Hyperloop: Voltage
The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. If we model the Hyperloop pod as a hollow metal cylinder with a radius of 1 m and a length of 1 0 m , what is the maximum voltage difference between the sides of the pod in Volts ?
Details and assumptions
- The magnetic field at the earth's surface is 0 . 5 Gauss .
- The speed of the pod is 3 0 0 m/s .
The answer is 0.03.
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4 solutions
Nice !
I think though that you mean 5 . 0 × 1 0 − 5 T instead of 5 . 0 × 1 0 5 T .
This is similar to an AP E&M question years ago where the moving object was a plane and the induced potential was across the wing. In that instance the planes direction and the orientation of earths fold was spelled out. This is a vector problem after all. Given conventional orientations of fields the can would have to be going roughly east-west with the potential difference measured from the top to bottom of the cylinder. Isn't this train proposed to be traveling roughly north-south though? ☺️
Did the ‘hard’ work and forgot the 2 for the diameter... How irritating!
To clarify, I mean the maximum distance between 2 points which, when a line is drawn in between each other is perpendicular to both the velocity and magnetic field vector
Emf = d t d Φ = B d t d A = B d t 2 r l = B d t 2 r v t
= B ∗ 2 r ∗ v = 5 ∗ 1 0 − 5 ∗ 2 ∗ 1 ∗ 3 0 0 = 0 . 0 3 V
Now we clearly know that motional emf (which is required in this case) is :
E = v . B . l
which in this case turns out to be
E = v . 2 r . l
now justing plugging in the values as v = 3 0 0 m / s magnetic field
B = 0 . 5 Gauss (convert it to Tesla) and then l = 2 r
multiplying we get the answer as 0 . 0 3 V
Apply Faradays Law:
Alternate text
(where, ϕ = B x Area)
This question can be solved by manipulating the equation for Lorentz force to find potential difference.
F= q × ( v × B ))
W= l × q × ( v × B )
q W = V = l × ( v × B )
B is the magnetic field, which in this case is 0.5 Gauss, or 5 . 0 × 1 0 5 Teslas, the latter being in SI units. v is 300m/s. Now in this case, the length refers to the maximum distance between points perpendicular to both the direction of the velocity and field, which for this question will be the 2 ends of the face of the train, rather than the front and back of the train. And now, you can plug in values to find your potential difference in volts.
V = ( 1 m × 2 ) × ( 5 . 0 × 1 0 5 T ) × 3 0 0 m / s = 0 . 0 3 V