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The magnetic field $\vec{\boldsymbol{B}}$ in a certain region is $0.128 \mathrm{T},$ and its direction is that of the $+z$ axis in Figure 21.47 (a) What is the magnetic flux across the surface abcd in the figure? (b) What is the magnetic flux across the surface befc? (c) What is the magnetic flux across the surface aefd? (d) What is the net flux through all five surfaces that enclose the shaded volume?

$0,-0.0115 W b, 0.0115 W b$

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

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Simon Fraser University

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and this problem, we're going to use the formula that the magnetic flux is equal to the magnetic field times the area of the surface they were looking at times the coast. And if I, We're fi is the angle between the magnetic field and the normal of the surface. So if I have the surface and it's lying flat than the normal points in that direction. And so let's say the magnetic fuel is going this way, five would be the angle between the normal to the surface and the magnetic field. And so we're just going to apply in this formula for each of the four parts. This problem and part, eh? The normal to the surface is parallel to the X axis, and so fi is equal to nine degrees. In this case, Number five is the angle between the magnetic fields in the normal to the surface and for part a for the surface. We're considering the normal to the surface, and the magnetic fields are perpendicular, and so this angle's Knight agrees. But this angle's nine degrees, the co sign of it is equal to zero. So for part A, there's no magnetic flux. So in Herbie the normal, the surfaces now parallel to the Z axis and so fi for Part B is equal to zero degrees. If icicle zero were our Costa, here is one and so five B It's just equal to the magnetic field times area, and so flooding these two things in is US 0.115 There's our answer, and in this case, the flux is into the volume, and you can get that just by looking at our volume and look at where the magnetic fields are going. And in this case, they're going into the wall for park Si the normal surface and the direction of bee is kind of hard to explain. But if you look at the figure, it's easy, see, and we can't get the angle Fi by saying that the tangent ify is equal to 40 centimeters. 40 Quinn's gross in years, over 30.0 centimeters. I'm just reading this off of the figure that they give us, and so five is equal to 53.1 degrees. If you take the inverse tangent of this once, we notify the angle between B and the normal to the area just like it in here. And so Ben's coach now, Bye. Hey, is going to end up doing 0.115 as the answer there. And the flocks is out of the volume in this case and again, you can get that just by looking at the situation by looking at the figure. Enjoy the magnetic field lines over it. And in part, me, they asked for the net by night field or the net magnetic flux. Rather and for Percy, we saw that it was this out of the line and part B. We saw that it was this into the line, but these are the same. And so the net flux, so I can indicate that like this, the Netflix is equal to zero because the amount going in is equal to the amount going out and that

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