e₀ = 8.85 X 10^-12 Fm^-1; q = 1.6 X 10^-19 C: e_r (of air) = 1.005; m_proton = 1.66 X 10^-26 kg

1. Two protons have been fired together in an accelerator. Their centres are 2.5 nm apart.

(a) Calculate the electrostatic force of repulsion between them.

(b) Calculate the gravitational force of attraction between them.

(c) Calculate the ratio of the electrostatic repulsion to the gravitational attraction. (This is why gravity is described as much weaker than electric, and magnetic, forces)

2. The spherical globe at the top of a Van der Graaf generator, of radius 0.2 m, is charged positively, such that the electric potential at its surface is +300 000 V.

(a) Draw the top of the generator. Sketch the rough positions of the equipotential surfaces of voltage +150 000 V and +100 000 V.

(b) A fast-moving sphere, carrying a charge of +3 mC, is moving towards the globe. How much electrical energy does it gain, as it moves from the +100 000 V equipotential surface to the +150 000 V surface?

(c) Assuming that the charge on the generator behaves as though it is all at the centre of the globe, how much charge is on the globe?

(d) Calculate the position of the 150 000 V equipotential surface.

3. Two metal plates are placed 8 cm apart. A high voltage supply is connected across them, switched to 4 000 V.

(a) Draw a side view of the plates. Mark on appropriate charges, and draw the electric field lines between the plates.

(b) What is the shape of the equipotential surfaces between the plates? Draw the position of the equipotential surface at +2 000 V, and the one at +1 000 V.

(c) What is the name of this kind of electric field?

(d) What is the size of the electric field strength between the plates?

(e) A singly-charged Oxygen ion drifts into the region between the plates. What is the size of the electric force on the ion?

4. An electron is located 2 X10^-9 m from a proton. At what minimum speed would it need to be travelling, if it was to be able to escape completely from the electric field of the proton? (Assume no collisions with other particles)

5. Air starts to conduct electricity if an electric field of about 20 kVcm^-1 exists in it.

(i) What is the minimum potential difference between the base of a cloud 800 m above the surface of the Earth, and the Earth, for a lightning bolt to jump directly from the cloud to Earth?

(ii) What is (approximately) the shape of the equipotentials between the cloud and the ground?

(iii) Suppose that the base of the cloud is negative. If a drop of water, just beneath the cloud, of mass 2 g, became charged, by friction, with three quanta of negative charge, what would its acceleration be?

(iv) How would calculation (iii) be affected if the drop was just above the surface of the Earth?

6. In a nucleus, the protons are typically about 10^-15 m away from each other.

(i) Calculate the electrostatic repulsive force between them.

(ii) Calculate the electric potential energy of one proton, as a result of this proximity to another.

(iii) There are 6 protons in ¹²C. Calculate the total electrical potential energy in a carbon nucleus.

(iii) To get a better impression of the size of this potential energy, calculate the amount of potential energy in the nuclei of 20 g of ¹²C.

7. A small pith ball is charged negatively with 2 mC of charge. 3 cm West of it there is a second pith ball, charged positively with 1.5 mC. 4 cm East of the original ball there is a third pith ball, charged positively with 3 mC.

(i) What is the resultant electrostatic force on the original ball? (magnitude and direction)

(ii) What is the electrostatic potential energy of the 3 mC sphere?

(iii)^* What approximately is the electric field strength, and electric potential, at a distance of 3 m from the original ball?

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