### Video Transcript

A 50.0 centimeter length of wire is in a 1.20 tesla magnetic field. The current in the wire is 8.00 amps. And a force of 2.40 newtons acts on the wire. What is the angle between the wire and the magnetic field? If the wire is rotated to be at an angle of 90.0 degrees to the magnetic field, what force is exerted on it?

We can call the length of wire, 50.0 centimeters, capital ๐ฟ. And the strength of the magnetic field, 1.20 teslas, weโll call ๐ต. The current in the wire of 8.00 amps, we can call ๐ผ. And the magnetic force magnitude of 2.40 newtons, weโll name ๐น sub ๐ต.

In part one, we want to solve for the angle between the wire and the magnetic field. Weโll call that angle ๐. And in part two, we want to solve for a magnetic force. Weโll call that force ๐น. We can start off solving for ๐ by drawing a diagram of the situation.

We have a wire of length ๐ฟ that carries a current ๐ผ. The wire exists in a magnetic field ๐ต which, relative to the wire, is pointed at an angle ๐. To solve for this angle, we can recall a mathematical relationship for the force on a current-carrying wire. The magnetic force, ๐น sub ๐ต, exerted on a current-carrying wire is equal to the product of the current in the wire, ๐ผ, times the length of the wire, ๐ฟ, times the magnitude of the magnetic field the wire is in, ๐ต, multiplied by the sine of the angle between the magnetic field and the wire direction. Considering this relationship for our scenario, we see weโve been given all the elements of this relationship, except for ๐.

When we rearrange to solve for ๐, we see that itโs the inverse sine of ๐น sub ๐ต over ๐ผ๐ฟ๐ต. When we plug in for the terms in these parentheses, being careful to use units of meters for the length of our wire, and plug them into our calculator, we find that ๐ is equal to 30.0 degrees. Thatโs the angle between the wire and the magnetic field.

Moving on to part two, where we solve for magnetic force. Weโre now told that our wire is rearranged with respect to the magnetic field ๐ต. Weโre told that now our wire is arranged at an angle of 90.0 degrees with respect to ๐ต. And under these new conditions, we wanna solve for the magnetic force ๐น. We can use the same basic equation as before to solve for ๐น. But now when we write it out, we write ๐น, the unknown we want to solve for, multiplied by ๐ผ๐ฟ๐ต times the sine of, we call it, ๐ sub ๐. Thatโs the final angle between the magnetic field ๐ต and the current-carrying wire, which weโre told is 90.0 degrees. Since ๐ผ, ๐ฟ, and ๐ต remain unchanged and weโre given ๐ sub ๐, weโre now ready to plug in and solve for ๐น.

With these values plugged in, once more being careful to use units of meters for our distance, we see that, because the sine of 90 degrees is equal to one, our expression simplifies to ๐ผ times ๐ฟ times ๐ต. When we enter these values on our calculator, we find that ๐น, to three significant figures, is 4.80 newtons. And thatโs the magnetic force on this current-carrying wire.