✱ ♣❡❧❛ ✐♥❝rí✈❡❧ ❝♦♠♣❛♥❤✐❛ ❢♦r♥❡❝✐❞❛❀

➚ ❆❧✐ss♦♥ ▲✉ís ❙♦❛r❡s ❚❡✐①❡✐r❛✱ ♣❡❧♦s ✐♥❝rí✈❡✐s ♠♦♠❡♥t♦s ♣r♦♣♦r❝✐♦♥❛❞♦s ❡ ♣♦r s✉❛ ♣r❡s❡♥ç❛ ú♥✐❝❛ ❡ ♠♦t✐✈❛❞♦r❛ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ♠❛r❝❛♥t❡s ❞❡st❡ ú❧t✐♠♦ ❛♥♦❀

➚ ▲✉❝❛s ❇❛r❜♦③❛ ▼❛rt✐♥s✱ ▼❛r✐♦ ❍❡♥r✐q✉❡ ●❡r♠♦❧✐❛t♦✱ ❇❡r❣ P❡ss♦❛✱ ❨✉r✐ ❇❡❧♠♦❝❦✱ ▲✉ís ▼❛t✐♥❤❛✱ ▼❛t❤❡✉s ❱✐❞❛❧✱ ▼❛t❤❡✉s ❘♦s❛✱ ▲✉❝❛s ▼❛✐❛✱ ●✐❧ ❈❛r❞♦s♦✱ ❍❡♥r✐q✉❡ ❋✉rq✉✐♠✱ ❆❧❡① ❇❛rr❡t♦✱ ❡♥tr❡ ♠✉✐t♦s ♦✉tr♦s✱ ♣♦r ♠♦s✲ tr❛r❡♠ ♦ ❣r❛✉ ❞❡ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❛♠✐③❛❞❡ ❡ ❛ ❢♦r♠❛ ❝♦♠♦ ❛ ♠❡s♠❛ s✉♣❡r❛ ♦❜stá❝✉❧♦s❀

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✐✐

❘❊❙❯▼❖

❖ s♣✐♥ ❞❡ ✉♠ ❡❧étr♦♥ ❝♦♥✜♥❛❞♦ ❡♠ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ✭❞♦ ✐♥❣❧ês✱ q✉❛♥t✉♠ ❞♦t ♦✉ ◗❉✮ ♦❢❡r❡❝❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❝♦❡rê♥❝✐❛ ❞❡ ❢❛s❡ ❡♠ ❡s❝❛❧❛s ❞❡ t❡♠♣♦ ♠✉✐t♦ ♠❛✐s ❧♦♥❣❛s ❞♦ q✉❡ ❛q✉❡❧❛s ❡♥❝♦♥tr❛❞❛s ❡♠ ❞✐s♣♦s✐t✐✈♦s ❝♦♥✈❡♥❝✐♦♥❛✐s✳ ❆ ♥❛t✉r❡③❛ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ss❛s ❡str✉t✉r❛s ♣♦❞❡ s❡r ❡①♣❧♦r❛❞❛ ❡♠ ❞✐s♣♦s✐t✐✈♦s ♦♣t♦❡❧❡trô♥✐❝♦s ❜❛s❡✲ ❛❞♦s ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ s♣✐♥ ♣❡❧❛ ❧✉③✱ t❛✐s ❝♦♠♦ ◗❉ ❧❛s❡rs✱ ❡♠✐ss♦r❡s ❞❡ ❢ót♦♥✲ú♥✐❝♦ ❡ tr❛♥s✐st♦r❡s ❞❡ ❡❧étr♦♥✲ú♥✐❝♦✳

(7)

✐✐✐

❆❇❙❚❘❆❈❚

❚❤❡ s♣✐♥ ♦❢ ❛♥ ❡❧❡❝tr♦♥ ❝♦♥✜♥❡❞ ✐♥ ❛ q✉❛♥t✉♠ ❞♦t ✭◗❉✮ ♦✛❡rs t❤❡ ♦♣♣♦rt✉✲ ♥✐t② t♦ st♦r❡ ❛♥❞ ♠❛♥✐♣✉❧❛t❡ ♣❤❛s❡ ❝♦❤❡r❡♥❝❡ ♦✈❡r ♠✉❝❤ ❧♦♥❣❡r t✐♠❡ s❝❛❧❡s t❤❛♥ ✐t ✐s t②♣✐❝❛❧❧② ♣♦ss✐❜❧❡ ✐♥ ❝❤❛r❣❡ ❜❛s❡❞ ❞❡✈✐❝❡s✳ ❚❤❡ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ♥❛t✉r❡ ♦❢ t❤❡s❡ ♥❛♥♦str✉❝t✉r❡s ❝❛♥ ❜❡ ❡①♣❧♦✐t❡❞ ✐♥ ♦♣t♦❡❧❡tr♦♥✐❝ ❞❡✈✐❝❡s✱ s✉❝❤ ❛s q✉❛♥t✉♠ ❞♦t ❧❛s❡r✱ s✐♥❣❧❡✲♣❤♦t♦♥ ❡♠✐tt❡rs✱ s✐♥❣❧❡✲❡❧❡❝tr♦♥ tr❛♥s✐s✲ t♦r ❛♥❞ s♣✐♥✲♠❛♥✐♣✉❧❛t✐♦♥✳

❚❤✉s✱ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ♣❤②s✐❝s ❜❡❤✐♥❞ ❧✐❣❤t ❝♦♥tr♦❧ ♦❢ ♠❛❣♥❡t✐s♠ ✐s ❡s✲ s❡♥t✐❛❧ t♦ ❛❞✈❛♥❝❡ t❤✐s ✜❡❧❞ ❛♥❞ ❞❡✈✐❝❡ ❛♣♣❧✐❝❛t✐♦♥s ❜❛s❡❞ ♦♥ ✐t✳ ■♥ ♣❛rt✐✲ ❝✉❧❛r✱ ♠❛❣♥❡t✐③❛t✐♦♥ ❣❡♥❡r❛t✐♦♥ ❝❛♥ ❜❡ ✐♥❞✉❝❡❞ ✐♥ ❛♥ ❡♥s❡♠❜❧❡ ♦❢ q✉❛♥t✉♠ ❞♦ts✱ ❡❛❝❤ ❝❤❛r❣❡❞ ✇✐t❤ ❛ s✐♥❣❧❡ ❡❧❡❝tr♦♥✱ ✇❤❡♥ ✐❧❧✉♠✐♥❛t❡❞ ✇✐t❤ ❛ s❤♦rt ❝✐r❝✉❧❛r❧② ♣♦❧❛r✐③❡❞ ❧✐❣❤t ♣✉❧s❡ r❡s♦♥❛♥t ✇✐t❤ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣❛♣ ♦❢ t❤❡ ◗❉s✳

(8)

❮♥❞✐❝❡

❮♥❞✐❝❡ ✐✈

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s ✹

✷✳✶ ■❧❤❛ ◗✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ◆í✈❡✐s ❞❡ ❊♥❡r❣✐❛ ❡ ❘❡❣r❛s ❞❡ ❙❡❧❡çã♦ Ó♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❡ ✉♠ ❊❧étr♦♥ ❡♠ ❈❛♠♣♦ ▼❛❣♥ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹✳✶ ❖ ❈❛♠♣♦ ▼❛❣♥ét✐❝♦ ❉✐♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹✳✷ ❆ ■♥t❡r❛çã♦ ◗✉â♥t✐❝❛ ❇ás✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✹✳✸ ❆ ■♥t❡r❛çã♦ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹✳✹ ❆ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ❡♠ ■❧❤❛s ◗✉â♥t✐❝❛s ✳ ✶✽ ✷✳✹✳✺ ❉❡❢❛s❛❣❡♠ ❞♦ ❙♣✐♥ ❊❧❡trô♥✐❝♦ ❞❡✈✐❞♦ às ❋❧✉t✉❛çõ❡s ❞♦ ❈❛♠♣♦

◆✉❝❧❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹✳✻ ❉❡❢❛s❛❣❡♠ ❞❡ ❙♣✐♥ ❊❧❡trô♥✐❝♦ ♥❛ ❋❧✉t✉❛çã♦ ❈♦♥❣❡❧❛❞❛ ❞♦ ❈❛♠♣♦

◆✉❝❧❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ ❘♦t❛çã♦ ❞❡ ❋❛r❛❞❛② ❘❡s♦❧✈✐❞❛ ♥♦ ❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸ ▼❛t❡r✐❛❧ ❡ ❆rr❛♥❥♦ ❊①♣❡r✐♠❡♥t❛❧ ✷✼

(9)

❮♥❞✐❝❡ ✈

✸✳✶✳✶ ▼♦♥t❛❣❡♠ ♣❛r❛ ❋♦t♦❧✉♠✐♥❡s❝ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✷ ▼♦♥t❛❣❡♠ ♣❛r❛ ❘♦t❛çã♦ ❞❡ ❋❛r❛❞❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❆♠♦str❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ❘❡s✉❧t❛❞♦s ❡ ❆♥á❧✐s❡s ✸✷

✹✳✶ ❖r✐❡♥t❛çã♦ Ó♣t✐❝❛ ❞❡ ❙♣✐♥ ❡♠ ❙❡♠✐❝♦♥❞✉t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✷ ❉❡s❝r✐çã♦ ◗✉❛❧✐t❛t✐✈❛ ♣❛r❛ ●❡r❛çã♦ ❞❡ ❈♦❡rê♥❝✐❛ ❞❡ ❙♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✸ ▼♦❞❡❧♦ ❱✐❣❡♥t❡ ♥❛ ❉❡s❝r✐çã♦ ❞❡ ❈♦❡rê♥❝✐❛ ❞❡ ❙♣✐♥ ♣❛r❛ ❚r❛♥s✐çõ❡s

es→Ss ❡ es →Sp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✹ ❊❢❡✐t♦ ❞❡ Pr❡❝❡ssã♦ ❞♦ ❇✉r❛❝♦ ♥❛ P♦❧❛r✐③❛çã♦ ❞♦ ❙♣✐♥ ❊❧❡trô♥✐❝♦ ✳ ✳ ✳ ✹✷ ✹✳✺ ▼♦❞❡❧♦ ❞♦ ❇✉r❛❝♦ ❊stát✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✻ ❘❡s✉❧t❛❞♦s ❊①♣❡r✐♠❡♥t❛✐s ❡ ❈♦♠♣❛r❛çã♦ ❝♦♠ ▼♦❞❡❧♦s ❚❡ór✐❝♦s ✳ ✳ ✳ ✳ ✺✶ ✹✳✼ ■♥t❡r♣r❡t❛çã♦ ❋ís✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✺ ❈♦♥❝❧✉sõ❡s ✺✻

