The task, which in the future should enter into the physics textbooks, that sounds about right. You are on the Planet's surface (a large sphere) and I know that it has a known static electric charge, which is mainly concentrated in its surface the conducting layer and is distributed there evenly. The question boils down to how you use the energy of the charge applying only materials of this Planet: conductors, semiconductors and insulators. Additional restrictions: you can not use points of support, contributing to the creation of the natural potential difference. For example, are not suitable atmospheric charge, lightning bolts or chemical potential difference. This also means that you can't use the potential difference between this Planet and any other object in the Universe.
In this work we will propose a few solutions to this problem, and that is why it is still there in the school books you, dear readers, will easily guess for yourself :)

To begin with, we introduce the readers into the swing of things and will introduce some interesting numbers that will treat the planet Earth. We'll need them in the future for assessment of energy in real devices. We consider our planet as a sphere of radius \(r_E = 6.37\cdot 10^6\) meters, and a negative charge \(q_E = 5.87\cdot 10^5\) Coulomb [1]. Because the charge distributed on the surface of a sphere evenly, then surface charge density will be the ratio of the total charge to the area of the sphere: \[\sigma = {q_e \over 4\pi r_E^2 } = 1.15\cdot 10^{9}\, [{C/m^2}] \qquad (1.1)\] This figure we show how the Pendant is in every square meter of the surface layer of the Earth. It's easy to remember: approximately 1 nanocolor on 1 square meter. Despite such a small magnitude we can use the second condition of the problem — conductivity of the surface of a sphere, for example, to specify the charges with any of its area, but more on that later. And while we believe these figures and simply substitute them into the classical formula for finding the electric field around the sphere (in this case Earth): \[E_E = {\sigma \over \varepsilon_0} = 130\, [{V/m}] \qquad (1.2)\] where: \(\varepsilon_0 = 8.85\cdot 10^{12}\) is the absolute dielectric permittivity of vacuum [2]. The resulting value of the tension is fully consistent with the official data on the planet: every meter above the surface adds 130 Volts [3]. This value is quite decent and seemingly you can just set the mast height of about ten meters, to consolidate there the metal plate and swing the free energy. And the voltage between the top of the mast and the ground is large, about 1300 Volts: you're welcome. But we forget that electric potential is not energy, for the latter requires at least a charge, but it is very weak (see formula 1.1). Therefore this method we will not use, especially because it prohibited additional terms of tasks.
Will get another value — a solitary capacity of the Land, which is based on the classical formula for the Orb [4]: \[C_E = 4\pi\, \varepsilon_0\, r_E = 7\cdot 10^{4}\, [F] \qquad (1.3)\] i.e. the capacity of our planet is only 700 microfarad. This seems a relatively small figure, but then you need to remember that in fact it is the capacity of the surface of the Earth relative to the rest of the neutral Universe. And a huge interplanetary (and interstellar) distances of our readers can well imagine, and without specific numbers.
By the way, the potential difference between the Earth's surface and the positively charged ionosphere, according to various estimates, from 400 to 600 kV. This great potential could be used, for example, for flights between the planets of the Solar system, simply by using the strength of the Coulomb [5]. But you need to know the electrical parameters of other planets, and learn how to change the potential or the charge of the spacecraft relative to them.
It may be another obvious question: where did Earth gets its electric charge and will it eventually recover? In [6] this is discussed in detail, but here we will just give you a small excerpt.
Since Feynman's teaching on the lightning has changed. It is now well understood by the fact that their ranks swell earth reserves of negative electricity [7].
Not so long ago was discovered another source of natural replenishment of the charge.
If you say quite simply, our task is to learn to "pump" electric charge from the Earth. By analogy with the extraction of other natural resources, we need the pump, only in this case it will be electrostatic. You can do it in different ways, but we will focus on electrostatic pump (EHF) with the parametric capacity, and later discuss EHF on parametric inductance. The solution of the problem can be divided into two parts, the first will present itself parametric capacitance and symbols, and the second — the methods and algorithms of its switching.
The materials used  Wikipedia. Earth.
 Wikipedia. Dielectric permeability.
 Feynman, Richard Phillips. Chapter 9. Electricity in the atmosphere.
 Wikipedia. The electrical capacity.
 Wikipedia. Coulomb's Law.
 V. I. Grigorev, G. Y. Myakishev Forces in nature. Chapter 4. Free charges and currents in nature. Charged particles above us and around us.
 Williams E. R. Lesson №4. Electrification of storm clouds. The negative charge of the earth's surface.
 Kuznetsov V. V. the potential of the geomagnetic field, the currents of the SCHMIDTBAUER and atmospheric electricity.