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## electric current

### (3 points)1. Series 33. Year - 2. battery issue on holidays

How long does it take for a fully charged car battery ($12 \mathrm{V}$, $60 \mathrm{Ah}$) to run out, when someone forgets to turn off the daytime running lights, locks the car and walks away? Specifically we are interested in a situation with two head lights H4 (each running with $55 \mathrm{W}$) and two rear lights P21/5W (each running with $5 \mathrm{W}$). For simplicity, assume no transport losses between the battery and the lights, that there is no other significant consumption of power and that the voltage on the battery stays constant.

### (12 points)3. Series 32. Year - E. indexed capacitor

Find an electrolytic capacitor and a resistor and measure their capacity and resistance, respectively. You cannot measure these quantities directly. We recommend a choice of parameters, such that $RC\approx 20 \mathrm{s}$.

Be aware of maximum allowed voltage on the capacitor and the capacitor's polarity.

Dodo was measuring resonance in labs.

### (10 points)3. Series 32. Year - P. personal power bank

Last battery percentages in your mobile phone are almost gone, your power bank is dead, or you left it at home and 230 is also not in the sight. Wouldn't it be awesome if you could have your own source of electrical energy with you all the time?

• Suggest several different tools, which would be able to produce electrical energy just from your body resources.
• Discuss their maximum power and efficiency. What devices could you supply with electricity using this method?
• Discuss its effects on your health and physical condition. Which body organs would fail first?

As a possible solution, consider a system of small turbines located in your bloodstream. Support all arguments with accurate calculations.

Jachym had a feeling that he is missing some energy.

### (8 points)1. Series 32. Year - 5. damned circuit

a) Determine the resistance between points A and B of the infinite grid in the picture. The point A is directly connected to two resistors with resistances $R_a$ and $R_b$. Each of these resistors is connected to two more resistors with $R_a$ and $R_b$ etc.

b) Replace all the resistors with capacitors of capacitances $C_a$ and $C_b$. What is the total capacitance of the circuit?

Yet again, Karel wanted something unendingly infinite.

### (3 points)6. Series 31. Year - 2. hot wire

Calculate the current, that needs to pass through a metal wire of a diameter $d = 0{,}10 \mathrm{mm}$ located in a vacuum bulb, so that its temperature stays at $T = 2 600 K$. Assume the surface of the wire radiates like an ideal black body and neglect any losses by heat conduction. The resistivity of the material of the wire at the given temperature is $\rho = 2{,}5 \cdot 10^{-4} \mathrm{\Ohm \cdot cm}$. \taskhint {Hint}{Use the Stefan-Boltzmann's law.}

Danka was contemplating the light bulb efficiency

### (7 points)4. Series 31. Year - 4. solve it yourself

We have a black box with three outputs (A, B, and C). We know that it consists of $n$ resistors with the same resistance but we don't know the circuit diagram. So we measure the resistance between each pair of outputs $R\_{AB} = 3 \mathrm{\Omega }$, $R\_{BC} = 5 \mathrm{\Omega }$ a $R\_{CA} = 6 \mathrm{\Omega }$. Your task is to find the minimum possible $n$ and calculate the corresponding resistance of one resistor.

Matěj solved it quickly.

### (3 points)2. Series 30. Year - 2. ultra high temperature superconducticvity

Many types of materials, mostly metals, have increasing dependence of resistivity on temperature. However, there are semiconductors or graphite which show a decreasing dependence. And you have also probably heard about superconductivity, the natural phenomenon when a cooled material shows almost no electrical resistance and becomes a perfect conductor. Our current state of knowledge says that the temperature of a superconductor must be well below room temperature, but let's assume that the equation defining the resistance is $R=$ R_{0} (1 + αΔt), where $R_{0}is$ the resistance at room temperature, $αis$ the temperature coefficient of resistance and $Δt$ is the temperature difference with respect to room temperature, and the equation holds for any temperature. Using this equation and coefficients $α_{C}=-0.5\cdot 10^{-3}K^{-1}$ for graphite and $α_{Si}=-75\cdot 10^{-3}K^{-1}$ for silicon, we obtain zero resistance for high temperatures. Determine these two temperatures and explain why the superconducting phenomenon does not work this way, i.e. neither carbon nor silicon are superconductors at high temperatures.

Karel se inspiroval nekonstantními konstantami.

### (4 points)6. Series 29. Year - 4. Fire in the hole

Neutral particle beams are used in various fusion devices to heat up plasma. In a device like that, ions of deuterium are accelerated to high energy before they are neutralized, keeping almost the initial speed. Particles coming out of the neutralizer of the COMPASS tokamak have energy 40 keV and the current in the beam just before the neutralization is 12 A. What is the force acting on the beam generator? What is its power?

### (2 points)5. Series 29. Year - 1. let it flow

Thin wire with resistance $R=100mΩ$ and length $l=1\;\mathrm{m}$, that is connected to the source of DC with voltage $U=3V$, contains in its volume $N=10^{22}$ free electrons, which contribute to the electric current. Determine what is the average speed (more accurately net velocity) of these electrons in the wire. 