✻ P❡rs♣❡❝t✐✈❛s ❋✉t✉r❛s ✺✽

❆ ❍❛♠✐❧t♦♥✐❛♥❛ ❉✐♣♦❧❛r ✺✾

❇ ■❞❡♥t✐❞❛❞❡ ❆❧❣é❜r✐❝❛ ✻✷

❈ ❙✐♥❛❧ ❚❘❋❘ ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ❞❡ ■❧❤❛s ◗✉â♥t✐❝❛s ✻✸

❉ ❆rt✐❣♦ P✉❜❧✐❝❛❞♦ ❡ ❆♣r❡s❡♥t❛çõ❡s ❡♠ ❈♦♥❢❡rê♥❝✐❛s ✻✼

(10)

(11)

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❆ ❢ís✐❝❛ ♥♦s ú❧t✐♠♦s ✷✺ ❛♥♦s ✈❡♠ s♦❢r❡♥❞♦ ♣r♦❢✉♥❞❛s tr❛♥s❢♦r♠❛çõ❡s✱ ❞❡❝♦rr❡♥t❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈❛s té❝♥✐❝❛s ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❞❛ ♠❛tér✐❛ ❡♠ ♥í✈❡✐s ♠♦❧❡❝✉❧❛r ❡ ❛tô♠✐❝♦✳ ❆té ♦ ✐♥í❝✐♦ ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✼✵✱ ♦s ♠♦❞❡❧♦s ♠✐❝r♦s❝ó♣✐❝♦s ❞❛ ♠❛tér✐❛ só ♣♦✲ ❞✐❛♠ s❡r t❡st❛❞♦s ❡♠ ❛♠♦str❛s ✈♦❧✉♠étr✐❝❛s✳ ❉❡♥tr♦ ❞❡st❡ ❝♦♥t❡①t♦✱ s❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ❛s t❡♦r✐❛s s♦❜r❡ ♦ ♠❛❣♥❡t✐s♠♦✱ ❛ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡✱ ❡t❝❬✶❪✳

◆♦ ❡♥t❛♥t♦✱ ❛ ♣❛rt✐r ❞❡ ♠❡❛❞♦s ❞♦s ❛♥♦s ✶✾✼✵ t♦r♥♦✉✲s❡ ♣♦ssí✈❡❧ ❛ ❢❛❜r✐❝❛çã♦ ❞❡ ❡s✲ tr✉t✉r❛s ✧❛rt✐✜❝✐❛✐s✧♥❛s q✉❛✐s át♦♠♦s ❡ ♠♦❧é❝✉❧❛s ❢♦ss❡♠ ♠❛♥✉s❡❛❞♦s ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦✱ ♣♦rt❛♥t♦✱ ♦ s✉r❣✐♠❡♥t♦ ❞❡ ♥♦✈♦s ♠❛t❡r✐❛✐s✱ ❛ ❡①❡♠♣❧♦ ❞❡ ❡str✉t✉r❛s ❝r✐st❛❧✐♥❛s ❞❡ ❜❛✐①❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✱ ♣♦ç♦s q✉â♥t✐❝♦s✱ ✐❧❤❛s q✉â♥t✐❝❛s ❡ ✜❧♠❡s✲✜♥♦s✳ ❆s ❝♦♥s❡q✉ê♥❝✐❛s t❡❝♥♦❧ó❣✐❝❛s ❞❡ss❡s ❡st✉❞♦s ❢♦r❛♠ ✐❣✉❛❧♠❡♥t❡ ♥♦✈❛s✱ ❝♦♠♦✿ ♦❜t❡♥çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ♠❡tá❧✐❝❛s✱ s❡♠✐❝♦♥❞✉t♦r❛s ♦✉ ✐s♦❧❛♥t❡s ❛tr❛✈és ❞❡ ✉♠ ú♥✐❝♦ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦❀ ❛❞✈❡♥t♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❡❧❡trô♥✐❝♦s✱ ❝♦♠♦ ❞✐♦❞♦s ❡ tr❛♥s✐st♦r❡s✱ ❡♠ ❞✐✲ ♠❡♥sõ❡s ♠♦❧❡❝✉❧❛r❡s❀ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♥❛♥♦♠❛❣♥❡t✐s♠♦ ✲ ♣r✐♥❝✐♣❛❧ ♣r♦♠❡ss❛ ❞❡ ✐♥♦✈❛çã♦ t❡❝♥♦❧ó❣✐❝❛ ♥❛s ♣ró①✐♠❛s ❞é❝❛❞❛s ✲ ❡♥tr❡ ♠✉✐t♦s ♦✉tr♦s ❬✶✱ ✷❪✳

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✷ ■♥tr♦❞✉çã♦

❛té ♦ ✉s♦ ❞❡ s♣✐♥s ✐s♦❧❛❞♦s ♣❛r❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛❬✼✱ ✽✱ ✾❪✳

❖ t❡r♠♦ ✧✐❧❤❛ q✉â♥t✐❝❛✧✱ ❛♠♣❧❛♠❡♥t❡ ❡♠♣r❡❣❛❞♦ ❛ ♣❛rt✐r ❞♦ ✜♥❛❧ ❞♦s ❛♥♦s ✽✵✱ r❡❢❡r❡✲s❡ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❛ ❡str✉t✉r❛s s❡♠✐❝♦♥❞✉t♦r❛s ❞❡ ❡s❝❛❧❛ ♥❛♥♦♠étr✐❝❛✳ ❖s t❛✲ ♠❛♥❤♦s tí♣✐❝♦s ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ❝♦♠♣r❡❡♥❞❡♠ ✉♠❛ r❡❣✐ã♦ q✉❡ ✈❛rr❡ ❞❡s❞❡ ♣♦✉❝♦s ♥❛♥ô♠❡tr♦s✱ ❡♠ ✐❧❤❛s ❝♦❧♦✐❞❛✐s ✭t❛♠❜é♠ r❡❢❡r✐❞❛s ❝♦♠♦ ♥❛♥♦✲❝r✐st❛✐s✮✱ ❛té ❛ ❛❧❣✉♠❛s ❝❡♥t❡♥❛s ❞❡❧❡s✱ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❡str✉t✉r❛s ❡❧❡tr♦stát✐❝❛s ❢❛❜r✐❝❛❞❛s ❧✐t♦❣r❛✜❝❛♠❡♥t❡✳ ❖ t❛♠❛♥❤♦ ❢ís✐❝♦ r❡❞✉③✐❞♦ ❞❡ss❛s ❝♦♠♣♦♥❡♥t❡s é ❛ ♣r✐♥❝✐♣❛❧ ❝❛r❛❝t❡ríst✐❝❛ ❡♠ ❝♦♠✉♠ ❞❛s ✐❧❤❛s q✉â♥t✐❝❛s ❢♦r♠❛❞❛s ♣♦r ❞✐❢❡r❡♥t❡s ♠❛t❡r✐❛✐s ❡ ❢❛❜r✐❝❛❞❛s ♣♦r ❞✐❢❡r❡♥t❡s té❝✲ ♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s✳ ❚❛❧ ❛s♣❡❝t♦ s✐♥❣✉❧❛r ❞á ♦r✐❣❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ❞❡ ♠❛✐♦r r❡❧❡✈â♥❝✐❛ ❞❛s ✐❧❤❛s q✉â♥t✐❝❛s✿ ❛ s✉♣r❡ssã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❡❧étr♦♥s ✭❡✴♦✉ ❜✉r❛❝♦s✮ ♣r❡s❡♥t❡s ♥❡ss❛s ❡str✉t✉r❛s✳ ■st♦ é✱ ♦ t❛♠❛♥❤♦ r❡❞✉③✐❞♦ ❞♦s ◗❉✬s ❛❝❛rr❡t❛ ♥♦ ❝♦♥✜♥❛✲ ♠❡♥t♦ ❡ ❧♦❝❛❧✐③❛çã♦ ❡s♣❛❝✐❛❧ ✭❡♠ t♦❞❛s ❛s três ❞✐r❡çõ❡s✮ ❞❛s ♣❛rtí❝✉❧❛s r❡s✐❞❡♥t❡s ❡♠ s❡✉ ✐♥t❡r✐♦r✶_{✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞♦ ❝♦♥✜♥❛♠❡♥t♦ ❡s♣❛❝✐❛❧ s❡❣✉❡ ❛ ❝♦♠♣❧❡t❛ q✉❛♥t✐③❛çã♦}

✭♦✉ ❞✐s❝r❡t✐③❛çã♦✮ ❞♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛ ❞♦s ♣♦rt❛❞♦r❡s ❞❡ ❝❛r❣❛ ❝♦♥✜♥❛❞♦s às ✐❧❤❛s q✉â♥t✐❝❛s✳

❖ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛ q✉❛♥t✐③❛❞♦ ❡ ❛ ❧♦❝❛❧✐③❛çã♦ ❡s♣❛❝✐❛❧ s❡ ❛ss❡♠❡❧❤❛♠ ❛♦ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞♦s ❡❧étr♦♥s ❡♠ ✉♠ át♦♠♦✱ ♦ q✉❡ ❞á ♦r✐❣❡♠ ❛ ❞❡s✐❣♥❛çã♦ ❞❛s ✐❧❤❛s q✉â♥t✐❝❛s ❝♦♠♦ ✧➪t♦♠♦s ❆rt✐✜❝✐❛✐s✧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡ss❛ ú❧t✐♠❛ ❞❡s❝r✐çã♦ ❡♥❢❛t✐③❛ ❡ ❝♦♥✜❣✉r❛ ♦ ❛✉♠❡♥t♦ ❞♦ ✐s♦❧❛♠❡♥t♦ ❞♦s ❡❧étr♦♥s ✭❡✴♦✉ ❜✉r❛❝♦s✮ ❝♦♥✜♥❛❞♦s às ✐❧❤❛s ❡♠ r❡❧❛çã♦ ❛♦ ❛♠❜✐❡♥t❡ ❡①t❡r♥♦ q✉❡ ♦s ❝❡r❝❛✳ ❚❛❧ ❞✐s♣♦s✐çã♦ é ✉♠❛ ❞❛s ♠❛✐s ❛tr❛❡♥t❡s ♣r♦♣r✐❡❞❛✲ ❞❡s ❡①♣❧♦r❛❞❛s ♣❛r❛ ❛♣❧✐❝❛çõ❡s❬✶✵❪✱ q✉❡ ✈❛rr❡♠ ❞❡s❞❡ ◗❉ ❧❛s❡rs ❞❡st✐♥❛❞♦s à ár❡❛ ❞❡ t❡❧❡❝♦♠✉♥✐❝❛çõ❡s✱ ❛té ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ s♣✐♥✲q✉❜✐t ✭✉♥✐❞❛❞❡s ❞❡ ✐♥❢♦r♠❛çã♦ ❝♦♠ q✉❛❧✐❞❛❞❡ q✉â♥t✐❝❛✮ ✈♦❧t❛❞♦s à t❡❝♥♦❧♦❣✐❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✷_✳

