# 14.6 Rlc series circuits  (Page 3/4)

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A rectangular toroid with inner radius ${R}_{1}=7.0\phantom{\rule{0.2em}{0ex}}\text{cm,}$ outer radius ${R}_{2}=9.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ , height $h=3.0$ , and $N=3000$ turns is filled with an iron core of magnetic susceptibility $5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}$ . (a) What is the self-inductance of the toroid? (b) If the current through the toroid is 2.0 A, what is the magnetic field at the center of the core? (c) For this same 2.0-A current, what is the effective surface current formed by the aligned atomic current loops in the iron core?

The switch S of the circuit shown below is closed at $t=0$ . Determine (a) the initial current through the battery and (b) the steady-state current through the battery.

a. 0 A; b. 2.4 A

In an oscillating RLC circuit, $R=7.0\phantom{\rule{0.2em}{0ex}}\text{Ω},L=10\phantom{\rule{0.2em}{0ex}}\text{mH},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C=3.0\phantom{\rule{0.2em}{0ex}}\mu \text{F}$ . Initially, the capacitor has a charge of $8.0\phantom{\rule{0.2em}{0ex}}\mu \text{C}$ and the current is zero. Calculate the charge on the capacitor (a) five cycles later and (b) 50 cycles later.

A 25.0-H inductor has 100 A of current turned off in 1.00 ms. (a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?

a. $2.50\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{V}$ ; (b) The voltage is so extremely high that arcing would occur and the current would not be reduced so rapidly. (c) It is not reasonable to shut off such a large current in such a large inductor in such an extremely short time.

## Challenge problems

A coaxial cable has an inner conductor of radius a, and outer thin cylindrical shell of radius b. A current I flows in the inner conductor and returns in the outer conductor. The self-inductance of the structure will depend on how the current in the inner cylinder tends to be distributed. Investigate the following two extreme cases. (a) Let current in the inner conductor be distributed only on the surface and find the self-inductance. (b) Let current in the inner cylinder be distributed uniformly over its cross-section and find the self-inductance. Compare with your results in (a).

In a damped oscillating circuit the energy is dissipated in the resistor. The Q -factor is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a lightly damped circuit the energy, U , in the circuit decreases according to the following equation.

$\frac{dU}{dt}=-2\beta U,$ where $\beta =\frac{R}{2L}.$

(b) Using the definition of the Q -factor as energy divided by the loss over the next cycle, prove that Q -factor of a lightly damped oscillator as defined in this problem is 

$Q\equiv \frac{{U}_{\text{begin}}}{\text{Δ}{U}_{\text{one cycle}}}=\frac{1}{R}\sqrt{\frac{L}{C}}.$

( Hint: For (b), to obtain Q , divide E at the beginning of one cycle by the change $\text{Δ}E$ over the next cycle.)

proof

The switch in the circuit shown below is closed at $t=0\phantom{\rule{0.2em}{0ex}}\text{s}$ . Find currents through (a) ${R}_{1}$ , (b) ${R}_{2}$ , and (c) the battery as function of time.

A square loop of side 2 cm is placed 1 cm from a long wire carrying a current that varies with time at a constant rate of 3 A/s as shown below. (a) Use Ampère’s law and find the magnetic field as a function of time from the current in the wire. (b) Determine the magnetic flux through the loop. (c) If the loop has a resistance of $3\phantom{\rule{0.2em}{0ex}}\text{Ω}$ , how much induced current flows in the loop?

a. $\frac{dB}{dt}=6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{T/s;}$ b. $\text{Φ}=\frac{{\mu }_{0}aI}{2\pi }\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\left(\frac{a+b}{b}\right)$ ; c. 4.0 nA

A rectangular copper ring, of mass 100 g and resistance $0.2\phantom{\rule{0.2em}{0ex}}\text{Ω}$ , is in a region of uniform magnetic field that is perpendicular to the area enclosed by the ring and horizontal to Earth’s surface. The ring is let go from rest when it is at the edge of the nonzero magnetic field region (see below). (a) Find its speed when the ring just exits the region of uniform magnetic field. (b) If it was let go at $t=0$ , what is the time when it exits the region of magnetic field for the following values: $a=25\phantom{\rule{0.2em}{0ex}}\text{cm},\phantom{\rule{0.2em}{0ex}}b=50\phantom{\rule{0.2em}{0ex}}\text{cm},\phantom{\rule{0.2em}{0ex}}B=3\phantom{\rule{0.2em}{0ex}}\text{T},\phantom{\rule{0.2em}{0ex}}\text{and}$ $g=9.8\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}?$ Assume the magnetic field of the induced current is negligible compared to 3 T.

Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
how was the check your understand 12.7 solved?
Who is ISSAAC NEWTON
he's the father of 3 newton law
Hawi
he is Chris Issaac's father :)
Ethem
how to name covalent bond
Who is ALEXANDER BELL
LOAK
what do you understand by the drift voltage
what do you understand by drift velocity
Brunelle
nothing
Gamal
well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
graviton
drift velocity
graviton
what is an electromotive force?
It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
Brunelle
How electromotive force is differentiated from the terminal voltage?
Danilo
in the emf power is generated while in the terminal pd power is lost.
Brunelle
what is then chemical name of NaCl
sodium chloride
Azam
sodium chloride
Brunelle
How can we differentiate between static point and test charge?
Wat is coplanar in physics
two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
0.0844kg
Humble
what is the difference between temperature and heat
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Abdul
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
Gamma
the average kinetic energy of molecules is called temperature. heat is the method or mode to transfer energy to molecules of an object but randomly, while work is the method to transfer energy to molecules in such manner that every molecules get moved in one direction.
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
A charge insulator can be discharged by passing it just above a flame. Explain.
of the three vectors in the equation F=qv×b which pairs are always at right angles?
what is an ideal gas?
What is meant by zero Kelvin ?
Justine
Why does water cool when put in the pot ?
Justine
when we pour the water in a vessel(pot) the hot body(water) loses its heat to the surrounding in order to maintain thermal equilibrium.Thus,water cools.
rupendra
when we drop water in the pot, the pot body loses heat to surrounded in order to maintain thermal equilibrium thus,water cool.
Srabon
my personal opinion ideal gas means doesn't exist any gas that obey all rules that is made for gases, like when get the temp of a gas lower, it's volume decreases.since the gas will convert to liquid when the temp get lowest.. so you can imagine it, but you can't get a gas at the lowest T
Edit An ideal gas is a theoretically gascomposed of many randomly moving point particles whose only interactions are perfectly elastic collisions.
Gamma
ideal gases are real gases at low temperature
Brunelle