Why is the answer to this GRE physics problem choice A?

[PsychonautQQ]

Homework Statement

I don't understand why the answer in this problem is choice A.
http://grephysics.net/ans/9277/57

I'm confused.
Lets say the semi circle is FULLY outside of the circle. For awhile after that, the semi circle would be accelerating the rate at which a certain % of it is inside the magnetic field.. So the change in magnetic flux with respect to time increases it's rate for awhile, and I don't feel that's reflected in choice A. Anyone help me out?

 

[Famwoor2]

The induced EMF is a function of CHANGE in flux. Once the semicircle begins entering the region with the constant magnetic field, the same amount of area is entering the field per unit time until it is fully in the magnetic field region. Likewise, the same amount of area per unit time exits the magnetic field region until the semicircle completely exits the region, and the process happens again and again (periodically).

For example, say the semicircle is revolving at a rate of a quarter revolution per second, and we start measuring the EMF right before the semicircle enters the magnetic field region. Then, in the first second, a quarter of a circle's area has entered the region. In the next second, another quarter of a circle's area enters the region. This is the same amount of CHANGE in area in the field for both seconds, hence the same change in flux for each second (since the magnetic field doesn't change). This is graphically represented by the first positive segment of the square wave. Apply the same logic for the next two seconds, but it is a negative change in area, hence a negative EMF, and the negative segment of the square wave. Rinse and repeat.

Good luck,
F2

 

[PsychonautQQ]

Once the semicircle begins entering the region with the constant magnetic field, the same amount of area is entering the field per unit time until it is fully in the magnetic field region.

This statement is true? It doesn't seem like it looking at the picture... i mean first only the tip is going in (haha) and then slowly after that it seems like a bigger and bigger "slice" of the circle is going in per unit time.

 

[Famwoor2]

While it is true that there is more of the semicircle in the magnetic field region as time goes by (until it is fully in), what is important to the EMF is how much MORE of the circle enters the magnetic field region per unit time. Since the circle is rotating at a uniform angular velocity, the same NEW amount of circle is entering (or exiting) the magnetic field per unit time.

The EMF doesn't care how much of the circle is already in the magnetic field. It only cares how much new area comes in per unit time.

 

[PsychonautQQ]

But what about this bar, where the x's represent a magnetic field
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[||||||||||||||||||||] velocity of bar (constant)--------->
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This bar is leaving the magnetic field at a constant rate. It would still produce an EMF? The circle is increasing it's area entering the magnetic field at an exponential rate (to a point).

and if this bar went back and forth after it exits the field I would expect the graph of induced EMF to look like answer A.

For the circle, the rate at which area from the circle enters the field is ACCELERATING.

Since the circle is rotating at a uniform angular velocity, the same NEW amount of circle is entering (or exiting) the magnetic field per unit time.

Is this statement true? Is there proof somewhere? At the very beginning when it first enters the field only a tiny tip with barely any diameter is being added, but by the time the center of the semi circle is being added that wholeee long slice is being added at once

 

[Famwoor2]

I am sending you these responses from my smartphone, so I am not sure if I am viewing your picture correctly. It looks as though the bar is completely within the magnetic field in both pictures. In this case, the induced EMF is a consequence of the moving electrons interacting with magnetic field. This is a result of the Lorentz force.

How can the amount of circle entering the magnetic field be accelerating if the circle has a uniform angular velocity?

 

[Famwoor2]

Is this statement true? Is there proof somewhere? At the very beginning when it first enters the field only a tiny tip with barely any diameter is being added, but by the time the center of the semi circle is being added that wholeee long slice is being added at once

I now believe that you are imagining the semicircle moving toward the field region translationally... this is not the case! It is spinning around like a record about point A in the plane of the screen.

 

[PsychonautQQ]

Wow I'm a noob >.< I went a run and then came back and looked at this and i feel like a cotton-headed-ninny-muggins >.< haha thanks for your patience :)

 

[Famwoor2]

No problem! I'm studying for the GRE too, and I think that explaining a question is the best form of study.