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✸

◆❡ss❡ ❝♦♥t❡①t♦✱ ✉♠❛ ❜♦❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❡st✉❞♦ q✉❛♥t♦ ❛♦s ♠❡❝❛♥✐s♠♦s ❞❡ ❣❡r❛çã♦ ❡ ♠❛♥✐♣✉❧❛çã♦ ✉❧tr❛rrá♣✐❞❛ ❞❛ ♦r❞❡♠ ♠❛❣♥ét✐❝❛ ❡♠ s❡♠✐❝♦♥❞✉t♦r❡s ❛tr❛✈és ❞❡ ❡stí♠✉❧♦s ❧✉♠✐♥♦s♦s ✭❞✐t♦ ♦♣t♦♠❛❣♥❡t✐s♠♦✮ ♠❡r❡❝❡ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡✱ ✉♠❛ ✈❡③ q✉❡ ❝♦♥❝✐❧✐❛ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥❢♦r♠❛çã♦ à ❛❧t❛s ✈❡❧♦❝✐❞❛❞❡s ♣❛r❛ ❝♦♠ ❛s ❝❛♣❛❝✐❞❛❞❡s ❛♠♣❧✐❛❞❛s ❞❡ tr❛♥s♠✐ssã♦ ❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳

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❈❛♣ít✉❧♦ ✷

❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

✷✳✶ ■❧❤❛ ◗✉â♥t✐❝❛

❯♠❛ ■❧❤❛ ◗✉â♥t✐❝❛ ✭❡♠ ✐♥❣❧ês✱ ◗✉❛♥t✉♠ ❉♦t✮ é ✉♠❛ ♥❛♥♦❡str✉t✉r❛ s❡♠✐❝♦♥❞✉✲ t♦r❛ q✉❡ ❧✐♠✐t❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❡❧étr♦♥s ❞❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦✱ ❜✉r❛❝♦s ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ♦✉ ❡①❝✐t♦♥s ✭♣❛r❡s ❧✐❣❛❞♦s ❞❡ ❡❧étr♦♥s ❞❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❡ ❜✉r❛❝♦s ❞❡ ✈❛❧ê♥❝✐❛✮ ❡♠ t♦❞❛s ❛s três ❞✐♠❡♥sõ❡s ❡s♣❛❝✐❛✐s✳ ❉❡✈✐❞♦ ❛♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥✜♥❛♠❡♥t♦✱ t❛✐s ❡str✉t✉r❛s sã♦ t❛♠❜é♠ ❝❤❛♠❛❞❛s ❞❡ át♦♠♦s ❛rt✐✜❝✐❛✐s✱ ❡ sã♦ ❝♦♥s✐❞❡r❛❞❛s ♣r♦✲ ♠✐ss♦r❛s ❝❛♥❞✐❞❛t❛s ♥♦ ❝❛♠♣♦ ❞❛ s♣✐♥trô♥✐❝❛ ❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❞✐s♣♦s✐t✐✈♦s ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉â♥t✐❝❛s✳

◆♦ ❝♦♥t❡①t♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ ✉♠ ❡❧étr♦♥ é ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛

Ψ ❛ q✉❛❧ ♦❜❡❞❡❝❡ ❛ ❞❡♥♦♠✐♥❛❞❛ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❬✶✶❪ ❞❛❞❛ ♣♦r ✿

i~∂

∂tΨ =HΨ ✭✷✳✶✮

♦♥❞❡~ é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❞✐✈✐❞✐❞❛ ♣♦r₂_π✱_t ♦ t❡♠♣♦ ❡ _H ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞♦

s✐st❡♠❛✳

❊♠ ♣❛rt✐❝✉❧❛r ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♣♦❞❡ s❡r s❡♣❛r❛❞❛ ❡♠ ❞✉❛s ❝♦♠♣♦s✐çõ❡s✿ ❛ ♣r✐♠❡✐r❛ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ T(t)✱ ❡ ❛ s❡❣✉♥❞❛ ❞❡♣❡♥❞❡♥t❡ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s✱Ψ(r)✳

(15)

✷✳✷ ◆í✈❡✐s ❞❡ ❊♥❡r❣✐❛ ❡ ❘❡❣r❛s ❞❡ ❙❡❧❡çã♦ Ó♣t✐❝❛ ✺

s✐st❡♠❛ ❡ ❞❛❞❛ ♣♦r✿

T(t) = exp

−iE

~t

✭✷✳✷✮ ❊♥q✉❛♥t♦ q✉❡ ❛ ♣❛rt❡ ❡s♣❛❝✐❛❧ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✲ ❞❡♥♦♠✐♥❛❞❛ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ✲ ❢♦r♠✉❧❛❞❛ ❝♦♠♦✿

− ~

2

2me

∇2 ₊_V₍_r₎

Ψ(r) =EΨ(r) ✭✷✳✸✮

♦♥❞❡ me é ❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥✱ ∇♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦ ❡V(r)❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞♦ s✐st❡♠❛✳

P❛rt✐❝✉❧❛r♠❡♥t❡✱ ❞❡✈✐❞♦ à ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝♦♥✜♥❛♠❡♥t♦ ❡❧❡trô♥✐❝♦✱ ♣♦❞❡♠♦s ❞❡s✲ ❝r❡✈❡r ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ❝♦♠♦ ✉♠❛ ✧❝❛✐①❛✧ ❝ú❜✐❝❛ ❝❛♣❛③ ❞❡ ❝♦♥✜♥❛r ✉♠ ❡❧étr♦♥ ❡♠ s❡✉ ✐♥t❡r✐♦r ❡♠ t♦❞❛s ❛s ❞✐♠❡♥sõ❡s ❡s♣❛❝✐❛✐s✱ ♥✉♠❛ ❡s♣é❝✐❡ ❞❡ ❡str✉t✉r❛ ③❡r♦ ❞✐♠❡♥✲ s✐♦♥❛❧✳ ❆ s♦❧✉çã♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ é✱ ♣♦rt❛♥t♦✱ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❜ás✐❝❛ ❡ ❞❛❞❛ ♣♦r ✿

E = ~

2_π2

2mea2

n2_x+n2_y +n2_z ✭✷✳✹✮

♦♥❞❡ a é ❛ ❞✐♠❡♥sã♦ ❞❛ ❝❛✐①❛ ❡ nx, ny, nz sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳

◆❛ ✈❡r❞❛❞❡✱ ♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ♥ã♦ sã♦ tã♦ s✐♠♣❧❡s ❝♦♠♦ ♥❛ ❡q✉❛çã♦ ✭✷✳✹✮✱ ♣♦ré♠ ❛s ❡♥❡r❣✐❛s ❝❛❧❝✉❧❛❞❛s ♣♦r ❡ss❛ ✈✐❛ ♥♦s ❞ã♦ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❝♦♥✜♥❛♠❡♥t♦ ❞♦s ❡❧étr♦♥s r❡s✐❞❡♥t❡s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛✳

✷✳✷ ◆í✈❡✐s ❞❡ ❊♥❡r❣✐❛ ❡ ❘❡❣r❛s ❞❡ ❙❡❧❡çã♦ Ó♣t✐❝❛

(16)

✻ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

♥♦ ❡♥t❛♥t♦✱ é ❣r❛♥❞❡ q✉❛♥❞♦ ❝♦♥❢r♦♥t❛❞❛ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ✐♥t❡r❛tô♠✐❝❛ tí♣✐❝❛ ♣r❡s❡♥t❡ ❡♠ ❡str✉t✉r❛s só❧✐❞❛s ✲ ❞❛ ♦r❞❡♠ ❞❡ ❛❧❣✉♥s â♥❣str♦♥s✳

P♦rt❛♥t♦✱ ♦ ❡st❛❞♦ ❡❧❡trô♥✐❝♦ ♥❛s ✐❧❤❛s q✉â♥t✐❝❛s ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ❇❧♦❝❤✱ ❝✉❥❛ ❡①t❡♥sã♦ ❡♥❝♦♥tr❛✲s❡ ♥❛ ❡s❝❛❧❛ ❛tô♠✐❝❛✱ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞♦ t✐♣♦ ❡♥✈❡❧♦♣❡✱ ❝✉❥❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ é ❞♦ t❛♠❛♥❤♦ ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛✳

❆s ✐❧❤❛s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❡ ♥♦ss♦ ✐♥t❡r❡ss❡ ❛♣r❡s❡♥t❛♠ ✉♠ ❣❛♣ ❞✐r❡t♦✱ ♥♦ q✉❛❧ ♦s ❡st❛❞♦s ♣ró①✐♠♦s ❛♦ ♠í♥✐♠♦ ❞❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❛♣r❡s❡♥t❛♠ ♣r♦♣r✐❡❞❛❞❡s ❡q✉✐✈❛✲ ❧❡♥t❡s ❛ ✉♠ ♦r❜✐t❛❧ ❛tô♠✐❝♦ t✐♣♦✲s✱ ❡♥q✉❛♥t♦ q✉❡ ♦s ❡st❛❞♦s ♣ró①✐♠♦s ❛♦ ♠á①✐♠♦ ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ❡①✐❜❡♠ s✐♠❡tr✐❛ ❞❡ ♦r❜✐t❛✐s t✐♣♦✲♣✳ ❆ ✜❣✉r❛ ✷✳✶✱ ❛ s❡❣✉✐r✱ ❡①✐❜❡ ❛ ♠✐❝r♦❣r❛✜❛ ♦❜t✐❞❛ ♣♦r ▼✐❝r♦s❝♦♣✐❛ ❊❧❡trô♥✐❝❛ ❞❡ ❚r❛♥s♠✐ssã♦ ✭❚❊▼✮ ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ❞❡ ■♥❆s✴●❛❆s ❝r❡s❝✐❞❛ ✈✐❛ ❊♣✐t❛①✐❛ ♣♦r ❋❡✐①❡ ▼♦❧❡❝✉❧❛r ✭▼❇❊✮✱ ❜❡♠ ❝♦♠♦ ✉♠ ❡sq✉❡♠❛ s✐♠♣❧✐✜❝❛❞♦ ❞♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ♣r❡s❡♥t❡s ❡♠ ✉♠ tí♣✐❝♦ q✉❛♥t✉♠ ❞♦t✳

