A PA-28's alternator belt became damaged in flight causing a strong electrical smell. The instructor thought the smell was an electrical fire and elected to land off airport as quickly as possible.

Narrative

My student and I were practicing maneuvers in the NW practice area recovering from a power off stall at 4000 feet MSL when we smelled a strong burning smell. I looked and saw that the alternator inoperative annunciator was illuminated and saw that the amps read 0000. Turned off the alternator switch and tried to recycle it. Turned it back off when it was still inoperative and still smelled a strong burning smell. I thought we had an electrical fire. Couldn't see any white smoke or any smoke in the cockpit; but thought we were on fire because the smell was stronger than I had smelled before. I didn't want to wait for smoke to come into the cockpit reading not long ago that an electrical fire could spread quickly. So I performed an emergency descent without declaring an emergency because I didn't think I had time; picked a landing spot and performed a landing on a back country road. There was one car coming head on; but knew they were no factor; no other cars on the road and performed a landing. I secured the airplane turning everything off and we jumped out to see if it was on fire. No fire; but still had a bad electrical burning smell. So we had maintenance drive out; they took off the cowling cover; and found the alternator belt had melted. No damage was done to anything or anybody else; nor was the airplane damaged in any way shape or form. I believe what I should do in the future is to turn the electrical equipment not needed off; to keep the alternator switch off and to wait longer to see if that extinguishes the fire or even the smell. I could have flown back to the base airport however; I really thought we were on fire due to the strong burning smell. I didn't want to wait only to make the problem worse; however what I know now is it was only a minor problem. It just smelled worse than what it was.