What is the total magnetic flux through the plastic of the soda bottle?

[mnafetsc]

Homework Statement

An open plastic soda bottle with an opening diameter of 2.7 cm is placed on a table. A uniform 1.95-{\rm T} magnetic field directed upward and oriented 21^\circ from vertical encompasses the bottle.

What is the total magnetic flux through the plastic of the soda bottle?

Homework Equations

flux=B*A*cos$$\theta$$

The Attempt at a Solution

I have the correct answer, but I don't understand why it is negative instead of positive, or in other words why to i subtract 21 from 180, instead of just plugging in 21 into cosine.

 

[collinsmark]

I have the correct answer, but I don't understand why it is negative instead of positive, or in other words why to i subtract 21 from 180, instead of just plugging in 21 into cosine.

It's negative because the problem statement wants you to find the total magnetic flux through the plastic, not the opening.

Sure, you could mathematically model the intricate, detailed shape of the plastic bottle, and then tediously calculate $\int B \cdot dA$ of the plastic. But unless you have a computer with some expensive modeling software, this approach will take awhile. Fortunately there is an easier way.

Gauss' law for magnetism states

$$\oint _S \vec B \cdot d \vec A = 0$$

Given what we know of the bottle (the plastic part and the opening part together form a closed surface), this can be written as

$$\int _{plastic} \vec B \cdot d \vec A + \int _{opening} \vec B \cdot d \vec A = 0$$

But we're only interested in the plastic part. But the opening part is the part that's easy to calculate. Fortunately, by rearanging the above terms, we can calculate the plastic part in terms of opening part.

$$\int _{plastic} \vec B \cdot d \vec A = -\int _{opening} \vec B \cdot d \vec A$$

And since you know

$$\int _{opening} \vec B \cdot d \vec A = A_oB \cos \theta$$

you can find the flux through the plastic part,

$$\int _{plastic} \vec B \cdot d \vec A = -A_oB \cos \theta$$

(Where A_o in the right side of the above two equations is the area of the opening -- not the area of the plastic)