❋✐❣✉r❛ ✷✳✶✿ ✭❛✮ ▼✐❝r♦❣r❛✜❛ ♦❜t✐❞❛ ♣♦r ♠✐❝r♦s❝♦♣✐❛ ❡❧❡trô♥✐❝❛ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ❞❡ ■♥❆s✴●❛❆s ❝r❡s❝✐❞❛ ✈✐❛ ❡♣✐t❛①✐❛ ♣♦r ❢❡✐①❡ ♠♦❧❡❝✉❧❛r❬✶✷❪❀ ✭❜✮ ❊sq✉❡♠❛ s✐♠♣❧✐✜❝❛❞♦ ❞♦s ❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛ ❞✐s❝r❡t♦s ❡♠ ✉♠ q✉❛♥t✉♠ ❞♦t✱ ♦♥❞❡ ❛ s❡♣❛r❛çã♦ ❞❡ ❡♥❡r❣✐❛ Eg ❝♦rr❡s♣♦♥❞❡ ❛♦ ❣❛♣ ❡♥❡r❣ét✐❝♦ ❡♥tr❡ ❛s ❜❛♥❞❛s ❞❡ ❝♦♥❞✉çã♦ ❡ ✈❛❧ê♥❝✐❛✳ ❆s ♥♦♠❡♥❝❧❛t✉r❛s Sc ❡ Pc r❡❢❡r❡♠✲s❡ ❛♦s ❡st❛❞♦s ❞❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ q✉❡ ❛♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s✐♠❡tr✐❛ t✐♣♦✲s ❡ t✐♣♦✲♣✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ ♦s ❡st❛❞♦s Sv ❡ Pv ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛❬✶✷❪✳

(17)

✷✳✷ ◆í✈❡✐s ❞❡ ❊♥❡r❣✐❛ ❡ ❘❡❣r❛s ❞❡ ❙❡❧❡çã♦ Ó♣t✐❝❛ ✼

♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦s ❡st❛❞♦s ♥❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❛♣r❡s❡♥t❛♠ ♣r♦♣r✐❡❞❛✲ ❞❡s s❡♠❡❧❤❛♥t❡s ❛ ♦r❜✐t❛✐s t✐♣♦✲s ✭♥ú♠❡r♦ q✉â♥t✐❝♦ ♦r❜✐t❛❧ ❧❂✵✮✱ ❡ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛✱ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧ ❞♦s ❡❧étr♦♥s ♣r❡s❡♥t❡s ♥❡ss❛ ❢❛✐①❛ é s✐♠♣❧❡s♠❡♥t❡ ms =±1₂ ✭↑♦✉↓✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦s ❡st❛❞♦s ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ❛♣r❡s❡♥t❛♠ s✐♠❡tr✐❛ t✐♣♦✲♣

✭♥ú♠❡r♦ q✉â♥t✐❝♦ ♦r❜✐t❛❧ ❧❂✶✮✱ ❡ ♣♦rt❛♥t♦✱ sã♦ ❞❡t❡r♠✐♥❛❞♦s ❛tr❛✈és ❞♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✳ ◆❡ss❡ ❝❛s♦ ❡s♣❡❝í✜❝♦✱ s❡❣✉❡ q✉❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧J =L+S ❡①✐❜❡

❛s s❡❣✉✐♥t❡s ♣r♦❥❡çõ❡s ✈❡t♦r✐❛✐s✶_✿ _Jh

z =±32 ✭⇑♦✉⇓✱ ❞❡♥♦♠✐♥❛❞♦s ✧❜✉r❛❝♦s ♣❡s❛❞♦s✧✮

❡Jh

z =±12 ✭❞❡♥♦♠✐♥❛❞♦s ✧❜✉r❛❝♦s ❧❡✈❡s✧✮✳

❉❡ss❛ ♠❛♥❡✐r❛✱ s❡❣✉❡ q✉❡ ❛ ❛❜s♦rçã♦ ❞❡ ✉♠ ❢ót♦♥ ♣♦❞❡ ❛✉♠❡♥t❛r ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧ ❞❡ ✉♠ ❡❧étr♦♥ ♥❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ♣❛r❛ ❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❡♠

Je

z = −Jzh + 1✱ s❡ ♦ ❢ót♦♥ ❢♦r ❝✐r❝✉❧❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ♣❛r❛ ❞✐r❡✐t❛✱ ♦✉ r❡❞✉③✐r ❡♠

Je

z =−Jzh −1✱ s❡ ♦ ❢ót♦♥ ❢♦r ❝✐r❝✉❧❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ♣❛r❛ ❡sq✉❡r❞❛✳ ❚❛✐s r❡❣r❛s ❞❡ s❡❧❡çã♦ ó♣t✐❝❛✷ _{❡♥❝♦♥tr❛♠✲s❡ ❡sq✉❡♠❛t✐③❛❞❛s ♥❛ ✜❣✉r❛ ✷✳✷ ❛❜❛✐①♦✳}

❋✐❣✉r❛ ✷✳✷✿ ❘❡❣r❛s ❞❡ s❡❧❡çã♦ ó♣t✐❝❛ ❛♣❧✐❝❛❞❛s às tr❛♥s✐çõ❡s ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥✲ t❛❧✱ ✐♥❝❧✉✐♥❞♦ ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛✳ P❛rt❡ s✉♣❡r✐♦r ❝♦rr❡s♣♦♥❞❡ às ♣r♦❥❡çõ❡s ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧ J ❞♦s ❡❧étr♦♥s ❢♦t♦❡①❝✐t❛❞♦s ♥❛ ❜❛♥❞❛ ❞❡ ❝♦♥✲

❞✉çã♦✱ ❡♥q✉❛♥t♦ ❛ ♣❛rt❡ ✐♥❢❡r✐♦r r❡♣r❡s❡♥t❛ ❛s ♣r♦❥❡çõ❡s Jz ❞♦s ❜✉r❛❝♦s ♥❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛✳❬✶✵❪

◆♦ ❝❛s♦ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♦s ❡st❛❞♦s ❞❡ ✈❛❧ê♥❝✐❛ ❞❡ ❜✉r❛❝♦s✲♣❡s❛❞♦s ❡ ❜✉r❛❝♦s✲❧❡✈❡s sã♦ s❡♣❛r❛❞♦s ♣♦r ✉♠❛ ❡♥❡r❣✐❛ ∆HL ❞❡✈✐❞♦ ❛♦s ❡❢❡✐t♦s ❞❡ t❡♥sã♦ ✭❡♠ ✐♥❣❧ês✱ str❛✐♥✮ ✶_{❖ ❡✐①♦ ③ ❞❡ q✉❛♥t✐③❛çã♦ é ❡s❝♦❧❤✐❞♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❛ ✐❧❤❛ q✉â♥t✐❝❛ ❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s} ❡①♣❡r✐♠❡♥t♦s t❛♠❜é♠ é ♣❛r❛❧❡❧♦ à ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ ❢❡✐①❡ ❞❡ ❡①❝✐t❛çã♦✳

(18)

✽ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

❡ ❝♦♥✜♥❛♠❡♥t♦ q✉â♥t✐❝♦✳ ◆❛ ♣rát✐❝❛✱ ♦s ❡❢❡✐t♦ ❞❡ t❡♥sã♦✱ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❢♦r♠❛✱ ❡♥tr❡ ♦✉tr♦s✱ ♣♦❞❡♠ ✐♥tr♦❞✉③✐r ♦ ❛❝♦♣❧❛♠❡♥t♦ ❞♦s ❜✉r❛❝♦s✲❧❡✈❡s ❛♦s ♣❡s❛❞♦s✱ ❞❡ ❢♦r♠❛ t❛❧ q✉❡ t♦❞❛s ❛s tr❛♥s✐çõ❡s ✐♥❞✐❝❛❞❛s ♥❛ ✜❣✉r❛ ✷✳✷ s❡❥❛♠ ♣♦ssí✈❡✐s ❡♥tr❡ ♦s ❡st❛❞♦s ♠✐st♦s✱ ❝♦♠ ❞✐❢❡r❡♥t❡s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♦❝♦rrê♥❝✐❛✳

✷✳✸ ❍❛♠✐❧t♦♥✐❛♥❛ ❞❡ ✉♠ ❊❧étr♦♥ ❡♠ ❈❛♠♣♦ ▼❛❣♥é✲

t✐❝♦

❯♠❛ ♣❛rtí❝✉❧❛ ❝❛rr❡❣❛❞❛ ❣✐r❛♥❞♦ ❡♠ ✉♠❛ ór❜✐t❛ ❝✐r❝✉❧❛r ❝♦♥st✐t✉✐ ✉♠ ❞✐♣♦❧♦ ♠❛❣✲ ♥ét✐❝♦✳ ❙❡✉ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦✱ µ✱ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r

✐♥trí♥s❡❝♦ ❞❡ s♣✐♥✱ S ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♠✉❧t✐♣❧✐❝❛t✐✈❛γ ❞❡♥♦♠✐♥❛❞❛ r❛③ã♦ ❣✐r♦♠❛❣✲

♥ét✐❝❛ ❬✶✶❪✱ ♦✉ s❡❥❛✿

µ=γS ✭✷✳✺✮

◗✉❛♥❞♦ ✉♠ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ µ é ♣♦s✐❝✐♦♥❛❞♦ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ B✱ ♦ ♠❡s♠♦ ❡①♣❡r✐♠❡♥t❛ ✉♠ t♦rq✉❡ ❞❛ ❢♦r♠❛µ_×B✱ ♦ q✉❛❧ t❡♥❞❡ ❛ ❛❧✐♥❤á✲❧♦ ♣❛r❛❧❡❧♦

