Which wingtip will be positive when airplane flies?

[songoku]

Homework Statement

When an aircraft flies from east to west its wings are electrical conductor cutting across the Earth magnetic flux.
a. In northern hemisphere, which wingtip will become positively charged?
b. Why will this wingtip be negative when in southern hemisphere?

Homework Equations

Faraday law
Lenz law
Fleming rule

The Attempt at a Solution

I think I need to clear up several things before I am able to answer the question

1. Why it can be said the airplane cuts the magnetic field of Earth so emf is induced in it? Maybe the answer will be the same if I ask why emf is induced in conductor moving with constant speed in uniform magnetic field.

I read this link:
https://courses.lumenlearning.com/physics/chapter/23-3-motional-emf/

It is written that when the rod moves, the area enclosed by the rod and rails increasses so there is change in magnetic flux, hence emf induced.
Now if I remove the rail, and the conductor still moves to the right with constant speed, will emf still be induced in it? Will there be "area" enclosed just like when the rail not removed?
Or maybe I use fleming's left hand rule to deterimine the direction of lorentz force acting on electrons in the conductor and when the electrons are being pushed by this force there is induced current and when the electrons stop moving (accumulated on one side of the conductor) the induced current becomes zero because there is no "external circuit" for the electrons to flow?

2. I read that Earth has vertical and horizontal component of magnetic field and at equator the vertical component is approximately zero and at north and south poles the horizontal component is approximately zero. I do not understand why. I google the pictures of Earth magnetic field and still can not understand. Which direction is taken as vertical and horizontal?

Thanks

 

[gneill]

Homework Statement

When an aircraft flies from east to west its wings are electrical conductor cutting across the Earth magnetic flux.
a. In northern hemisphere, which wingtip will become positively charged?
b. Why will this wingtip be negative when in southern hemisphere?

Homework Equations

Faraday law
Lenz law
Fleming rule

The Attempt at a Solution

I think I need to clear up several things before I am able to answer the question

1. Why it can be said the airplane cuts the magnetic field of Earth so emf is induced in it? Maybe the answer will be the same if I ask why emf is induced in conductor moving with constant speed in uniform magnetic field.

Because the field lines are curved (research: "magnetic dip"), there is a vertical component of the field lines passing through the plane, and yes, an emf is induced in a conductor that is cutting through magnetic field lines.

I read this link:
https://courses.lumenlearning.com/physics/chapter/23-3-motional-emf/

It is written that when the rod moves, the area enclosed by the rod and rails increasses so there is change in magnetic flux, hence emf induced.
Now if I remove the rail, and the conductor still moves to the right with constant speed, will emf still be induced in it? Will there be "area" enclosed just like when the rail not removed?

Or maybe I use fleming's left hand rule to deterimine the direction of lorentz force acting on electrons in the conductor and when the electrons are being pushed by this force there is induced current and when the electrons stop moving (accumulated on one side of the conductor) the induced current becomes zero because there is no "external circuit" for the electrons to flow?

Your second explanation is the way to go. Equilibrium will be reached (the charges stop moving along the wings) when there's established an electric field between the wingtips such that the Lorentz force on the charges is zero from one wingtip to the other.

2. I read that Earth has vertical and horizontal component of magnetic field and at equator the vertical component is approximately zero and at north and south poles the horizontal component is approximately zero. I do not understand why. I google the pictures of Earth magnetic field and still can not understand. Which direction is taken as vertical and horizontal?

Thanks

Horizontal would mean the local horizontal for the Earth's surface. Vertical is the local vertical.

 

[songoku]

Horizontal would mean the local horizontal for the Earth's surface. Vertical is the local vertical.

I am not sure I get this. Maybe like this: if I stand on equator facing west, there will be horizontal component of Earth's magnetic field passing through me in direction from my left to my right? If I stand on north poles, there will be Earth's magnetic field directed vertically downward?

Because the field lines are curved (research: "magnetic dip"), there is a vertical component of the field lines passing through the plane

So in northern hemisphere there will be horizontal component of Earth's magnetic field directed from south to north geographic poles and vertical component of Earth's magnetic field directed vertically downwards to Earth's surface?

The vertical component of Earth's magnetic field won't produce induced current because the direction of the current will be vertically downward (perpendicular to the plane) which does not make sense?

Your second explanation is the way to go. Equilibrium will be reached (the charges stop moving along the wings) when there's established an electric field between the wingtips such that the Lorentz force on the charges is zero from one wingtip to the other.

Can the lorentz force be zero? I thought lorentz force won't be zero because from formula F = Bqv sin theta there will always be value of F and the electrons will reach equilibrium because the electric force cancels out the magnetic force?

When equilibrium is reached, induced current = 0?

and yes, an emf is induced in a conductor that is cutting through magnetic field lines.

I learn that based on faraday's law, emf will be induced if there is change in magnetic flux and magnetic flux = BA cos theta. The value of B is constant (uniform magnetic field), theta is constant (no change in orientation of the rod), so area is changing to produce change in magnetic flux? I can not imagine the area because the rod moving without rail so where is the "area enclosed"?

Thanks

 

[gneill]

I am not sure I get this. Maybe like this: if I stand on equator facing west, there will be horizontal component of Earth's magnetic field passing through me in direction from my left to my right? If I stand on north poles, there will be Earth's magnetic field directed vertically downward?

Yes, that's correct for all intents and purposes for this problem. There are a few "reality" details that would modify this a bit, but not important at all for this problem.

So in northern hemisphere there will be horizontal component of Earth's magnetic field directed from south to north geographic poles and vertical component of Earth's magnetic field directed vertically downwards to Earth's surface?

Yes.

The vertical component of Earth's magnetic field won't produce induced current because the direction of the current will be vertically downward (perpendicular to the plane) which does not make sense?

No, that's exactly wrong
It's the vertical component that will produce a leftward or rightward Lorentz force on the electrons in the (horizontal) wings. Revisit your right-hand-rule for charges moving in an electric field.

Can the lorentz force be zero? I thought lorentz force won't be zero because from formula F = Bqv sin theta there will always be value of F and the electrons will reach equilibrium because the electric force cancels out the magnetic force?

Yes. The Lorentz force is the sum of the electric and magnetic forces on a charge. When the electric and magnetic forces cancel, equilibrium will be achieved. But first you need to establish a potential difference across the wingspan to create the electric field by moving some charges. Hence the potential difference between the wing tips.

When equilibrium is reached, induced current = 0?

Yes!

I learn that based on faraday's law, emf will be induced if there is change in magnetic flux and magnetic flux = BA cos theta. The value of B is constant (uniform magnetic field), theta is constant (no change in orientation of the rod), so area is changing to produce change in magnetic flux? I can not imagine the area because the rod moving without rail so where is the "area enclosed"?

No area here, just Lorentz force on the charges in the wire. As it turns out the EMF generated by a segment of wire moving through a magnetic field is the same as though the wire and velocity were conspiring to increase the "enclosed area" (which doesn't exit here) by B_⊥Lv. You can research this on the web with search terms such as "emf on wire moving though a magnetic field". It's actually a pretty neat result, showing that the experiment with the wire sliding on the rails through a magnetic field can be interpreted in two different ways as to how/why the emf is being produced.

 

[songoku]

No, that's exactly wrong
It's the vertical component that will produce a leftward or rightward Lorentz force on the electrons in the (horizontal) wings. Revisit your right-hand-rule for charges moving in an electric field.

Ah my bad. I mean horizontal component of Earth's magnetic field won't produce induced current because the induced current will be directed perpendicular to the plane (vertically downwards towards Earth's surface)

Yes. The Lorentz force is the sum of the electric and magnetic forces on a charge. When the electric and magnetic forces cancel, equilibrium will be achieved. But first you need to establish a potential difference across the wingspan to create the electric field by moving some charges. Hence the potential difference between the wing tips.

Oh I see. The term "Lorentz force" means total force of electric and magnetic. I thought it is only for magnetic force. If I use the formula F = Bqv sin theta, it means that I only calculate magnetic force part of Lorentz force?

 

[gneill]

Ah my bad. I mean horizontal component of Earth's magnetic field won't produce induced current because the induced current will be directed perpendicular to the plane (vertically downwards towards Earth's surface)

Yes. It may allow you to tell whether the charges will be on the upper or lower wing surface, though.

Oh I see. The term "Lorentz force" means total force of electric and magnetic. I thought it is only for magnetic force. If I use the formula F = Bqv sin theta, it means that I only calculate magnetic force part of Lorentz force?

Yes. The Lorentz force actually combines the electric and magnetic contributions:

$$\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$$

For equilibrium conditions where the net force on any charge is zero, you could find $\vec{E}$ such that it balances $q\vec{v} \times \vec{B}$.

But here we're only interested in the direction of the field (which wing tip is more positive than the other). So don't get bogged down in the math, just determine which direction the E-field has to be directed in order to counteract the magnetically induced force on the electron in the wing metal.

 

[songoku]

Yes. It may allow you to tell whether the charges will be on the upper or lower wing surface, though.

Oh ok ok. I get it

Thank you very much for the help

 

[gneill]

Glad to help!