Series 5, Year 22

Choosing two different vectors in a plane and shifting the origin, an infinite grid of nodes can be created (see picture). Using the same approach in 3D will make a crystal lattice. If such grid is shifted by one of the vectors, we will get identical grid. Also rotation of the grid by some angle will generate identical grid. Find out all angles which can be basis for rotationaly symetrical grid and draw how such grid looks like.

During a Battle of Britain a strong search light were used to find enemy planes. What will be the deflection from the direct direction of light beam, assuming the linear decrease of refractive index of the atmosphere. How this deflection depends on the angle of the beam towards vertical direction?

What was the speed of Earth rotation during solidifying of Earth to have the difference between Equator radius and pole radius same as today?

Assume a straight optical fibre. The incoming light enters fibre at maximum angle $α$ from the axis of fibre (higher angles will not be guided by fibre). What is the minimum length of the light pulse to guarantee no overlap between 0 and 1 bit (e.g. at least for short interval the intensity of signal must be minimal or maximal). The length of fibre is $d$.

Estimate, what energy is absorbed by the body of visitor on a rock concert. Give reasons for your answer.

Measure, how high the water will splash during fall of different bodies onto a water level. Observe the dependence of height, shape and mass of falling object. What percentage of energy is used to create waves on the water level?

• The second method to measure specific electron charge used by J. J. Thomson is observation of deflection of cathode ray by electric field. Assume apparatus as on the figure. How depends deflection of the beam observed on the screen on the right on the electrical voltage, speed and geometrical dimensions of apparatus?

• The one of problems which was J. J. Thomson facing during measurement of specific electron charge was following: After the beam entered magnetic field, the beam has spread into a bigger area (see figure). The dispersion is causing some error in measuring position of electrons (Thomson mentioned up to 20%). How can be this dispersion explained? How can be this inaccuracy improved (this is for bonus point)?

• By using the data from table calculate charge of electron in case, that the oil had density of 920 kg\cdot m^{ − 3}, air density was 1,2 kg\cdot m^{ − 3} and viscosity 17,1 \cdot 10^{−7} Pa\cdot s. Used electric field was 250 kV\cdot m^{ − 1}.