Video Transcript
Which of the following is the correct formula for the magnitude of the force experienced by a current-carrying wire in a uniform magnetic field? 𝐹 is the force experienced by the wire. 𝑎 is the acceleration of the wire. 𝐿 is the length of the wire. 𝐼 is the magnitude of the current within the wire. And 𝐵 is the strength of the magnetic field. (A) 𝐹 is equal to 𝐵 times 𝐼 squared times 𝐿. (B) 𝐹 is equal to 𝐵 times 𝐼 divided by 𝐿. (C) 𝐹 is equal to 𝐵 divided by 𝐼 times 𝐿. (D) 𝑎 is equal to two times 𝐵 divided by 𝐼 times 𝐿. And (E) 𝐹 is equal to 𝐵 times 𝐼 times 𝐿.
In this situation, we’re imagining a wire of length 𝐿 carrying a current of magnitude 𝐼 that exists in a magnetic field of strength 𝐵. In this situation, the current-carrying wire will experience a force and that force will increase. The longer the wire is, the greater the current in the wire is, and the stronger the magnetic field 𝐵. This fact that 𝐹 increases as any one of these three variables increases lets us eliminate any answer options where any of these three variables are in the denominator. For example, answer option (B) claims that as the length of the wire 𝐿 increases, 𝐹 decreases. We know that that’s not true, that rather as 𝐿 grows, so does 𝐹. So we can eliminate answer choice (B). And for the same reason, we can also eliminate answer choice (C).
Taking a look at answer choice (D), we see that this option also has 𝐼 and 𝐿 in the denominator, but now we’re considering the acceleration of our wire rather than the force on it. Recalling Newton’s second law of motion that the net force on an object equals the mass of that object times its acceleration, we know that in general the net force on an object is proportional to its acceleration. Therefore, we would expect that as 𝐼 and 𝐿 increase, the acceleration of the wire would increase as well rather than decrease. We can also cross out, therefore, answer choice (D). The remaining answer options (A) and (E) are only different in the power of the current 𝐼.
At this point, we can note that this force 𝐹 on a current-carrying wire in a magnetic field comes down to a force that’s experienced by the moving charges in the current of the wire. That is, it is individual charged particles that experience this force. If we double the amount of charge passing through this wire in a given interval of time, in other words, if we double the current 𝐼, then that means there are twice as many charges to experience this force. And therefore, the overall force is twice as big. We can say then that the overall force 𝐹 is proportional to the current 𝐼. That is, it’s not the case that 𝐹 is proportional to 𝐼 squared as answer option (A) would claim.
For our final answer, we choose option (E). The force experienced by the wire is equal to the magnetic field strength 𝐵 times the magnitude of the current 𝐼 times the length of the wire 𝐿.