❛♦ ❝❛♠♣♦✳

P❛r❛ ❝❛❧❝✉❧❛r♠♦s ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❛ss♦❝✐❛❞❛ ❛ ❡ss❡ s✐st❡♠❛ ♣r❡❝✐s❛♠♦s ♣r✐♠❡✐r♦ ❢❛③❡r ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ❝❛♠♣♦B✉♥✐❢♦r♠❡ ❡ ❛♣♦♥✲

t❛♥❞♦ ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦ y✱ ❝♦♥❢♦r♠❡ ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✷✳✸✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡✲

r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ✈✐♥❞♦ ❞♦ ✐♥✜♥✐t♦ ❡♥❝♦♥tr❛✲s❡ ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ x ❡ q✉❡ ❛♦ s❡r ❧❡✈❛❞♦ ❛té ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s s♦❢r❡

✉♠❛ r♦t❛çã♦ ❛té ❝❤❡❣❛r ♥✉♠❛ ♣♦s✐çã♦ ❞❡ â♥❣✉❧♦ θ ♣❛r❛ ❝♦♠ ♦ ❡✐①♦ ✈❡rt✐❝❛❧✳

◆❡ss❛ s✐t✉❛çã♦ ♦ t♦rq✉❡ ❡①❡r❝✐❞♦ ♣♦r B é τ =µ_×B =µBsenθzˆ ❡ ❝♦♥s❡q✉❡♥t❡✲

♠❡♥t❡ ♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ❛té ❛ ❝❤❡❣❛❞❛ ♥❡ss❛ ❝♦♥✜❣✉r❛çã♦ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r✿

U = Z θ

π/2

µBsenθ′dθ′ =µB(₋cosθ′)θπ/2 =−µB(cosθ−cos(π/2))

(19)

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✾

❋✐❣✉r❛ ✷✳✸✿ ❋✐❣✉r❛ ❡sq✉❡♠át✐❝❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❡ ✉♠ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❡♠ ✉♠ ❝❛♠♣♦ ✉♥✐❢♦r♠❡ B✳ ❖ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ µ ♣❛rt❡ ❞♦ ✐♥✜♥✐t♦

❛♣♦♥t❛♥❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡ ❛♦ ❡✐①♦ x ❡ s♦❢r❡ ✉♠❛ r♦t❛çã♦ ♠❡❞✐❛♥t❡ ♦ t♦rq✉❡ ❡①❡r❝✐❞♦

♣♦rB ❛té ❝❤❡❣❛r ♥✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ â♥❣✉❧♦ θ ♣❛r❛ ❝♦♠ ♦ ❡✐①♦ ✈❡rt✐❝❛❧ ❬✶✹❪✳

❋✐♥❛❧♠❡♥t❡✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❡♠ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦B é ❞❛❞❛ ♣♦r✿

H =₋µ_·B ✭✷✳✼✮

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛

❆ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ é ♣r♦✈❡♥✐❡♥t❡ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❞♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❡❧❡trô✲ ♥✐❝♦ ♣❛r❛ ❝♦♠ ♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ♥✉❝❧❡❛r✱ ❡ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ❝♦♠♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ❡❧étr♦♥ ♥❛ ♣r❡s❡♥ç❛ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞♦ ♥ú❝❧❡♦✳ ◆❡ss❛s ❝♦♥❞✐çõ❡s✱ é r❛③♦á✲ ✈❡❧ s✉♣♦r q✉❡ ❛ ✐♥t❡r❛çã♦ ♦❝♦rr❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛ ✉♠ ♣❛r ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s✱ ❝✉❥❛ ❛❜♦r❞❛❣❡♠ s❡rá ♠❡❧❤♦r ❞❡s❡♥✈♦❧✈✐❞❛ ♥❛s ♣❛ss❛❣❡♥s s❡❣✉✐♥t❡s✳

✷✳✹✳✶ ❖ ❈❛♠♣♦ ▼❛❣♥ét✐❝♦ ❉✐♣♦❧❛r

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❝❧áss✐❝♦✱ ♦ t❡r♠♦ ❞♦♠✐♥❛♥t❡ ♥❛ ❡①♣❛♥sã♦ ❡♠ ♠✉❧t✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❞♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦rA♣♦❞❡ s❡r ❡①♣r❡ss♦✸❡♠ t❡r♠♦s ❞♦ ♠♦♠❡♥t♦

♠❛❣♥ét✐❝♦ µ❝♦♠♦✿

A= µ0 4π

µ_×r

r3 ✭✷✳✽✮

(20)

✶✵ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

♦♥❞❡µ0 ❝♦♥s✐st❡ ❞❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛ ♥♦ ✈á❝✉♦✱ ❡r ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❡t♦r

♣♦s✐çã♦ ❞♦ ♣♦♥t♦ ❞❡ ♦❜s❡r✈❛çã♦ ❬✶✺❪✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡✈✐❞♦ ❛♦ ❞✐♣♦❧♦ µ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦

❝♦♥s✐❞❡r❛♥❞♦✲s❡ ♦ r♦t❛❝✐♦♥❛❧ ❞❛ ❡q✉❛çã♦ ✭✷✳✽✮✳ P❛r❛ t❛♥t♦✱ ✉s❛♥❞♦✲s❡ ❞❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡ ✈❡t♦r✐❛❧ ∇_×₍_A_×_B_{) =}_A₍∇_·_B₎₋_B₍∇_·_A_{) + (}_B_·∇₎_A₋₍_A_·∇₎_B✱

t❡♠♦s q✉❡

∇_×₍_µ_×_r_{) = (}∇_·_r₎_µ_{+ (}_r_·∇₎_µ₋₍∇_·_µ₎_r₋₍_µ_·∇₎_r

= 3µ₋(µ_·∇₎_r ✭✷✳✾✮

✉♠❛ ✈❡③ q✉❡∇_·_r _{= 3}✱ ❡ q✉❡ q✉❛❧q✉❡r ❞❡r✐✈❛❞❛ ❡s♣❛❝✐❛❧ ❞❡ _µé ✐❣✉❛❧ ❛ ③❡r♦✳

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦

(µ_·∇₎_r ₌

µx

∂ ∂x +µy

∂ ∂y +µz

∂ ∂z

(xxˆ+yyˆ+zzˆ)

= µxxˆ+µyyˆ+µzzˆ= µ ✭✷✳✶✵✮ ❙❡❣✉❡ q✉❡∇_×₍_µ_×_r_{) = 3}_µ₋_µ_{= 2}_µ✱ ❞♦ q✉❛❧ ♦❜té♠✲s❡ q✉❡ ✹

B = ∇_×_A₌ µ0

4π

−_r35r×(µ×r) + 2µ

r3

= µ0 4π

1

r5

−3r_×(µ_×r) + 2r2µ ✭✷✳✶✶✮

❋✐♥❛❧♠❡♥t❡✱ ✉s❛♥❞♦✲s❡ ❞❡ q✉❡ r_×(µ_×r) = (r_·r)µ₋(r_·µ)r =r2_µ₋₍_r_·_µ₎_r

❝♦♥❝❧✉í♠♦s q✉❡ ♦ ❝❛♠♣♦ ❞❡✈✐❞♦ ❛ ✉♠ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ é ❞❡s❝r✐t♦ ♣♦r✿

B = µ0 4π

1

r5

3(r_·µ)r₋r2µ ✭✷✳✶✷✮

❊♠ ♣♦ss❡ ❞❡ss❡ r❡s✉❧t❛❞♦ ✭✷✳✶✷✮✱ s❡❣✉❡ q✉❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦E ❡♥tr❡

❞♦✐s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s µ1 ❡ µ2 é ❞❛❞❛ ♣♦r ♠❡✐♦ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞♦ ❝❛♠♣♦

✹_{❆q✉✐ ❡♠♣r❡❣♦✉✲s❡ ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡ ✈❡t♦r✐❛❧✿}

(21)

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✶✶

♠❛❣♥ét✐❝♦ ❣❡r❛❞♦ ♣❡❧♦ ♣r✐♠❡✐r♦ ✭B1✮ ♣❛r❛ ❝♦♠ ♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ❞♦ s❡❣✉♥❞♦

✭µ2✮✱ ♦✉ s❡❥❛✱ E =−µ2·B1 ✲ ✈✐❞❡ ❡q✉❛çã♦ ✭✷✳✼✮✳ P♦r ❝♦♥s❡q✉ê♥❝✐❛✱ ❝♦♥❝❧✉í♠♦s q✉❡

E = µ0 4π

µ1·µ2

r3 −

3(µ1·r)(µ2·r)

r5

✭✷✳✶✸✮

♦♥❞❡ r ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❡t♦r q✉❡ ❝♦♥❡❝t❛ ❛♠❜♦s ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❡♥❝✐♦♥❛✲

❞♦s✳

✷✳✹✳✷ ❆ ■♥t❡r❛çã♦ ◗✉â♥t✐❝❛ ❇ás✐❝❛

❆ ❡q✉❛çã♦ ✭✷✳✶✷✮ ❡①♣r❡ss❛ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❣❡r❛❞♦ ♣♦r ✉♠❛ ♣❛rtí❝✉❧❛ q✉❡ ♣♦ss✉✐ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞✐♣♦❧❛r ♣✉r♦✳ ❚❛❧ r❡❧❛çã♦ t❡♠ s❡✉ r❡❣✐♠❡ ❞❡ ✈❛❧✐❞❛❞❡ r❡str✐t♦ ❛ r❡✲ ❣✐õ❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ❞✐st❛♥t❡s✱ ♥❛s q✉❛✐s ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❛♦ ❞✐♣♦❧❛r ❛♣r❡s❡♥t❛♠ ❝♦♥tr✐❜✉✐çõ❡s ❞❡s♣r❡③í✈❡✐s ❡♠ r❡❧❛çã♦ ❛ ❡ss❡ ú❧t✐♠♦✳ ❙❡♥❞♦ ❛s✲ s✐♠✱ s❡❣✉♥❞♦ ✭✷✳✶✸✮✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ q✉❡ ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ✉♠ ♥ú❝❧❡♦ ❛tô♠✐❝♦ ❡ ✉♠ ❡❧étr♦♥✱ s✉✜❝✐❡♥t❡♠❡♥t❡ ❛❢❛st❛❞♦s ✉♠ ❞♦ ♦✉tr♦✱ é ❞❛❞❛ ♣♦r✿

