# 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.

please why is the first law of thermodynamics greater than the second
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
explain the lack of symmetry in the field of the parallel capacitor
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe
does your app come with video lessons?
What is vector
Vector is a quantity having a direction as well as magnitude
Damilare
tell me about charging and discharging of capacitors
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
equivalent capacitance is 9nf nd pd across each capacitor is 500v
santanu
four effect of heat on substances
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
the study of heat an other form of energy.
John
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
ing
It is abranch of physical chemistry which deals with the interconversion of all form of energy
Vishal
what is colamb,s law.?
it is a low studied the force between 2 charges F=q.q`\r.r
Mostafa
what is the formula of del in cylindrical, polar media
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
what is flux
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine