Induced EMF due to a moving magnet

[phantomvommand]

Homework Statement

See picture below

Relevant Equations

Faraday's Law, Lenz's Law

I am asking about (c) (i). To me, there should be 0 current induced because the change in flux is 0. Specifically, there is 0 (net) flux initially and 0 (net) flux as the magnet is pulled to the right.

The correct answer, however, is current into page for both wires. I am not sure how this is obtained. Interestingly, it seems to agree with my view that there is 0 current, since having current flowing into page for both wires (which the question states forms a closed circuit) is equivalent to having 0 net current, although I am not sure if this might be a coincidence.

Can anyone explain why current would flow into page for both wires? And why is my argument for 0 induced current incorrect?

Many thanks.

 

[TSny]

Suppose you go to the reference frame moving with the magnet. In this frame the magnet is at rest and the rods are moving. What is the direction of the magnetic force on the free electrons in the two rods in this frame? If the free electrons moved in this direction, what would be the direction of the conventional current in the two rods?

Can you see a way to hook up wires to the rods to make a complete circuit so that current can flow in the same direction in the two rods?

 

[phantomvommand]

Suppose you go to the reference frame moving with the magnet. In this frame the magnet is at rest and the rods are moving. What is the direction of the magnetic force on the free electrons in the two rods in this frame? If the free electrons moved in this direction, what would be the direction of the conventional current in the two rods?

Can you see a way to hook up wires to the rods to make a complete circuit so that current can flow in the same direction in the two rods?

Thanks. I can see how this gives the answer. However, why is it incorrect to apply this same argument to the textbook moving rod example, in which case by switching to the frame of the moving rod, the rod would be stationary and therefore have no induced current?

 

[TSny]

why is it incorrect to apply this same argument to the textbook moving rod example, in which case by switching to the frame of the moving rod, the rod would be stationary and therefore have no induced current?

In the frame where the rod is at rest and the magnet is moving, the movement of the rod causes the magnetic field at points within and around the rod to change with time. An “induced” electric field is always associated with a time varying magnetic field. This E field causes the current in the circuit in this frame.

 

[phantomvommand]

In the frame where the rod is at rest and the magnet is moving, the movement of the rod causes the magnetic field at points within and around the rod to change with time. An “induced” electric field is always associated with a time varying magnetic field. This E field causes the current in the circuit in this frame.

What if the rod were moving in a uniform magnetic field?

 

[TSny]

What if the rod were moving in a uniform magnetic field?

Good question. In the "original frame" (in which the rod is moving), assume that the magnetic field is uniform and time-independent. Also, assume that the rod moves with uniform velocity in a direction that is not parallel to the magnetic field. There will be a magnetic force on the electrons in the moving rod that will cause equal and opposite charge to build up at the ends of the rod.

When you switch to the frame moving with the rod, the magnetic field will still be uniform and time-independent. But, in this frame, there will also be a uniform, time-independent electric field, even though the magnetic field is static. In this frame, the accumulation of opposite charge at the ends of the rod is due to this electric field.

Special relativity accounts for how the fields transform between inertial frames.

 

[phantomvommand]

Good question. In the "original frame" (in which the rod is moving), assume that the magnetic field is uniform and time-independent. Also, assume that the rod moves with uniform velocity in a direction that is not parallel to the magnetic field. There will be a magnetic force on the electrons in the moving rod that will cause equal and opposite charge to build up at the ends of the rod.

When you switch to the frame moving with the rod, the magnetic field will still be uniform and time-independent. But, in this frame, there will also be a uniform, time-independent electric field, even though the magnetic field is static. In this frame, the accumulation of opposite charge at the ends of the rod is due to this electric field.

Special relativity accounts for how the fields transform between inertial frames.

Would you have a qualitative explanation for why a uniform time independent electric field emerges (even though the magnetic field is static)? thanks

 

[TSny]

Would you have a qualitative explanation for why a uniform time independent electric field emerges (even though the magnetic field is static)? thanks

There are ways to see why an electric field can exist in one frame but not in another. I’ll just quickly outline one popular way. Any explanation must be in terms of concepts that are accepted as valid. In this case, we’ll need the concept of relativistic length contraction.

Consider two infinitely long rods, one (red) with uniform positive charge and the other (blue) with equal but opposite (negative) charge density.

They are initially placed essentially on top of each other so that their charge balances to zero. There will not be any appreciable electric or magnetic field.

Now imagine the rods move in opposite directions with the same constant speed.

The charge densities of the two rods will increase due to relativistic length contraction. But, since the two speeds are the same, the change in charge densities is the same. So, the positive and negative charge cancel again. There is still no electric field. However, the motion of the negative charge to the left and the positive charge to the right constitute net electric current to the right. This creates a magnetic field. So, in this frame of reference we have a static magnetic field and no electric field.

Now switch to a frame of reference moving with the negative charge.

The negatively charged rod is now at rest and the positively charged rod has increased its speed to the right. We still have a net current to the right, so in this frame there will still be a static magnetic field. But now we also have an electric field! This is because, in this frame, there is no length contraction of the negative rod while there is greater contraction of the positive rod. So, now the positive charge density is greater than the negative charge density. Therefore, in this frame there is a radial electric field pointing away from the rods. In this frame there is both a static magnetic field and a static electric field.

You could replace the rods with infinite, flat sheets of charge if you want the fields to be uniform in the two frames.