While I was DMing a 5th Edition game, one of my player's sorcerers cast the spell Silence while they were 5 feet away from an enemy caster. They centered the spell on the space that the enemy caster was in, and a round of combat finished. On the sorcerer's turn, they asked me if they could use a verbal spell while in the Silence area. Their reasoning is that they had cast the spell, so they could prevent it from affecting them.
In this situation, I decided to not allow it because it would unbalance the spell. But I want to know what other people would think about this.
So would the user of the Silence spell be able to prevent it from affecting themselves?