Finding the flux along the curved part of a surface

[Pablo]

Homework Statement

The figure shows a closed surface. Along the flat top face, which has a radius of 4.5 cm, a perpendicular field of magnitude 0.18 T is directed outward. Along the flat bottom face, a magnetic flux of 0.60 mWb is directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

Relevant Equations

Flux on a surface = 0 (by Maxwell's equations)

I have this problems I am trying to figure out:

So I know that $$\int B\, dA = 0 = \phi_{total}$$
$$\phi_{total} = \phi_{bottom} + \phi_{top} + \phi_{side} = 0$$

$\phi_{side}$ must be equal to the other two fluxes, since they are both outwards:

$$\phi_{side} = \phi_{bottom} + \phi_{top}$$
$$\phi_{side} = 0.6*10^{-3} + \pi*(0.045)^2*0.18=1.745Wb$$

However, I am getting that 1.745Wb is the wrong answer. Does anyone know why?

 

[collinsmark]

Homework Statement: The figure shows a closed surface. Along the flat top face, which has a radius of 4.5 cm, a perpendicular field of magnitude 0.18 T is directed outward. Along the flat bottom face, a magnetic flux of 0.60 mWb is directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?
Homework Equations: Flux on a surface = 0 (by Maxwell's equations)

I have this problems I am trying to figure out:
View attachment 252397

So I know that $$\int B\, dA = 0 = \phi_{total}$$
$$\phi_{total} = \phi_{bottom} + \phi_{top} + \phi_{side} = 0$$

$\phi_{side}$ must be equal to the other two fluxes, since they are both outwards:

$$\phi_{side} = \phi_{bottom} + \phi_{top}$$
$$\phi_{side} = 0.6*10^{-3} + \pi*(0.045)^2*0.18=1.745Wb$$

However, I am getting that 1.745Wb is the wrong answer. Does anyone know why?

Your approach looks OK to me.

What did you get regarding the "inward" vs "outward" part of the problem?

For what it's worth, there is a convention that is sometimes used that states outward going flux is positive and flux directed inward is negative. I'm not sure if that applies here though since the problem statement just says "magnitude." (So I'm assuming that the expected "magnitude" part of the answer is non-negative.)

 

[Pablo]

Your approach looks OK to me.

What did you get regarding the "inward" vs "outward" part of the problem?

For what it's worth, there is a convention that is sometimes used that states outward going flux is positive and flux directed inward is negative. I'm not sure if that applies here though since the problem statement just says "magnitude." (So I'm assuming that the expected "magnitude" part of the answer is non-negative.)

So, I actually got the direction of the magnetic flux through the curved part of the surface correctly (inward). However, this shouldn't matter for the magnitude of the flux as long as I treat the flux of the side a different direction from the flux on the top and bottom. I don't understand why I am getting the magnitude incorrectly.

 

[haruspex]

1.745Wb is the wrong answer

Try mWb!

 

[collinsmark]

So, I actually got the direction of the magnetic flux through the curved part of the surface correctly (inward). However, this shouldn't matter for the magnitude of the flux as long as I treat the flux of the side a different direction from the flux on the top and bottom. I don't understand why I am getting the magnitude incorrectly.

Again, I agree with you. By that, I at least came up with the same answer that you did.

For what it's worth, the reason that inward flux is often treated as negative is because the surface area vector is conventionally pointed in the outward direction. So when you take the dot product between the magnetic field and the surface area vector, the result will be negative on patches where the magnetic field has a net "inward" component.

$d \phi = \vec B \cdot \vec{dA}$

Although $d \phi$ is a scalar, it can be positive or negative.

I was just throwing it out there that maybe the homework program is expecting a negative value for the magnitude. I don't know. The problem statement did say "magnitude," so I wouldn't think the negative sign is necessary. But, lacking some other explanation, maybe that's what it is expecting.

[Edit: Oop! Haruspex figured it out.]

 

[collinsmark]

Try mWb!

That's it. (or bring back the "$\times 10^{-3}$" that got lost somewhere in the submitted answer.)

 

[Pablo]

tried doing mWb (both 1.745 and -1.745), but still wrong:

 

[haruspex]

tried doing mWb (both 1.745 and -1.745), but still wrong:

View attachment 252411

You are quoting more digits than in the given data. Try rounding to 2 or 3. Beyond that, I give up.

 

[Pablo]

You are quoting more digits than in the given data. Try rounding to 2 or 3. Beyond that, I give up.

that shouldn't be a problem because there is a margin of error of 2% for answers.

 

[haruspex]

that shouldn't be a problem because there is a margin of error of 2% for answers.

Then we are down to guessing the error in the question. Maybe it was meant to say that both flat faces have a flux upward.

 

[Pablo]

I also put the negative value, and it was wrong:

 

[haruspex]

I also put the negative value, and it was wrong:
View attachment 252414

No, I am suggesting that of the two fluxes through the flat surfaces it is supposed to say that one is in and the other out.