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An open plastic soda bottle with an opening diameter of 2.5 $\mathrm{cm}$ is placed on a table. A uniform 1.75 T magnetic field directed upward and oriented $25^{\circ}$ from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

$$-7.8 E-4 \mathrm{Wb}$$

Physics 102 Electricity and Magnetism

Chapter 21

Electromagnetic Induction

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Inductance

Alternating Current

Cornell University

Rutgers, The State University of New Jersey

Simon Fraser University

University of Winnipeg

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this problem. We have a plastic soda bottle sitting on the table. It's open, and as an opening diameter of 2.5 centimeters, it's sitting in a uniform magnetic field at 1.75 Tessa, and that's directed up at 25 degrees from the vertical. Our goal is to find the total magnetic flux through the plastic soda bottle, the plastic of the soda bottle. So in doing that, we can actually get most of the way qualitatively before we laid out any calculations. Let's think about this lower region here. So when we have a magnetic field factor going in, it's gonna enter, and our convention, when it enters, is that will be a negative flux. And when it exits, we have a positive flux. So everything that's entering in below this opening just doing a little cone here. Everything entering in below this line is just exiting out after it enters. So those negative flux contributions air canceled out by the positive looks contributions want to passes through. So this entire region has no net flux contribution through the plastic contribution through the plastic. How about the other part? If we don't have to worry about that whatsoever, then we only have. This is a region up here, and if we look at one of the field vectors when that goes into the bottle, we have some negative flux. Now, when it continues, it doesn't exit out of any plastic because it's open. That's the key thing of this problem is we have an open bottle. So everything entering in to this region down here exits out above into this little bring. This also makes our calculation easier, because if we look, if we were to calculate the flux on this left hand side, that's gonna be pretty challenging. Because our area are surface is not constant, it's curved, but the opening of this bottle is just a disc. So let me let me blow up the opening here and now we have uniform field lines. These were directed at 25 degrees, and this is a uniform cone, so these field vectors are in this entire region and the perpendicular of this circular area. I'll drive with this red vector, so we have our angle between the perpendicular of the opening and our field, and now we just have to call on the relevant equation. So we know in a constant a magnetic field. We have our magnetic flux being equal to the perpendicular of our Byfield, multiplied by our area or be a times are co signed fee where that sea is the angle between our Byfield and the perpendicular of our area. Here we have a circular region. So our area is pi r squared of Seinfield. Or since we have, we're given a diameter here. That would be a D over, too. But we'll just keep doing that now. We have all the information we need and we have our sign convention here because they're all entering into the bottle. This value that we calculate here is going to be a negative flux through the plastic. Let's see what that gives us. We have a 1.75 magnetic field multiply, and by our diameters, 2.5 star radius, it's 1.25 but that in the meters square that and then we have co sign of 25 degrees. Do you enter in those numbers? We get 7.7 nine, but actually, let's let's just keep this to significant two significant figures because that's the lowest one were given. So we have 7.8 times 10 to the negative four weavers. Or, if you want to put this in tow, Ah ah, nice convention are just different. Smaller units. We have Millie Weber's. So that's the flux exiting out of this circular region. And as we just said, going into the plastic, our convention is negative. So the flux through the plastic is going to be minus 0.8 million burgers, and there you go. So I think this demonstrates why it's so useful to really qualitatively think through a problem before tackling it. Ah, we eliminated one entire region, the whole region on the bottom. We then went up at the top. We noted that everything going in is going to exit out of our opening. And calculating the flux on a constant service is much easier than Ben. What we would have had to do if we wanted to calculate it on that curved opening or the curved part of the bottom before the opening. And yep,

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