H= µ0 4π

µe·µn

r3 −

3(µe·r)(µn·r) r5

✭✷✳✶✹✮

❖♥❞❡ µe = −geµBIe/~ é ♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞♦ ❡❧étr♦♥❀ µn = gnµnIn/~ é ♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞♦ ♥ú❝❧❡♦❀ µB é ♦ ♠❛❣♥❡t♦♥ ❞❡ ❇♦❤r❀ µn é ♦ ♠❛❣♥❡t♦♥ ♥✉❝❧❡❛r❀

r ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❡t♦r q✉❡ ❝♦♥❡❝t❛ ❛♠❜♦s ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♠❡♥❝✐♦♥❛❞♦s❀ Ie r❡♣r❡s❡♥t❛ ♦ ♦♣❡r❛❞♦r ❞❡ s♣✐♥ ❞♦ ❡❧étr♦♥❀In ❝♦rr❡s♣♦♥❞❡ ❛♦ ♦♣❡r❛❞♦r ❞❡ s♣✐♥ ♥✉❝❧❡❛r❀ ge ❡gn sã♦✱ ♥❡ss❛ ♦r❞❡♠✱ ♦s ❢❛t♦r❡s ❣✐r♦♠❛❣♥ét✐❝♦s ❡❧❡trô♥✐❝♦ ❡ ♥✉❝❧❡❛r ❬✶✻❪✳

❊s❝r❡✈❡♥❞♦✲s❡ µe ❡ µn ❡♠ ❝♦♠♣♦♥❡♥t❡s ❝❛rt❡s✐❛♥❛s✱ s❡❣✉❡ q✉❡ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✭✷✳✶✹✮ ❛♣r❡s❡♥t❛ t❡r♠♦s ❞❛ ❢♦r♠❛✿

−gegn µBµn

~2 IexInx

1

r3 , −gegn

µBµn ~2 IexIny

xy

r3 ✭✷✳✶✺✮

❙❡ ❡①♣r❡ss❛r♠♦s Iex ❡ Iey ❡♠ t❡r♠♦s ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ❛❜❛✐①❛♠❡♥t♦ ❡ ❧❡✈❛♥t❛♠❡♥t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ I−

(22)

✶✷ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

I−

n✱ ❡ ❡s❝r❡✈❡r♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ①✱ ② ❡ ③ ❡♠ t❡r♠♦s ❞❛s ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s✱ ♣♦❞❡♠♦s ❞❡s❡♥✈♦❧✈❡r ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✭✷✳✶✹✮ ❡♠ ✉♠❛ ❢♦r♠❛ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❝♦♥✈❡♥✐❡♥t❡✺

H =₋µ0 4π

gegnµBµn

~2_r3 (A+B+C+D+E+F) ✭✷✳✶✻✮

♥❛ q✉❛❧                 

A=IezInz(1−3 cos2θ)

B =₋1 4(I

+

e I − n +I

−

e In+)(1−3 cos2θ)

C=₋3 2(I

+

e Inz+IezIn+) senθcosθe −iφ                   

D=₋3 2(I

−

e Inz+IezIn−) senθcosθeiφ

E =₋3 4I

+

e In+sen2θe −2iφ

F =₋3 4I

− e I

−

n sen2θe2iφ

✭✷✳✶✼✮ ❉❡st❛ ♠❛♥❡✐r❛✱ ❝❛❧❝✉❧❡♠♦s ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦ ♠❛❣♥ét✐❝❛ ❝♦♥s✐❞❡r❛♥❞♦✲ s❡ ✉♠ s✐st❡♠❛ ❡❧étr♦♥✲♥ú❝❧❡♦✱ ♥♦ q✉❛❧ ✭✷✳✶✹✮ ❛t✉❛ ❝♦♠♦ ✉♠❛ ♣❡rt✉r❜❛çã♦ s♦❜r❡ ♦s ♦r✲ ❜✐t❛✐s ❛tô♠✐❝♦s✳ ◆❡ss❛ s✐t✉❛çã♦✱ ❡♥q✉❛♥t♦ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❡❧❡trô♥✐❝❛ ❢♦r ✉♠ ❡st❛❞♦ t✐♣♦✲♣✱ t✐♣♦✲❞✱ ♦✉ q✉❛❧q✉❡r ♦✉tr♦ ❡st❛❞♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♥ã♦ ♥✉❧♦✱ ❡s♣❡r❛♠♦s q✉❡ ✭✷✳✶✹✮ s❡❥❛ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ♥♦ tr❛t❛♠❡♥t♦ ❞❛ ✐♥t❡r❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ♥❡ss❡s ❝❛s♦s✱ ❛s ❢✉♥çõ❡s ❡❧❡trô♥✐❝❛s s❡ ❛♥✉❧❛♠ ❡♠ r = 0✱ ❡ ♣♦rt❛♥t♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦

❡❧étr♦♥ s❡r ❡♥❝♦♥tr❛❞♦ ♥❛ r❡❣✐ã♦ ♣ró①✐♠❛ ❛♦ ♥ú❝❧❡♦ é ❜❛st❛♥t❡ ❞✐♠✐♥✉t❛✳ P❛r❛ ❡st❛❞♦s t✐♣♦✲s✱ ❡♥tr❡t❛♥t♦✱ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❡❧❡trô♥✐❝❛ é ♥ã♦✲♥✉❧❛ ♥❛ ♦r✐❣❡♠ ❡ ♥❡ss❛ s✐t✉❛çã♦ ❞❡ ❝✉rt❛s ❞✐stâ♥❝✐❛s ❛ ❛♣r♦①✐♠❛çã♦ ❞✐♣♦❧❛r t♦r♥❛✲s❡ s✉s♣❡✐t❛✳

❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s ❢♦r ♣❡q✉❡♥❛ ❛ ♣♦♥t♦ ❞❡ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r t♦r♥❛r❡♠✲s❡ r❡❧❡✈❛♥t❡s✱ ❛ ✐♥t❡r❛çã♦ ❞❛❞❛ ♣♦r ✭✷✳✶✹✮ t♦r♥❛✲s❡ ✐♥❡①❛t❛ ♥❛ ❞❡s❝r✐çã♦ ❞❛ ✐♥t❡r❛çã♦ ♠❛❣♥ét✐❝❛ ❡♥tr❡ ♦ ♥ú❝❧❡♦ ❛tô♠✐❝♦ ❡ ♦ ❡❧étr♦♥✳ ❯♠❛ ♦❜s❡r✈❛çã♦ ♠❛✐s ❝✉✐❞❛❞♦s❛ ❞❡ss❡ ❢❛t♦ ❡✈✐❞❡♥❝✐❛ ❛ ✐♠♣r❡❝✐sã♦ ❞❡ ✭✷✳✶✹✮✳ P❛r❛ t❛♥t♦✱ s✉♣♦♥❤❛♠♦s ✉♠ ✈❛❧♦r ♠é❞✐♦ ❞❡Hs♦❜r❡ ✉♠❛ ❢✉♥çã♦ ❡❧❡trô♥✐❝❛ t✐♣♦✲s✱ ✉✭r✮✱

(23)

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✶✸

❞❡st❛ ♠❛♥❡✐r❛✱ ❤á ✉♠ ♥ú♠❡r♦ ❞❡ t❡r♠♦s ❞❡ ✭✷✳✶✹✮ s✐♠✐❧❛r❡s ❛♦s t❡r♠♦s ❆✱ ❇✱ ❈✱ ❉✱ ❊ ❡ ❋ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦s ♥❛ ❡q✉❛çã♦ ✭✷✳✶✻✮✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡s❝♦❧❤❛♠♦s ♦ t❡r♠♦ ❆✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ q✉❛❧ ❞❡♣❡♥❞❡ ❞♦s ♣❛râ♠❡tr♦s ❞❡ â♥❣✉❧♦ ❡ ❞✐stâ♥❝✐❛✳ ▲♦❣♦✱ ❛ ♠❡♥♦s ❞❡ ❝♦♥st❛♥t❡s ♠✉❧t✐♣❧✐❝❛t✐✈❛s✱ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❡ t❛❧ t❡r♠♦ é ❞❛❞♦ ♣♦r✿

Z

u2(r)

r3 (1−3cos

2_θ₎_r2_{dr d}_Ω _{✭✷✳✶✽✮}

♦♥❞❡ dΩé ♦ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ â♥❣✉❧♦ só❧✐❞♦✳

◆❡ss❡ ♠♦♠❡♥t♦ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ✉♠❛ s✐t✉❛çã♦ s✐♥❣✉❧❛r✳ ❙❡ ❝❛❧❝✉❧❛r♠♦s ❛ ✐♥t❡✲ ❣r❛çã♦ ❛♥❣✉❧❛r ♣r✐♠❡✐r♦✱ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ♣❛r❛ ✭✷✳✶✽✮ é ③❡r♦✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡

Z Z

(1₋3cos2θ) senθ dθ dφ= (2π) Z π

0

(1₋3cos2θ)dθ = (2π) Z 1

−1

(1₋3x2)dx= 0

✭✷✳✶✾✮ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ✐♥t❡❣r❛r♠♦s ♣r✐♠❡✐r♦ s♦❜r❡ r✱ ❡♥❝♦♥tr❛♠♦s ✉♠ ♣r♦❜❧❡♠❛ ♥❛

♣r♦①✐♠✐❞❛❞❡ ❞❡ r_≃0✱ ♦♥❞❡ u2₍₀₎₆_{= 0}_{✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡✱ ♥❡ss❛ r❡❣✐ã♦✱ ❛ ✐♥t❡❣r❛çã♦}

r❡s✉❧t❛ ❡♠ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❡ ♦❝❛s✐♦♥❛ ❡♠ ✉♠ r❡s✉❧t❛❞♦ ✐♥✜♥✐t♦ ♣❛r❛ ✭✷✳✶✽✮✳

❉❡st❛ ♠❛♥❡✐r❛✱ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜t❡r ③❡r♦ ♦✉ ✐♥✜♥✐t♦ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♦r❞❡♠ ❞❡ ✐♥t❡❣r❛çã♦ ❡♠♣r❡❣❛❞❛✱ t♦r♥❛✲s❡ ❝❧❛r♦ q✉❡ ♥ã♦ ♣♦❞❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ✐❣♥♦r❛r ❛s ❝♦♥✲ tr✐❜✉✐çõ❡s ❞❡ q✉❛♥❞♦ r é ♣❡q✉❡♥♦✳

P♦rt❛♥t♦✱ t♦r♥❛✲s❡ ❡✈✐❞❡♥t❡ q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ ❞✐♣♦❧❛r é ✐♥❛❞❡q✉❛❞❛ ♣❛r❛ ❛ ❞❡s❝r✐✲ çã♦ ❞❛ ✐♥t❡r❛çã♦ ♠❛❣♥ét✐❝❛ ❡❧étr♦♥✲♥ú❝❧❡♦ ♥❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❛ ❢✉♥çã♦ ❡❧❡trô♥✐❝❛ ♥ã♦ s❡ ❛♥✉❧❛ ♥❛ ♣♦s✐çã♦ ♥✉❝❧❡❛r✳ ◆❡ss❡ ❝❛s♦✱ ❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥t♦r♥❛r t❛❧ ✐♥❝♦♥s✐stê♥❝✐❛ é ❛♣r❡s❡♥t❛❞❛ ♥❛ s❡çã♦ ✷✳✹✳✸ ❛❜❛✐①♦✳

