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

• mnafetsc
In summary, the problem involves finding the total magnetic flux through the plastic of a soda bottle, given its opening diameter and a uniform magnetic field. The solution involves using Gauss' law for magnetism, and finding the flux through the opening of the bottle, then using the relationship between the flux through the opening and the flux through the plastic to solve for the latter. The negative sign in the solution is due to the fact that the problem asks for the flux through the plastic, not the opening.
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.

mnafetsc said:
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.

Guass' 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 Ao in the right side of the above two equations is the area of the opening -- not the area of the plastic)

## 1. What is magnetic flux?

Magnetic flux is the measure of the total magnetic field that passes through a given area. It is represented by the symbol Φ and is measured in units of webers (Wb).

## 2. How is magnetic flux calculated?

Magnetic flux is calculated by multiplying the magnetic field strength (B) by the area (A) that the field passes through, and by the cosine of the angle between the field and the area. The formula for magnetic flux is Φ = B*A*cos(θ).

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

The total magnetic flux through the plastic of the soda bottle depends on the strength and direction of the magnetic field passing through it, as well as the area of the plastic surface that the field passes through.

## 4. Can magnetic flux pass through non-magnetic materials?

Yes, magnetic flux can pass through non-magnetic materials, such as plastic. However, the amount of flux that passes through the material will depend on its permeability, which is a measure of how easily a material can be magnetized.

## 5. How does the total magnetic flux through the plastic of the soda bottle affect the soda inside?

The magnetic flux passing through the plastic bottle will not have any direct effect on the soda inside. However, if the magnetic field is strong enough, it can induce electrical currents in the soda, which can cause it to heat up slightly. This is known as magnetic induction and is the principle behind induction cooktops.

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