cahier physique - physics workbook

good link for details about diff equ. + exercices Also see my page about solving diff. equ.
in construction, please be patient.

PART I: circuit (R, C)

A capacitor is made of 2 conductive plates that can store charges. No charge can
flow between the plate. Air can be used as an insulator between the plate.

source: britannica

A resistor decays electric currents through heat loss.

In electronics, another insulating material is used between the plates so the capacitor can
be rolled down and takes less space.

Consider a capacitor in series with a resistor and a generator of constant tension U = Uo.

The current produced by the batteries will charge the capacitor. The charges q(t) will increase at both ends of the capacitor.
(+Q(t) and - Q(t) ). As the tension of the capacitor increases toward Uo, the current through the resistor decreases.
(less potential difference between the capacitor and the battery terminal). When the capacitor is charged, its tension is equal to
Uo and there is no current flowing in the circtuit. If we remove the generator, the capacitor will be discharged.
The time it takes for the capacitor to be charged depends on the resistance R (greater R, greater is the charge time
because the current encounters more resistance to its flow)
and also on C (greater is C, more charges it can absorbed, more time it takes).

Let's write the differential equation that describe the charge.
Uc = q(t) / C (the relation between q(t) and Uc is linear, the slope is C )
Q = C Uc
C depends on the capacitor. unit is Farads (F)

Ur = R i(t) and i(t) = dq(t) / dt
So Uo = R dq/dt + q / c or q' + 1/RC q = Uo/R

This is a 1st order, linear differential equation. The integrating factor is e ^t/RC
we get: (q e ^t/RC ) ' = Uo/R e ^t/RC integrate : q e ^t/RC = Uo/R (RC) e ^t/RC + A
q(t) = UoC + A e ^{- t/RC}
A is a constant and depends on the initial conditions. If t=0, q(t) = 0 then A = - UoC
so q(t) = UoC - UoC e ^{- t/RC}= UoC ( 1 - e ^t/RC)
At the beginning of the charge, - UoC e ^{- t/RC} is the dominant part. (exponential growth), then as time increases this term
becomes very small and the solution tends to UoC = Qo = Max charge for the capacitor.
q(t) is the response to the input Uo.

RC has the dimension of the time. It is the intersection between the slope of q(t) at t=0 (q = Uo/C (1/RC) t and q=Qo )
To find the equation of the tangent, derive with respect of the time q(t) = Uo/C ( 1- e ^{- t/RC} ) and find the value at t = 0.

RC is called the time constant and is about the time it takes for the capacitor to be 63% charged.
Indeed when t = RC, q = UoC ( 1 - e^-1 ) = UoC 0.63 so q(t) / qmax = 0.63 with qmax = Uo/C
When t = 5 RC, the transcient phase is over.

finding Uc(t) is easy. (tension of the capacitor ). Uc = q(t) /C = Uo ( 1 - e ^-t/RC)
The tension across the capacitor tends to Uo once the tanscient phase is over.

The tension across the resistance decreases with time because the current decreases. The tension across the resistance
is proportional to the current.
UR = R i(t) = R dq/dt = RUoC d/dt (1 - e ^-t/RC) = Uo e ^-t/RC
The tension across the resistance decreases with time.

source of diagram.
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Consider now a capacitor connected, in parallel to a low frequency generator
and to a resistance.

U = Uo sin(wt). w is called the angular frequency. It gives the number of oscillations
in one 2pi time.

The capacitor will be charged (current in circuit decreases as charges accumulate on plates)
when the tension of the generator increases (positive tension) or the capacitor will be discharged when the tension decreases.

i1(t) is the current going through the resistor and i2(t) the current going through the capacitor.
i1(t) = Uosin(wt) / R .

But i2(t) = dq / dt = C dUc/dt with Uc = Uosinwt so i2(t) = C Uo w cos (wt)
i2(t) = C Uo w sin (wt + 90 degrees )

There is a time lag between i1 and i2 of 90 degrees)

The curves indicate that, for a capacitor, the voltage reaches its maximum value one quarter
of a cycle after the current reaches its maximum value. The voltage lags the current through the
capacitor by 90 degrees.

To find the current i(t) produced by the generator we can write i(t) = i1(t) + i2(t)
i(t) = Uo/R sin(wt ) + C Uo w sin (wt+90)

A Fresnel digram can be used to add i1 and i2.

i1 + i2 = A sin(wt + P ) with A = square root ( i1² + i2² ) tan(P) = CUow / Uo/R = CwR

Here some algebraic relations:
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ENERGY IN A CAPACITOR

A capacitor can stock energy since it can store charges. During the discharge, these charge can produce work.
By definition P = U . i(t) and P = dE / dt
so dE/dt = U i and Uc = q / C so dU/dt = i / C or ic = C dU/dt (with i =dq/dt)
dE/dt = C U dU/dt
During charge E increases when U tends to Uc if we integrate: EC= 0.5 C U²

more
even more

PART II: SELF-INDUCTANCE

Imagine a simple circuit of a switch, a coil, and a battery.

