A Boeing 747 with a 65-m wingspan is cruising northward at 250 m / s toward Alaska. The B⃗ field

A Boeing 747 with a 65-m wingspan is cruising northward at...

Key Concepts

Vector Decomposition of Magnetic Field Components

When analyzing the generation of motional EMF, it is important to consider the effective component of the magnetic field that is perpendicular to the direction of the conductor's motion. This involves decomposing the magnetic field vector into components relative to the direction of travel. Only the component perpendicular to the motion contributes to the induced EMF, hence proper vector decomposition is crucial for accurately determining the potential difference.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field, and it is given by F = q(v × B). In the context of a moving conductor, this force acts on the free electrons (or charge carriers), pushing them to one end and creating a voltage difference. The direction and magnitude of this force depend on both the velocity of the conductor and the magnetic field, making it essential for understanding induced EMF.

Motional Electromotive Force (EMF)

Motional EMF refers to the voltage generated across a conductor moving through a magnetic field. As the conductor cuts through magnetic field lines, free charges within it experience forces that cause charge separation. This separation creates a potential difference along the length of the conductor, depending on the magnetic field strength, the speed of the conductor, and the orientation of its motion relative to the field.

Transcript

00:01 In this question, we have a boeing 747 with a 65 meter wingspan going north to alaska at a speed of 250 meters per second.

00:12 And to simplify the question, i have represented the wings of this airplane with a green block and just pretend that this has the same length as the wingspan.

00:23 And the problem also tells us that the magnetic field at this location is, 5 times 10 to the negative 5th tesla and it points in the direction of 60 degrees below the airplane's direction of travel.

00:38 So i have drawn a side view right here to show you what that means.

00:43 So pretend like the green line is the airplane and the red line here is the magnetic field.

00:48 Then the degree or the angle between the plane and the field here is 60 degrees.

00:54 And because the magnetic field is pointing down from a top view perspective, it is going into the page, which is why i have represented the magnetic field with this symbol right here, representing into the page.

01:11 So the problem asks us to determine the potential difference between the tips of its wings.

01:16 So i'm just going to say that this right here is the top of the wing, and here is the bottom of the wing, just from our perspective.

01:27 You can name it 1 -2 -2 -1.

01:30 It doesn't matter what you name it.

01:31 It's just for our reference for now.

01:34 So how do we approach this question? well, first, you have to recognize that as it goes through this magnetic field, there is going to be a gazillion electrons in the wingspan that will be subject to a magnetic force.

01:54 So let's first look at how this magnetic force impacts our problem.

02:02 So say if i have three electrons here, and if you use the right -hand rule and remember electron because it's negative, the right -hand rule applies.

02:17 You just have to remember it goes in the opposite direction as what the right -hand rule tells you.

02:25 So in this case, you know that the formula for magnetic force is equal to the charge times the velocity of the charge and then times the magnetic field experienced by the moving charge.

02:45 The right -hand rule tells you that if you put your index at the velocity vector in the direction of velocity vector and then your middle finger in the direction of the magnetic field, then your thumb will tell you the direction of the magnetic force as experienced by this moving electron right here within the wingspan metal.

03:13 So once you do that, you notice the right -hand rule tells you that the force should be moving downward from this perspective.

03:23 So moving towards the bottom.

03:25 But because it's an electron that we're talking about, it's the opposite of what your right -hand rule tells you.

03:31 So the electron actually moves up as a result of this magnetic force.

03:37 And you guessed it, because of the magnetic force, there will be an accumulation of electrons on the top of the wingspan.

03:46 Now, as you know, protons don't move.

03:49 So as electrons freely move to the top, that leaves a lack of electrons on the bottom, and that gives you a positive net charge on the bottom as a result.

04:05 So when will this moving of electrons stop? that is where our second force comes in.

04:14 Because you have a separation of charge, you will now establish.

04:24 An electric field.

04:31 So notice you have positive charges down here and negative charges up there and to an electron that can move.

04:40 Remember, electron will be attracted to the positive charges because the force that the electron experiences in an electric field will always be in the opposite the direction as the direction of the field.

04:56 So in this case, direction of the field is from positive to negative...

Please give Ace some feedback