What is the purpose of Macbeth's letter to his wife? (Act 1, scene v)
In Shakespeare's Macbeth, there is a clear purpose to the letter our "hero" sends to his wife.
Macbeth, first and foremost, has wondrous news to tell Lady Macbeth. First, he met witches who offered him predictions. One has already come true, which means they will have more lands, money and honor in the King's court; the other promises that he will be King, and she will be Queen one day—it seems obvious he believes the old women after the first prediction came to be. In learning how excited he is about being Scotland's sovereign, we get the briefest glimpse of his "vaulting ambition"—his desire to become King that will eclipse everything else of value in his life.
We learn that Macbeth and his wife are very close. Not only because of how he addresses her, but also because he shares his news immediately rather than letting her wait until he returns home.
The third and perhaps most important purpose of the letter is to see how Lady Macbeth reacts to this news. We come to see how hard and calculating she is, and find that she worries that her husband is too kind to do what must be done if they are to secure the throne for themselves: which means murdering the King, for she sees no other way in which the crown can come to Macbeth.
Then she wishes Macbeth home quickly so that she can tell him what he needs to hear in order to do what he must:
Hie thee hither, that I may pour my spirits in thine ear. (I, v, 20-21)
It almost sounds as if she is casting her own spell: come quickly so I can pour my evil intentions ("spirits") into your ear, so you won't hesitate to follow through.
It could logically be argued that without Lady Macbeth's encouragement, Macbeth might have been satisfied with the rewards he had been given, and not kill the King: he says as much to his wife when he returns. In essence, this letter galvanizes the plot forward at the hands of Lady Macbeth's machinations.