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## electric current

### (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.

### (10 points)2. Series 34. Year - S. series 2

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 \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?

Non-mechanical oscillations are oscillations as well.

### (10 points)5. Series 33. Year - P. there will be light

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.

Dodo throws the circuit breaker.

### (10 points)5. Series 33. Year - S. min and max

*We are sorry. This type of task is not translated to English.*

They had to wait a lot for Karel.

### (10 points)4. Series 33. Year - S.

*We are sorry. This type of task is not translated to English.*

### (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.

Karel. Don't even ask.

### (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