**TAP 408-3: Non-uniform electric fields**
**Data required:**
charge of electron = 1.6 10^{-19} C
k = 1/ (4_{} = 9.0 10^{9} N m^{2 }C^{-2}
1) In an experiment to test Coulomb’s law, two expanded polystyrene spheres, each with a charge of 1.0 nC were 0.060 m apart (measured from their centres). Calculate the force acting on each sphere.
2) The electric field strength field at a distance of 1.0 10^{-10} m from an isolated proton is 1.4410^{11} N C^{–1} and the electrical potential is 14.4 V.
a) Calculate the electric field strength at a distance of 2.0 10^{-10} m from the proton.
b) Calculate the electrical potential at a distance of 2.0 10^{-10} m from the proton.
3) A simple model of a hydrogen atom can be thought of as an electron 0.50 10^{-10} m from a proton.
a) Calculate the electrical potential 0.50 10^{-10} m from a proton.
b) The electron is in the electric field of the proton. Calculate the electrical potential energy of the electron and proton in joules.
4) When a uranium nucleus containing 92 protons and rather more neutrons emits an alpha particle of charge + 3.2 10^{-19} C the remaining nucleus then behaves like a sphere of charge of magnitude + 1.4 10^{-17} C.
a) Assuming that the alpha particle is 2.0 10^{-14} m from the centre of the nucleus on release, calculate the electric field experienced by the alpha particle.
b) Calculate the force on the alpha particle when at this distance.
c) Calculate the maximum acceleration of the alpha particle (of mass 6.6 10^{-27} kg).
5) The graph shows the variation of potential with distance from the charged dome of a van de Graaff generator.
U se the graph, together with the equation to find the electric field strength at a distance of 0.3 m from the dome. (You may like to check your answer with an alternative calculation).
**Answers and worked solutions**
1) F=kQ_{1}Q_{2}/r^{2} = 9.0 x 10^{9} x 1.0 x 10^{-9} x 1.0 x 10^{-9} / 0.060^{2} = 2.5 x 10^{-6} N
2
a) Field strength is proportional to 1/r^{2}. Therefore doubling the distance decreases field strength by a factor of 4. Thus, answer is 1.44 x 10^{11} / 4 = 3.6 x 10^{10} N C^{-1}
b) Potential is proportional to 1/r. Thus doubling the distance halves the potential. The answer is therefore 14.4 / 2 = 7.2 V
3
a)
V = kQ/r = 9.0 x 10^{9} x 1.6 x 10^{-19} / (0.50 x 10^{-10}) = 28.8 V
b) EPE = VQ = 28.8 x 1.6 x 10^{-19} = 4.61 x 10^{-18} J
4)
a) E = kQ/r^{2} = 9.0 x 10^{9} x 1.4 x 10^{-17} / (2.0 x 10^{-14})^{2} = 3.2 x 10^{20} N C^{-1}
b) F = EQ = 3.2 x 10^{20} x 3.2 x 10^{-19} = 101 N (3sf)
c) a =F/m = 101 / 6.6 x 10^{-27} = 1.5 x 10^{28} m s^{-2}
The field strength will be the negative of the gradient at r = 0.3m. This should give a value of around 1.7 x 10^{5} N C^{-1}.
To calculate this we note that at 0.3 m the potential is 50 kV. Now V=kQ/r, and we note that E=kQ/r^{2}, so numerically, we can see that E = V/r. In other words, in this case
E = 50,000 / 0.3 = 1.7 x 10^{5} N C^{-1}.
**External reference**
This activity is taken from Advancing Physics chapter 16 |