[Alcuin]

We might be thinking about Gandalf’s broken staff in the wrong way.

Gandalf had been using the staff as a focus, or at least as an extension of himself: lighting the fire on Caradhras, for instance. That doesn’t mean it’s “magic”. To the person who understands it, it’s just a tool. A shepherd’s crook may seem “magic” to the sheep rescued from a ledge, for instance, but to the shepherd, it’s just a useful tool.

If Gandalf on Durin’s Bridge is already tired and wants to keep his distance between himself and his opponent, he might use the staff as means to do that, channeling whatever he’s doing to break the bridge through the staff. Think of the expression, “I wouldn’t touch that Balrog with a ten-foot pole!” Gandalf could have been as surprised as the rest of the onlookers that the staff burst. For that matter, he was pretty smart: he might have done it for effect, to surprise the Balrog; and the Balrog does seem to have been surprised.

Maybe the staff burst the same way the door to the Chamber of Mazarbul burst: we know, or think we know, that Gandalf intended to break the bridge. Perhaps the Balrog tried to stop him. If Gandalf was using the staff as a tool, then just like the door on which each of them had his hand, it could have broken under the strain between them when the bridge broke.

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Another idea presents itself as well. If the staff has symbolic meaning – Gandalf, you are to unite Men and Elves in opposition to Sauron – breaking the staff might be interpreted as, Gandalf, you’re done with the first assignment. Don’t leave without killing the Balrog.

If Gandalf had escaped and the Balrog survived, the Umaia might well have helped Sauron in his three assaults upon Lórien. Celeborn seemed uncertain that the little kingdom would have survived, and probably for good reason! It would not mean that Gandalf had either exceeded his authority or abandoned his charge: he was confronted with something unforeseen. It doesn’t appear that Eönwë took a head-count of Morgoth’s balrogs at the end of the War of Wrath, the Balrog was a problem that had to be dealt with, and Gandalf was the only one in the Fellowship capable of dealing with it. (Glorfindel had already successfully dealt with balrogs, but he stayed behind in Rivendell when Merry and Pippin filled up Elrond’s count.)