When The switch is closed, the current through the coil sets up a magnetic field.
As the current is increasing, the magnetic flux through the coil is also changing. This changing magnetic field flux generates
an emf e (voltage across the coil) opposing that of the battery. THat means there is an induced current in the coil that opposes the increasing current
that produced the changing magnetic field. This occurs only while the current is either increasing to its steady state value immediately
after the switch is closed or decreasing to zero when the switch is opened. This effect is called self-inductance.
The proportional constant between the self-induced emf and the time rate of charge of the current is called the inductance (L)
and is given by : e = - L di/dt
The units for L is henry (H)
The voltage across the coil is RL - e where Rl is the resistance of the coil.

The induced emf e always opposes the effect (like increasing or decreasing current) that gave rise to it.
So the induced current has a direction opposite to the current in the circuit. This is called Lenz law.
Otherwise, you would get energy for nothing. This would violate the conservation of energy.

induced emf and induced current phenomenon is used in transformers. The changing current I1 inside the coil 1 (connected to a source of energy) , induces a current I2 in the coil2 (not connected to a source of energy). A core of soft iron is used to guide the magnetic field. By changing the number of loops N2, you can vary the amplitude of I2. Because energy is conserved if I2 increases, the voltage across coil2 decreases.	magnetic braking system also relies on this effect. If a wheel, made of a conductive material, moves inside a magnetic field, current will be induced in the wheel that will opposes the motion and stops the vehicle. This is nicely shown with a pendulum. The mass swings through a domain where a magnetic field can be switched on or off. Switching the magnetic field on, will stop the pendulum. SEE THE DEMONSTRATION in this lecture move to 38:27 minutes of the video. watch to the end
	Induction heating also uses this effect. The changing magnetic field produced by the coil will induce currents in the bottom of metal pans. The glass between the circuit and the pan is not hot and can be touched. see this website

With a battery of tension V , the differential equations is : V = UR + UL = Ri - e = Ri + L di/dt
or V = Ri + L di/dt

i' + R/L i = V / L integrating factor is e ^{R/L t}
It's a first order, linear, differential equation. i' + R/L i = V/L
i = V/R + C e ^-R/Lt C depends on the initial conditions. if t= 0 i = 0
then i = V/R ( 1 - e ^{-R/L t} )

The solution has 2 part. one transcient solution - V/R e ^{R/L t} and the steady state V/R .

L/R is called the time constant. When t = L/R i(t) = V/R ( 1 - e^-1 ) = V/R 0.63
i(t) / imax = i(t) / V/R = 0.63, the current in the circuit reachess 63% ot its max value.
Also L/R is the intersection between the tangent at t=0 of i(t) (that is i = V/Lt and i = V/R = imax , the lines intersects at t = L/R)

Larger L is and larger is the time constant. When the time is 5 times the time constant, the transcient phase is considered as over.

It is easy to find UL(t) tension across the inductance. UL = L di/dt = V e^{-R/L t} The tension of the inductance tends to 0 when time is large.

The tension across the resistance increases with time and tends to Uo: (Ur=Ri)

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If the inductance is connected to a low frequency generator : U = Uo sin wt
UL = L di/dt

L di/dt = Uo sin wt
di dt = Uo/L sin wt integrate
i(t) = - Uo/Lw cos wt = Uo/Lw sin (wt - 90)
The current is behind the tension , the time lag is 90 degree (1/4 of an oscillation)

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The energy stocked in the coil is EL = 0.5 L i²

The ratio L/R is the time constant. (R is the total resistance of the circuit).
It gives an idea of the time it takes for the current to reach its maximum.

PART 3: CURCUIT (R, L, C)

1)

Suppose first the circuit includes only an inductor and a capacitor. Suppose the capacitor is charged when we close the switch.
If there is no loss of energy, the charges oscillate between the 2 dipoles. The circuit behaves like a linear oscillator.
The graph below shows the current (orange) in the circuit and the tension U (blue). The tension increases
and reaches its maximum when the current reaches zero. The 2 quantities are out of phase and their variations
are periodic. (sine curves)

The energy is conserved (we neglect the resistance and the loss of energy) and it switches
between the electric energy in the condensator and the magnetic energy in the inductor.
The natural period To of the circuit depends on C and L such as To = 2 pi sqrt ( LC)

This is an ideal situation as the oscillations are always damped.
If the damping is not too strong, there is a pseudo-period close to the natural period.

2 )You could inject energy to compensate the loss by introducing an operational amplificator.
The period is the natural one.

This circuit allow to compensate the loss of energy.
Ro = R for the oscillations to take place.
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3) circuit (R,L,C) driven oscillations

Driven oscillations, For the right frequency, resonnance is reached but stays under controlled because of the resistance. An AC circuit can be tuned in resonnance with the frequency of a broadcast. In that case the current is large.

The differential equation that describes the oscillations is the same for a mass on a spring. The inertia m opposes the change in velocity and the spring constant opposes the change in displacement, the motion.

The capacitor opposes the motion of charges and stop it. Like a spring opposes the motion of the mass.

The inductor opposes the change in current. Slow down the discharge and
the charge of the capacitor.
Like wise a mass (inertia) opposes the change in the velocity of the oscillations.

The frequency is now the one imposed by the generator. It is not anymore the natural frequency.
The current reaches a maximum when the frequency of the generator = natural frequency = resonant frequency
fo=fr = 1 / (2 pi sqrt (LC) )
The maximum current is given by Io = U / R U is the max value for the voltage of the generator.
The impedance of the circuit is minimum R = U / Io
The tension and the current are in phase.

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to be continued