I think the OP was asking about .357 magnum in particular, wasn't he?
Link above is for a heavier than typical (for a .357 magnum) 180 grain bullet and the foot pounds of energy is 783 at the muzzle. Pretty potent. (For stopping power, a heavier bullet is nearly alwasy better. So a 'heavy for caliber' bullet is typically a good selection for a situation like this, with specific bullet design being very important as well. )But the bullet in the link is a hardcast bullet which won't deform and mushroom like a typical lead core bullet. However, because of the fact that it won't deform like typical lead bullets, it will likely penetrate farther, which may be a benefit in a bear/self defense situation. Most 'dangerous game' hunters in Africa, for instance, will use heavy, solid bullets that deform little, opting instead for deep penetration in dangerous game situations. This type of 'hard cast' bullet may be more like the solid bullet mentioned for dangerous game hunters and so may be appropriate for a close bear encounter, but I don't have personal experience with them, so...
If you do a little research, you'll see that handgun (because that was what the OP talked about) black bear hunters, in general, have a some unoffical items to consider when hunting bears. Generally, they'll look for a .40 caliber or larger, which the .357 magnum is not. (A larger frontal area to a bullet means more tissue upset, in general, with potential for a quicker killing power, which is the point in this situation.) In terms of kinetic energy (KE), they look for a cartridge/bullet/powder/gun combination that develops at least 600 ft. lbs. of energy (By contrast, the medium power of a .30-06 round will have just under 3000 ft. lbs. of energy at the muzzle and the typical 9mm handgun round is around or just under 400 ft. lbs.). And, they will look for a premium bullet designed for the purpose at hand. Many will select the Hornady XTP bullet as it is a good compromise between expansion and penetration. The variables in making selections liket this can make your mind spin, but knowing how to select the right gun and bullet/cartridge/powder combination for the purpose is pretty important.
That being said, hunting a black bear is much different from stopping a charging black bear. In the latter, you may want more power than the minimums listed above, which is why I mentioned the .44 magnum and others in a previous post. Problem is these 'hand cannons' get difficult to shoot accurately.
Bear spray is likely the best answer after just learning about wildlife and how to avoid issues.
If you clearly understand the issues surrounding handloading (loading your own rounds--not nearly as difficult and dangerous as many are lead to erroneously believe--and it's much cheaper than buying factory ammo), you can easily then select the best combination, among all the variables of bullet/powder and charge of powder, for your purpose, instead of being locked into the factory offerings that may not fit the purpose you may have.