✷✳✹✳✸ ❆ ■♥t❡r❛çã♦ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐

❖ ❢❛t♦ ❞♦ ❝❛♠♣♦ ❞✐♣♦❧❛r ✭✷✳✶✷✮ s❡r ✐♥❞❡✜♥✐❞♦ ♥❛ ♦r✐❣❡♠ ✭r = 0✮ r❡s✉❧t❛ ❡♠ ✉♠❛

(24)

✶✹ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

✷✳✹✳✷✱ ❛ ✐♥t❡❣r❛❧ ❛♥❣✉❧❛r r❡s✉❧t❛ ❡♠ ③❡r♦ ❝♦♠♦ s♦❧✉çã♦✱ ❡♥q✉❛♥t♦ ❛ ♣❛rt❡ r❛❞✐❛❧ t♦r♥❛✲ s❡ ✐♥✜♥✐t❛ ♠❡❞✐❛♥t❡ ❛ ✐♥t❡❣r❛çã♦✳

P❛r❛ ❡♥❝♦♥tr❛r ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛ss♦❝✐❛❞♦ ❛♦ s♣✐♥ ♥✉❝❧❡❛r ♥✉♠❛ r❡❣✐ã♦ ♣ró①✐♠❛ ❛♦ ❝❡♥tr♦ ❞♦ át♦♠♦✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ♥ú❝❧❡♦ ❝♦♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❛r❜✐trár✐❛ ❞❡ ❝♦rr❡♥t❡s ❡st❛❝✐♦♥ár✐❛s✱ ❧✐♠✐t❛❞❛s ❛ ✉♠❛ r❡❣✐ã♦ ❡s❢ér✐❝❛ ❞❡ r❛✐♦ R✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦s

♣❛ss♦s s❡❣✉✐♥t❡s✱ ❝❛❧❝✉❧❛r❡♠♦s ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❝❧áss✐❝♦ ❛ss♦❝✐❛❞♦ ❛ ❡st❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦rr❡♥t❡s✱ ❡ ✉s❛r❡♠♦s ♦ ✈❛❧♦r ♦❜t✐❞♦ ♥♦ ❧✐♠✐t❡ ❞❡R _→0❝♦♠♦ ❡st✐♠❛t✐✈❛ ❞♦ ❝❛♠♣♦

♠❛❣♥ét✐❝♦ ♥✉❝❧❡❛r ♣ró①✐♠♦ ❛♦ ❝❡♥tr♦ ❛tô♠✐❝♦✳

❖ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♠é❞✐♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ♥ú❝❧❡♦ é✱ ♣♦r ❞❡✜♥✐çã♦✱ ❞❛❞♦ ♣♦r✿

Bmedio =

1

τ Z

Bdτ′

✭✷✳✷✵✮

❖♥❞❡ τ = 4π 3 R

3 _{✭✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✮ ❡} _dτ′ _{♦ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✈♦❧✉♠❡✳} ❊s❝r❡✈❡♥❞♦✲s❡ B❡♠ t❡r♠♦s ❞♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r A❡ ✉s❛♥❞♦✲s❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ♠❛t❡♠á✲

t✐❝❛ ❛❧❣é❜r✐❝❛ ✻

Z

V olume

(∇_×_A₎_dτ′ ₌₋

I

Superf icie

A_×da ✭✷✳✷✶✮

❙❡❣✉❡ q✉❡

Bmedio =−

1

τ Z

A_×da ✭✷✳✷✷✮

❖♥❞❡ da ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ár❡❛ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ❡s❢❡r❛✱

❛♣♦♥t❛♥❞♦ ♥❛ ❞✐r❡çã♦ r❛❞✐❛❧ ❞❛ ♠❡s♠❛✳ ❆❣♦r❛✱ ❝♦♠♦ ♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r A ♣♦r s✐ só

❝♦♥s✐st❡ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ s♦❜r❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦rr❡♥t❡s ❬✶✹❪✱ s❡❣✉❡ q✉❡✿

A= µ0 4π

Z _J

(r′

)dτ′

|r₋r′

| ✭✷✳✷✸✮

❖♥❞❡ r′ _{❞❡✜♥❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❛s ❝♦rr❡♥t❡s ♥❛ ❡s❢❡r❛ ❡}

r ❞❡✜♥❡ ♦ ♣♦♥t♦ ❞❡ ♦❜s❡r✲

✈❛çã♦ ♥❛ ❝❛s❝❛ ❡s❢ér✐❝❛ ✭|r_|=R✮✱ ❝♦❢♦r♠❡ ❛ ❣❡♦♠❡tr✐❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✳✹✳

(25)

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✶✺

❋✐❣✉r❛ ✷✳✹✿ ●❡♦♠❡tr✐❛ ❜ás✐❝❛ ♣❛r❛ ❡q✉❛çã♦ ✭✷✳✷✸✮✱ ♦♥❞❡ ❞a ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡❧❡♠❡♥t♦

✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ár❡❛ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ❡s❢❡r❛❀ dτ′ _{❝♦rr❡s♣♦♥❞❡ ❛♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡} ✐♥t❡r♥♦ à ❡s❢❡r❛❀ r′ ❞❡✜♥❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦rr❡♥t❡s ♥❛ ❡s❢❡r❛ ❡ _r ❞❡✜♥❡ ♦ ♣♦♥t♦ ❞❡

♦❜s❡r✈❛çã♦✳ ❆❞❛♣t❛❞♦ ❞❡ ❬✶✼❪✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡❣✉❡ q✉❡

Bmedio = −

1

τ µ0 4π

I Z

J(r′

)dτ′

|r₋r′

|

×da

= ₋1

τ µ0 4π

Z

J(r′)_× I

da

|r₋r′

|

dτ′

✭✷✳✷✹✮ ✉♠❛ ✈❡③ q✉❡ J ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞❛s ❢♦♥t❡s ✭❝♦rr❡♥t❡s✮ ❡✱ ♣♦rt❛♥t♦✱

❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ r′ _{❡ ♥ã♦ ❞❛ ♣♦s✐çã♦}

r ❞❡ ♦❜s❡r✈❛çã♦✳

P❛r❛ ❢❛③❡r♠♦s ❛ ✐♥t❡❣r❛❧ ❞❡ s✉♣❡r❢í❝✐❡✱ ❡s❝♦❧❤❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❛s ❝♦♦r❞❡♥❛❞❛s ✭①✱②✱③✮ ❡♠ t❡r♠♦s ❞❛s ❝♦♦r❞❡♥❛❞❛s ❡s❢ér✐❝❛s ✭r✱θ✱φ✮ ❡ ❡st❛❜❡❧❡❝❡♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡

❣❡♥❡r❛❧✐❞❛❞❡✱ ♦ ❡✐①♦ ♣♦❧❛r ❥✉♥t♦ ❛ ❞✐r❡çã♦ ❞❡r′_{✳ ❆ss✐♠✱}

|r₋r′

|2 ₌_R2₊_z′2₋₂_Rz′

cosθ✱

❡♥q✉❛♥t♦ q✉❡da=Rsenθdθdφrˆ✳

I _d_a

|r₋r′

| =

I _(cos_θ

ˆ

z+ senθcosφxˆ+ senθsenφyˆ)R2_sen_θdθdφ √

R2₊_z′2₋₂_Rz′_cos_θ ✭✷✳✷✺✮ ❆s ❝♦♠♣♦♥❡♥t❡s ① ❡ ② ❛♥✉❧❛♠✲s❡ ♠❡❞✐❛♥t❡ ❛ ✐♥t❡❣r❛çã♦ ❡♠ φ✱ r❡st❛♥❞♦ s♦♠❡♥t❡

❛q✉❡❧❛ ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦ zˆ

I

da

|r₋r′

| = 2πR 2

ˆ

z

Z

senθcosθ dθ √

R2₊_z′₂

−2Rz′_cos_θ ✭✷✳✷✻✮ P♦r s✉❜st✐t✉✐çã♦ ❞❛ ❢♦r♠❛ u= cosθ _⇒du=₋senθdθ ❡♠ ✭✷✳✷✻✮

= 2πR2zˆ

Z

u du √

R2₊_z′₂

(26)

✶✻ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

◆♦✈❛♠❡♥t❡ ♣♦r ✉♠❛ ♥♦✈❛ s✉❜st✐t✉✐çã♦ v = R2 ₊_z′2 ₋₂_Rz′

u _⇒ dv = ₋2Rz′

du

❝♦♥❝❧✉í♠♦s q✉❡

= 2πR2zˆ

1 (2Rz′₎2

Z

(v₋R2₋z′2₎_dv √

v

= 2πR2zˆ

1 (2Rz′₎2

2 3v

3/2

−2(R2+z′2₎_v1/2

= 2πR2zˆ 1

(2Rz′₎2

2 3v

1/2_v

−3(R2+z′2₎

= 2πR2zˆ 1

(2Rz′₎₂

−2₃2(R2+z′2_{) + 2}_Rz′

u√R2₊_z′2 ₋₂_Rz′_u

✭✷✳✷✽✮ ❆♣❧✐❝❛♥❞♦✲s❡ ♦s ❧✐♠✐t❡s ❞❡ ✐♥t❡❣r❛çã♦✱ ♦❜t❡♠♦s q✉❡✿

=₋ 2πR 2

3(Rz′₎2zˆ n

[R2+z′2₊_Rz′

]√R2₊_z′₂

−2Rz′

−[R2+z′2 −Rz′

]√R2₊_z′₂_{+ 2}_Rz′o

=₋ 2π 3z′₂zˆ

(R2+z′2₊_Rz′

)_|R₋z′

| −(R2+z′2 −Rz′

)(R+z′

) ✭✷✳✷✾✮

❙❡ r′

< R✱ ♦✉ s❡❥❛✱ ③✬ ❁ ❘✱ s❡❣✉❡ q✉❡ I

da

|r₋r′

| = −

2π

3z′2zˆ

(R2+z′2₊_Rz′

)(R₋z′

)₋(R2+z′2 −Rz′

)(R+z′

)

