In The Crucible, what are Elizabeth Proctor's internal and external conflicts?
In Act Two, we see Elizabeth's conflict with her husband, John. He asks if she is sad again, and "she doesn't want friction, yet she must" say,
You come so late I thought you'd gone to Salem this afternoon.
She seems to still suspect him of having ulterior motives for going to Salem. Elizabeth is somewhat justified in her suspicion since John shortly reveals that he spoke with Abigail, his former lover, alone, when he'd led Elizabeth to believe that he was never alone with the girl.
We also see evidence of Elizabeth's conflict with her employee, Mary Warren, in Act Two. Even though John has forbidden Mary to go into the courts in Salem anymore, Elizabeth feels she could not stop her. She says,
I couldn't stop her [....]. She frightened all my strength away.
Despite their age difference, and the fact that Mary is employed by the Proctors, Elizabeth still felt overpowered by Mary, especially given how powerful the court has become in town. Mary claims to be an official of the court, and Elizabeth is afraid to interfere, it seems, though she's also trying to obey her husband.
Then, in Act Four, we see evidence of Elizabeth's internal conflict, a conflict she's had much time to mull over during her three months in jail for witchcraft. When John asks her forgiveness again, she says,
I have read my heart this three month, John. I have sins of my own to count. It needs a cold wife to prompt lechery [....]. John, I counted myself so plain, so poorly made, no honest love could come to me! Suspicion kissed you when I did; I never knew how I should say my love. It were a cold house I kept!
Elizabeth has felt conflicted by her love for John and her simultaneous distrust of that love. It sounds like low self-esteem has caused her to doubt his love since its beginning, and she blames herself and her behavior for John's infidelity. Only now can she really put that conflict into words.
In the play, Elizabeth has quite a bit to be conflicted about. She faces external conflict of her husband having had an affair with their servant, Abigail Williams. Their affair was something outside of herself that created tension and unhappiness in her life. She had to figure out how to deal with that situation, and how to behave after it was over. It created conflict in her marriage, and a huge rift that was evidenced even months after the affair was over. Another external conflict that she faces is later on in the play when Danforth asks her if her husband committed an affair. Admit to it, and her family's reputation would be ruined; so, she doesn't. Earlier, when Reverend Hale comes to visit their home, he presents an external conflict for her when he questions her beliefs about witchcraft, and about the town's proceedings of late.
Internally, Elizabeth also struggles quite a bit. She feels like she was a bad wife somehow, a cold wife, and that was part of the reason that John turned from her to Abby. She was internally conflicted about her potential role in that entire affair. In act two, we see her struggle with her internal conflict of being angry and bitter over the affair still; she still struggles with forgiveness. That is a very personal, internal struggle that she battles with every day. She was also internally conflicted about what to recommend to her husband at the end of the play when everyone wants her to convince John to confess. She admits that she wants him to live, but also that she wants him to be happy. That is an internal struggle--her selfish desire for him to live life with her, or the more noble route of not lying.
I hope that those thoughts helped; good luck!