Tolkien uses the word "power" rather ambiguously, I've noticed. And it seems to have made it quite difficult in determining the significance of the staff to the user. These quotes I find confusing:
The staff in the hand of a wizard may be more than a prop for age', said Hama. He looked hard at the ash-staff on which Gandalf leaned. 'Yet in doubt a man of worth will trust to his own wisdom. I believe you are friends and folk worthy of honour, who have no evil purpose. You may go in.' -- The King of the Golden Hall, Book 3, TTT.
I'm pretty sure Gandalf requires his staff when he strikes down Grima Wormtongue.
"He has power still, I think, while in orthanc, to resist the nine riders. He may try to do so. He may try to trap the Nazgul, or at to slay the thing on which it now rides in the air. In that case let Rohan look to its horses!" -- Gandalf talking about Saruman in The Palantir, book 3, TTT.
Saruman's staff has been broken, and he's safe within Orthanc. But what kind of power would Gandalf be referring to if Saruman can indeed imprison one of the Nine or kill a Fell beast?
I guess it would depend upon what you mean by magic. Sauruman still was able to convince Treebeard to let him out of his Orthanc prison. He was still able to take over the Shire without his staff. I can't think of any other instance when a wizard was without it, so I've given all I got right now.
I was reminded that Saruman calls the Staves of the Wizards "Rods" perhaps this is an indication that the staff, in this context, is somewhat of a conduit to focus some of the Wizards' power? Gandalf uses the staff on the Bridge of Khazad Dum by striking the bridge causing it to crack. He thrusts it into the corridor outside the Chamber of Marzabul in Moria causing a blinding flash of light which dismays the attacking Orc raiders and then he does it again to shatter the eastern door to Balin's tomb cutting off their pursuers after the brief battle in the Chamber of Marzabul. He uses it before then to ignite the wet kindling in the Pass of Caradhras. To me it always seems like he uses it to powerful affect.
This is unofficial but I think when the Wizards each chose their own particular staff in the Lands of Valinor that they imbued them each with their own brand of power to aid them in their task. Perhaps the material from the Undying Lands gave them unique qualities like: Vigor, Light Radius and so forth.
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I think the staves are more symbolic. When Gandalf 'broke' Saruman's staff he removed his authority but not his power (which was decreasing anyway after the Ents invaded Isengard).
No, I don't think there is any 'magical' power in the staff. It's possible they are required by the Wizards in order to exercise their own power, but I doubt even that much is true.
This is a question that's often been asked and there doesn't seem to be any certain answer to it.
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