- #1

rofldude188

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- 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.

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.