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## electric current

### (9 points)5. Series 37. Year - 5. tuning a circuit

Consider a series circuit with a resistor of resistance $R$, a coil, and a capacitor with the capacitance $C$. AC voltage sources with identical amplitudes $U$ are connected in series with these components. These sources vary in frequency by being multiples of $\omega _0$, where $n$ represents an integer. What frequency, denoted by $\omega _0$, would allow us to find a coil possessing an inductance $L$, such that voltages with frequencies different from $N \omega _0$ are suppressed by at least $90 \mathrm{\%}$ on the resistor? $N$ is a positive natural number known in advance (i.e., the value of $\omega _0$ may depend on it), and we do not want to suppress the voltage with frequency $N\omega _0$ by more than $90 \mathrm{\%}$.

Jarda wanted to have as many different sources in the circuit as possible.

### (10 points)5. Series 37. Year - S. we are spending electricity

- The aluminum smelter annually produces $160~000 t$ of aluminum, which is produced by electrolysis of alumina using a DC voltage of $U=4{,}3 \mathrm{V}$. Determine how many units of nuclear power plant with a net electrical output of $W_0=500 \mathrm{MW}$ are equivalent to the energy consumed by the aluminum smelter.
- In the second half of the last century, conventional electrical units were based on the values of the frequency of the cesium hyperfine transition $\nu \_{Cs}=9~192~631~770 Hz$, the von Klitzing constant $R\_K=25~812.807 \ohm $ and the Josephson constant $K_J=483~597.9e9 Wb^{-1}$. Determine the value of the coulomb $1 \mathrm{C}$ using these constants. {enumerate}

Dodo has dead batteries.

### (9 points)3. Series 37. Year - P. by a flash

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

Karel stumbled upon a claim about the Sky Tower lightning rod.

### (5 points)6. Series 36. Year - 3. repellent resistive bipyramides

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

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

Karel wanted N-gonal bipyramides.

### (3 points)4. Series 36. Year - 1. discharging the battery

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 $\eta =90 \mathrm{\%}$?

Robert's headlamp was not working.

### (8 points)5. Series 35. Year - 5. alternating triangle

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 $N = 15$.

Honza likes fractals.

### (5 points)3. Series 35. Year - 3. two solenoids

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?

Jindra played games with paper rolls.

### (10 points)6. Series 34. Year - P. more dangerous corona

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.

Karel was at a conference and then he saw a video on the same topic.

### (12 points)5. Series 34. Year - E. do they deceive us?

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

### (10 points)2. Series 34. Year - 5. magnetic non-stationarities detector

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.

Vašek thought that an electromagnetic induction problem would be welcome.