Example 23.2 from College Physics, 23.3 Motional Emf

Calculating the Large Motional Emf of an Object in Orbit

Figure shows a tethered satellite in Earth orbit. The Earth magnetic field is given as B Earth directed toward the plane of the paper. A tether satellite is a satellite connected to another by a space tether. An aircraft is shown flying at distance l below the tethered satellite. A current path is shown from the aircraft flying in the ionosphere to the tethered satellite and back.

Figure 23.12 Motional emf as electrical power conversion for the space shuttle is the motivation for the Tethered Satellite experiment. A 5 kV emf was predicted to be induced in the 20 km long tether while moving at orbital speed in the Earth’s magnetic field. The circuit is completed by a return path through the stationary ionosphere.

Calculate the motional emf induced along a 20.0 km long conductor moving at an orbital speed of 7.80 km/s perpendicular to the Earth’s $5 . 00 \times 10^{- 5} T$ magnetic field.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
This is a straightforward application of the expression for motional emf— $emf = Bℓv$ .
Solution
Entering the given values into $emf = Bℓv$ gives

$\begin{array}{l} emf & = & Bℓv \\ = & (5.00 \times 10^{- 5} T) (2 . 0 \times 10^{4} m) (7 . 80 \times 10^{3} m/s) \\ = & 7.80 \times 10^{3} V. \end{array}$
23.10
Discussion
The value obtained is greater than the 5 kV measured voltage for the shuttle experiment, since the actual orbital motion of the tether is not perpendicular to the Earth’s field. The 7.80 kV value is the maximum emf obtained when $θ = 90º$ and $sin θ = 1$ .

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