# Faraday's Law - Balloon Problem

• rofldude188
In summary: Also, keep in mind that the flux through a surface element is the dot product of the element and the field, not the cross product.
rofldude188
Homework Statement
A scheme for power generation proposes to use conducting balloons that expand and contract in waterfalls (with adequate insulation). Consider a balloon of volume V at a given time, with water flowing into it with a flow rate F (m3/s). The water falls down into the spherical balloon causing it to expand. The balloon is in a region where the magnetic field is uniform and vertical of magnitude B.
Relevant Equations
$$\phi = B \cdot dS$$
a) Calculate the proposed induced emf along the equator of the balloon. (horizontal
equator), at the moment indicated above.

$$V(t) = V + Ft \implies \frac{4 \pi r^3(t)}{3} = V + Ft \implies r(t) = \sqrt[3]{\frac{3V+3Ft}{4 \pi}}$$
$$\phi = B \pi r^2(t) = B\pi (\frac{3V+3Ft}{4 \pi})^{2/3}$$
$$V_{ind} = -\frac{d \phi}{dt} = 2FB \pi (\frac{3V+3Ft}{4 \pi})^{-1/3}$$

b) Indicate the direction of flow of the current around equator if B points down

By Lenz's Law, since area of equator is expanding that means more flux lines are entering downwards so there must be an induced current counterclockwise to oppose this.

c) The scheme then allows for the mouth of the balloon to be closed and water to leak
out through holes in the bottom when the volume of the balloon becomes 4V. The
initial leak rate is 2.5 F. Calculate the induced emf along a horizontal ring on the
balloon a vertical distance z away from the centre of the balloon

For this part, would the method not be the exact same as part a)? i.e. z = r and we find r(t) and proceed from there in the same manner as above?

d) Calculate the induced emf along the largest vertical circle on the balloon for c).

Not sure what to do here. Any help would be appreciated.

rofldude188 said:
a) Calculate the proposed induced emf along the equator of the balloon. (horizontal
equator), at the moment indicated above.

$$V(t) = V + Ft \implies \frac{4 \pi r^3(t)}{3} = V + Ft \implies r(t) = \sqrt[3]{\frac{3V+3Ft}{4 \pi}}$$
$$\phi = B \pi r^2(t) = B\pi (\frac{3V+3Ft}{4 \pi})^{2/3}$$
$$V_{ind} = -\frac{d \phi}{dt} = 2FB \pi (\frac{3V+3Ft}{4 \pi})^{-1/3}$$
This looks correct so far. But the question asks for the emf "at the moment indicated above". So, what value of the time ##t## should you use that corresponds to this moment?

b) Indicate the direction of flow of the current around equator if B points down

By Lenz's Law, since area of equator is expanding that means more flux lines are entering downwards so there must be an induced current counterclockwise to oppose this.
OK if you mean counterclockwise as viewed looking down from above the balloon.

c)

For this part, would the method not be the exact same as part a)? i.e. z = r and we find r(t) and proceed from there in the same manner as above?
Why do you say z = r? Also, note that the radius of the ring will be different than the radius of the balloon.

d) Calculate the induced emf along the largest vertical circle on the balloon for c).

Not sure what to do here. Any help would be appreciated.
Think about the flux through a vertical circle.

TSny said:
This looks correct so far. But the question asks for the emf "at the moment indicated above". So, what value of the time ##t## should you use that corresponds to this moment?

OK if you mean counterclockwise as viewed looking down from above the balloon.

Why do you say z = r? Also, note that the radius of the ring will be different than the radius of the balloon.

Think about the flux through a vertical circle.

Yeah you're right for the first part I need to plug in t = 0.

Sorry I misread I thought z was along the horizontal axis but I now see it's along the vertical axis. So in this case you would find the new radius $$r_z^2 = r^2 + z^2$$ and then find area of that circle with respect to time and go from there in a similar fashion to part a) right?

Largest vertical circle would have dS element perpendicular to B right so there would be no V_ind right?

rofldude188 said:
Yeah you're right for the first part I need to plug in t = 0.

Sorry I misread I thought z was along the horizontal axis but I now see it's along the vertical axis. So in this case you would find the new radius $$r_z^2 = r^2 + z^2$$ and then find area of that circle with respect to time and go from there in a similar fashion to part a) right?

Largest vertical circle would have dS element perpendicular to B right so there would be no V_ind right?
All of that sounds good except for your equation for ##r_z##. Looks like a sign error.

##\phi=B.dS## is wrong, ##d\phi=B.dS## is the flux element.

## 1. What is Faraday's Law?

Faraday's Law is a fundamental principle in electromagnetism that explains the relationship between electric fields and magnetic fields. It states that a changing magnetic field will induce an electric field, and a changing electric field will induce a magnetic field.

## 2. What is the "Balloon Problem" in Faraday's Law?

The "Balloon Problem" is a thought experiment used to demonstrate Faraday's Law. It involves a balloon filled with conductive gas being placed in a changing magnetic field. The result is that an electric field is induced in the balloon, causing it to move in the opposite direction of the magnetic field.

## 3. How does Faraday's Law relate to everyday technology?

Faraday's Law is the basis for many technologies that we use in our daily lives, such as electric generators, transformers, and electric motors. It is also essential in understanding how electromagnetic radiation, such as radio waves and light, can be transmitted and received.

## 4. What are the practical applications of Faraday's Law?

Faraday's Law has numerous practical applications, including power generation, electromagnetic compatibility testing, and electromagnetic shielding. It is also used in medical technology, such as MRI machines, and in the production of electronic devices.

## 5. Who is Michael Faraday and why is he important to Faraday's Law?

Michael Faraday was a British scientist who made significant contributions to the fields of electromagnetism and electrochemistry. He is credited with formulating Faraday's Law and is considered one of the most influential scientists in history. His experiments and discoveries laid the foundations for modern physics and technology.

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