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What determines the width of a lightning channel in a thunderstorm? Create a quantitative model.

We have a model of a regular $N$-gonal bipyramid made of conducting wires. The connections in the plane of symmetry each have resistance $R_2$, whereas the connections going from one of the vertices to a point in the base have resistance $R_1$. Determine the resistance between

1. main vertices (above and below the base plane),
2. adjacent vertices in the base plane,
3. opposite vertices in the base plane (the most distant ones) for even values of $N$.

Robert found out that he had to put 3 batteries with capacity $1~000~\mathrm{mAh}$ and voltage $U=1{,}5 \mathrm{V}$ into his new headlamp. In the headlamp, the batteries are connected in series. How long does it take for the batteries to discharge if they power a headlamp of output power $P=5 \mathrm{W}$ and efficiency $\eta =90 \mathrm{\%}$?

Let us construct the finite Sierpiński triangle of a degree $N$ (for $N = 1$ it is a single triangle, in case of $N = 2$ it is four triangles, etc.). The bases of small triangles (that the Sierpiński triangle is made of) consist of a resistor with resistance $R = 150 \mathrm{\ohm}$, the left legs are coils of inductance $L = 0{,}4 \mathrm{H}$ and the other legs are capacitors of capacitance $C = 20 \mathrm{\micro F}$. We measure the impedance between the triangle's bottom left and right corners. The angular frequency of the source is $\omega = 50 \mathrm{s^{-1}}$. Find the recurrent relation for the measured impedance and find its value for $N = 7$. What does the recurrent formula looks like if we replace coils and capacitors with resistors $R$? Determine its numerical value for $N = 15$.

Consider two coils wound around a common paper roll. First coil has a winding density of $10 \mathrm{cm^{-1}}$ and the second coil has a winding density of $20 \mathrm{cm^{-1}}$. The paper roll is $40 \mathrm{cm}$ long and has $1 \mathrm{cm}$ in diameter. Both coils are wound along the whole length of the roll, with the second coil wound around the first one. Considering the dimensions of the roll, we can neglect the boundary effects and assume that the coils behave as perfect solenoids. Now consider connecting the coils in series. This configuration can be substituted by a circuit with a single coil. What is the inductance of the substituting coil?

When there is a coronal mass ejection from the Sun, the mass will start to propagate with high velocity through the space. Sometimes the mass can hit the Earth and affect its magnetic field. Estimate the magnitude of the electric currents in the electric power transmission network on Earth which could be generated by such ejection. What parameters does it depend on? Comment on what effects would such event have on the civilisation.

Measure the capacity of an arbitrary battery (e.g. AA battery) and compare it with the declared value.

The electrical circuit shown in the figure can serve as a non-stationary magnetic field detector. It consists of nine edges of a cube formed by electric wire. The electrical resistance of one edge is $R$. If this construction lies in a non-stationary homogeneous magnetic field, which has, for simplicity, a constant direction, and its magnitude changes slowly, then there are currents $I_1$, $I_2$, $I_3$ flowing at the marked spots. With the knowledge of these currents, determine the direction of the magnetic field in space and also the dependence of its magnitude on time.

Consider a circuit with a coil, a capacitor, a resistor and a voltage source connected in series (i.e. they are not parallel to each other). The coil has an inductance $L$, the capacitor has a capacitance $C$ and the resistor has a resistance $R$. The voltage source creates a voltage $U = U_0 \f {\cos }{\omega t}$. Assume all devices to be ideal. Using the law of conservation of energy, write the equation relating the charge, the velocity of the charge (current $I$) and the acceleration of the charge (rate of change of the current $I$). This is an equation of a damped oscillator. Compared to the equation of damped oscillations of a mass on a spring, what are the quantities analogous to mass, stiffness of the spring and friction? Find the natural frequency of these oscillations.

Furthermore, using the quantities $L$, $R$ and $\omega $, find the capacity $C$ which causes a phase shift of the voltage on the capacitor equal to $\frac {\pi }{4}$. What is the amplitude of the voltage on the capacitor, assuming this phase shift?

Estimate the time that passes between the flip of a light switch and the turning on of the light source. Make independent estimates for a light bulb, fluorescent lamp, LED light bulb and Neon tube light. Discuss as many factors influencing the time as you can.