= ₋ 2π 3z′2zˆ

−2R2z′

−2z′3_{+ 2}_R2_z′

= 4π 3 z

′ ˆ

z= 4π 3 r

′ _{✭✷✳✸✵✮}

❋✐♥❛❧♠❡♥t❡✱ t♦♠❛♥❞♦✲s❡ ❡ss❡ r❡s✉❧t❛❞♦ ❡ ❛♣❧✐❝❛♥❞♦✲♦ ♥❛ ✐♥t❡❣r❛çã♦ ✈♦❧✉♠étr✐❝❛ ❞❡ ✭✷✳✷✹✮ ❝♦♥❝❧✉í♠♦s q✉❡

Bmedio =−

3µ0 (4π)2_R3

4π

3 Z

J(r′)_×r′dτ′

= µ0 4π

2µ

R3 ✭✷✳✸✶✮

❖♥❞❡ µ= 1 2

R

r′

×J(r′

)dτ′ _{é ♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ t♦t❛❧ ❞❛ ❡s❢❡r❛}✼_✳

❖ ❝❛♠♣♦ ❢♦r❛ ❞❡ t❛❧ ❡s❢❡r❛ é ❞❛❞♦ ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✶✷✮✱ ♦✉ s❡❥❛✱

Bf ora(r) =

µ0 4π

1

r3 [3(µ·rˆ)rˆ−µ] parar > R ✭✷✳✸✷✮

✼_{P♦r ❡①❡♠♣❧♦✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠❛ ❝♦rr❡♥t❡} _I _{❛tr❛✈és ❞❡ ✉♠❛ ❡s♣✐r❛ ❞❡ r❛✐♦} _R_{✱ ❡ ❞❡✜♥✐r✲} ♠♦s ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ♦ ❝❡♥tr♦ ❞❛ ❡s♣✐r❛✱ t❡♠♦s✱ ❡♠ ❝♦♦r✲ ❞❡♥❛❞❛s ❝✐❧í♥❞r✐❝❛s✱ q✉❡ ❛ r❡❧❛çã♦ ❡♥✉♥❝✐❛❞❛ s❡ r❡❞✉③ ❛ µ = 1/2Rr′ _×₍_Id_l′₎ ✭❝❛s♦ ✉♥✐❞✐♠❡♥✲

s✐♦♥❛❧ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ✜♦ J(r′₎_dτ′ _→ _Id_l′✮✱ ♦♥❞❡ _r′ ₌ _R_r_ˆ ❡ _d_l′ ₌ _Rdθ_θˆ✳ ■ss♦ ✐♠♣❧✐❝❛ ❡♠

µ = 1 2

R

(Rrˆ)×(IRdθθˆ) = I(πR2_)ˆ_z ₌ _IA_z_ˆ_{✱ s❡♥❞♦ ❆ ❛ ár❡❛ ✐♥t❡r♥❛ à ❡s♣✐r❛✳ ❚❛❧ ❡①❡♠♣❧✐✜❝❛çã♦} ♣❡r♠✐t❡ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡❧❛çã♦ ❞✐s♣♦st❛✱ ❥á q✉❡ ❝♦♥❝♦r❞❛ ❝♦♠ ♦ r❡s✉❧t❛❞♦ ❤❛❜✐t✉❛❧ ❞❡

(27)

✷✳✹ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ✶✼

❊♥q✉❛♥t♦ q✉❡ ♦ ❝❛♠♣♦ ♥♦ ✐♥t❡r✐♦r ❞❛ ❡s❢❡r❛ é ✉♥✐❢♦r♠❡ ❡ ❞❛❞♦ ♣♦r ✭✷✳✸✶✮✿

Bdentro(r) =

µ0 2π

µ

R3 parar < R ✭✷✳✸✸✮

◆♦ ❧✐♠✐t❡ ❞❡ ✉♠ ❞✐♣♦❧♦ ✐❞❡❛❧ ✭R _→ 0✮✱ ❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❛ ❡s❢❡r❛ ❝♦♥tr❛✐ ❛té ③❡r♦✱

♠❛s ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡♠ s✐ t❡♥❞❡ ❛ ✐♥✜♥✐t♦✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ ✐♥t❡❣r❛❧ ❞❡Bdentro s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛

Z

Bdentrodτ =

_µ

0 2π

µ

R3 4 3πR

3

= 2

3µ0µ ✭✷✳✸✹✮

♣❡r♠❛♥❡ç❛ ❝♦♥st❛♥t❡✱ ♥ã♦ ✐♠♣♦rt❛♥❞♦ ♦ q✉ã♦ ♣❡q✉❡♥♦ ♦ r❛✐♦ s❡ t♦r♥❡✳ ◗✉❛♥❞♦

R _→ 0✱ ♣♦rt❛♥t♦✱ ♦ ❝❛♠♣♦ ♥♦ ✐♥t❡r✐♦r ❞❛ ❡s❢❡r❛ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❛tr❛✈és ❞❡ ✉♠❛

❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❉✐r❛❝✱ ♥❛ ❢♦r♠❛✿

Bdentro(r) =

2 3µ0µδ

3₍_r₎ _{✭✷✳✸✺✮}

t❛❧ t❡r♠♦ é ❞❡♥♦♠✐♥❛❞♦ ■♥t❡r❛çã♦ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ❡ ❞❡✈❡ s❡r ❛❞✐❝✐♦♥❛❞♦ ❛ ✭✷✳✶✷✮ ❛ ✜♠ ❞❡ ❝♦rr✐❣✐r ❛ ✐♥❞❡✜♥✐çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♥❛ ♣♦s✐çã♦ ♥✉❝❧❡❛r✳

P♦rt❛♥t♦✱ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡ ✉♠ ❞✐♣♦❧♦ ✐❞❡❛❧ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

B(r) = µo 4π

1

r3 [3(µ·rˆ)rˆ−µ] + 2 3µ0µδ

3₍_r₎ _{✭✷✳✸✻✮}

❝♦♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ q✉❡ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ s❡ ❛♣❧✐❝❛ ❛♣❡♥❛s ❛ ✉♠❛ r❡❣✐ã♦ ❡①t❡r♥❛ ❛ ✉♠❛ ❡s❢❡r❛ ✐♥✜♥✐t❡s✐♠❛❧ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠✳

▲♦❣♦✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ✐♥t❡r❛çã♦ ❡♠ ✉♠ s✐st❡♠❛ ❡❧étr♦♥✲♥ú❝❧❡♦ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ❝♦♠♦✿

H=₋µe·Bn H= µ0

4π

1

r3 [µe·µn−3(µe·rˆ)(µn·rˆ)]− 2

3µ0µe·µnδ

(28)

✶✽ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s

❋✐❣✉r❛ ✷✳✺✿ ❉✐❛❣r❛♠❛ ❡sq✉❡♠át✐❝♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❞❡♥tr♦ ❡ ❢♦r❛ ❞❡ ✉♠ ♥ú❝❧❡♦ ♠❛❣♥ét✐❝♦ ✭❛ss✉♠✐❞♦ ❡s❢ér✐❝♦ ❡ s♦♠❜r❡❛❞♦ ♥♦ ❞✐❛❣r❛♠❛✮✳ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❝❛✉s❛❞❛ ♣♦r ✉♠❛ ❝♦rr❡♥t❡ q✉❡ ✢✉✐ ❛♦ r❡❞♦r ❞♦ ❡q✉❛❞♦r ❞❛ ❡s❢❡r❛ ♠♦str❛❞❛ ❡♠ s♦♠❜r❡❛❞♦ ♥❛ ✜❣✉r❛✳ ❋♦r❛ ❞❛ ❡s❢❡r❛✱ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❞❡✈✐❞♦ ❛ ✉♠ ❞✐♣♦❧♦ ♣✉♥t✐❢♦r♠❡ ❧♦❝❛❧✐③❛❞♦ ♥♦ ❝❡♥tr♦ ❞❛ ❡s❢❡r❛ ❡ ❝✉❥♦ ✈❛❧♦r ♠é❞✐♦ s♦❜r❡ ✉♠ ✈♦❧✉♠❡ ❡s❢❡r✐❝❛♠❡♥t❡ s✐♠étr✐❝♦ r❡s✉❧t❛ ❡♠ ③❡r♦✳ ❉❡♥tr♦ ❞♦ ♥ú❝❧❡♦✱ ♦ ❝❛♠♣♦ ❡stá ❡♠ ♠é❞✐❛ ❛♣♦♥t❛♥❞♦ ♣❛r❛ ❝✐♠❛✱ ❞❡ ❢♦r♠❛ q✉❡ ♦ ✈❛❧♦r ♠é❞✐♦ s♦❜r❡ ✉♠ ✈♦❧✉♠❡ ❡s❢❡r✐❝❛♠❡♥t❡ s✐♠étr✐❝♦ ♥ã♦ r❡s✉❧t❛ ❡♠ ③❡r♦✳ ❊st❛ é ❛ ♦r✐❣❡♠ ❞♦ t❡r♠♦ ❞❡ ■♥t❡r❛çã♦ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ❬✶✽❪✳

✷✳✹✳✹ ❆ ■♥t❡r❛çã♦ ❍✐♣❡r✜♥❛ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ❡♠ ■❧❤❛s

◗✉â♥t✐❝❛s

❊♠ ❝♦♥tr❛st❡ ❛ ✉♠ át♦♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧✱ ✉♠ ❡❧étr♦♥ ❝♦♥✜♥❛❞♦ ❛ ✉♠❛ ✐❧❤❛ q✉â♥t✐❝❛ ✐♥t❡r❛❣❡ ❝♦♠ ♠✉✐t♦s sít✐♦s ♥✉❝❧❡❛r❡s✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ q✉❡ ❞❡s❝r❡✈❡ ❛ ■♥t❡r❛çã♦ ❞❡ ❈♦♥t❛t♦ ❞❡ ❋❡r♠✐ ✲ ❝♦♠ t♦❞♦s ♦s ♥ú❝❧❡♦s ❞❛ r❡❞❡ ✲ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❝♦♥tr✐❜✉✐çã♦ ❝♦♥❥✉♥t❛ ❞❛ ✐♥t❡r❛çã♦ ❞♦ s♣✐♥ ❡❧❡trô♥✐❝♦ ♣❛r❛ ❝♦♠ ❝❛❞❛ s♣✐♥ ♥✉❝❧❡❛r ❞♦ s✐st❡♠❛✱ ♦✉ s❡❥❛✱

Hhf =

2µ0

3 gegnµnµB X

i