Monday, July 31, 2017

Current Electricity2

Rheostat as a potential divider

A rheostat is a resistance element with considerable length. It is usually metallic in nature. The defining equation is R=ρlA

Where;
ρ= resistivity of the element.
l= length of the wire.
A= area of cross section of the wire.
The electrical component rheostat, have 3 terminals,two terminals are the begining and end of the wire (T1 and T2), while the 3rd terminal is the movable contact which can be moved along the length of the wire(T3).  ρ and A remains the same throughout; thus the resistance between terminals is decided by the length of wire between them.
The length T1 to T3 is l1 while T3 to T2 is l2. This results in two resistances R1 and R2; causing an equivalent circuit.By adding a voltage source across T1 and T2 ; desired voltage can be obtained across T3
and ground by tuning the point of contact C on the rheostat.
52
example

Circuits involving electrical instruments

To convert a galvanometer into an ammeter a small resistance S
(called the shunt) is connected in parallel with a galvanometer. A galvanometer has resistance G and full scale deflection current i. To convert this galvanometer into an ammeter of range 10i, a shunt S=G9 is connected in parallel with GNow we want to measure the potential difference with the help of this ammeter. The maximum value of potential difference which can be measured with the help of this is:The voltage across galvanometer is:
Vg=iG
When the galvanometer is converted into an ammeter of range 10i by applying a shunt resistance S=G9, the current is divided in inverse ratio of their resistance 19:1  between shunt and ammeter .The current will be in ratio 9:1.Current in shunt is i=910×10i=9i
The voltage across shunt resistance is:
Vs=9i(G9)


Since the galvanometer(ammeter) and shunt are in parallel voltage across them is same, hence the maximum value of potential difference which can be measured with the help of this ammeter is iG
.
53
example

Solving circuit of Wheatstone bridge

Example:
In a Wheatstone's bridge, three resistances P,Q and R are connected in three arms and the fourth arm is formed by two resistances S1
and S2 connected in parallel. Find the condition for the bridge to be balanced.
Solution:
The condition for Wheatstone bridge is:
PQ=RS

S=S1S2S1+S2[S1,S2 in parallel]

PQ=R(S1+S2)S1S2
 
54
definition

Meter bridge

A meter bridge consists of a wire of length 1 m
and of uniform cross-sectional area stretched taut and clamped between two thick metallic strips bent at right angles with two gaps across which resistors are to be connected. The end points where the wire is clamped are connected to a cell through a key. One end of a galvanometer is connected to the metallic strip midway between the two gaps. The other end of the galvanometer is connected to a jockey which moves along the wire to make electrical connection. R is an unknown resistance connected across one of the gaps. Across the other gap, we connect a standard known resistance S. The jockey is connected to some point D on the wire, a distance l1 cm from the end A.The portion AD of the wire has a resistance Rcml1, where Rcm is the resistance of the wire per unit centimeter. The portion DC of the wire similarly has a resistance Rcm(100l1).
The meter bridge works on the principle of Wheatstone bridge. At balance condition:
RS=Rcml1Rcm(100l1)=l1100l1
55
definition

Null point

The null point or balanced point is an arrangement of resistors across the arms of a Wheatstone bridge or meter bridge such that the deflection in the galvanometer is zero. A,B,C and D are four coils of wires of 2,2,2
and 3 ohm resistances respectively and are arranged to form a Wheatstone bridge.The resistance which the coil 'D' must be shunted in order that the bridge may be balanced is given by:Wheatstone condition:

RARC=RBRD
RB=RD
RB=3×RS3+RS
6+2RS=3RS
RS=6Ω

56
example

Problems using meter bridge

When an unknown resistance and a resistance of 4Ω
are connected in the left and right gaps of a meter bridge, the balance point is obtained at 50cm. The shift in the balance point, if a 4Ω resistance is now connected in parallel to the resistance in the right gap, isCase 1

R4=11
R=4
42=x100x
200=3x
x=2003
Shift =200350=503=16.7
 cm
57
definition

Comparision of emfs using a potentiometer

Comparison of Emfs:
The driving circuit of a potentiometer is set up with a strong battery so that the potential difference V0
across AB is larger than the emf of either battery. One of the batteries is connected between the positive end A and the galvanometer. The jockey is adjusted to touch the wire at a point C1 so that there is no deflection in the galvanometer. The length AC1=l1 is noted. Now, the first battery is replaced by the second and the length AC2=l2 for the balance is noted. If L is the length AB=L, the emf of the first battery is,
E1=l1LV0
and that of the second battery is,
E2=l2LV0

Thus E1E2=l1l2
58
definition

Internal resistance using potentiometer

We can also use a potentiometer to measure internal resistance of a cell. For this the cell (emf E
) whose internal resistance (r) is to be determined is connected across a resistance box through a key K2. With key K2 open, balance is obtained at length l1 (AN1). Then,
E=ϕl1

When key K2 is closed, the cell sends a current (I) through the resistance box (R). If V is the terminal potential difference of the cell and balance is obtained at length l2(AN2),
V=ϕl2

so,   EV=l1l2=I(r+R)IR
r=R(l1l21)
which is the internal resistance of the given cell.
59
law

Joule's Law of heating

According to Joule's Law of heating, heat generated in a conductor is given by:
H=VIt=V2Rt=I2Rt
60
law

Joule's Law of heating

According to Joule's Law of heating, heat generated in a conductor is given by:
H=VIt=V2Rt=I2Rt
61
definition

Electrical power

Electrical power is given by: P=H/t=VI=I2R

Its unit is W
(Watt).
62
example

Example of a problem on Joule's Law

Example:
The internal resistance of a primary cell is 4
ohm. It generates a current of 0.2 ampere in an external resistance of 2 ohm. Find the rate at which  chemical energy is consumed in providing the current.Solution:
The rate of consumption of chemical energy = Rate of generation of electrical energy.
The energy generated is calculated from the formula P=VI. Substituting V=IR (Ohm's law) in the above equation we get,

Power=VI=I2R.

In this case, there are external and internal resistances. So, the equation is rewritten as I2(R+r).
Therefore, Chemical energy = 0.22(4+2)=0.24J/s.
Hence, t
he rate at which  chemical energy is consumed in providing the current is 0.24 J/s
.
63
example

Power dissipated in resistors

Three equal resistors connected in series across a source of emf together dissipate 10 W
power. The power dissipated if the same resistors are connected in parallel is:Case 1
P=V23R=10 Watt
Case 2
P=3V2R
=9(V23R)
=90 W

64
definition

Maximum power transfer theorem

The maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.
(PR)max=ϵ24r

when R=r where ϵ is the emf of the cell and r is the internal resistance.
Example : Two identical batteries each of e.m.f. 2 V and internal resistance 1 Ω are available to produce heat in an external resistance by passing a current through it. The maximum power that can be developed across R using these batteries isFor maximum current, the two batteries should be connected in series. The current will be maximum when external resistance is equal to the total internal resistance of cells i.e R=r+r=1+1=2Ω
Thus, E+E=I(R+r+r)I=2ER+2r=2×22+2=1A
Maximum power, P=I2R=12×2=2W

65
definition

Power supplied by a cell

Power supplied by a cell in a circuit is given as
P=VI=V2r+R

where V is the emf of the cell, r is the internal resistance and R
is the load of the circuit.
66
definition

Energy conservation in a circuit

In an electrical circuit, total power supplied equals the total power delivered at all instants.
Consider a circuit consisting of an ideal voltage source with emf ε
and a resistance R.
Power in the emf source is Ps=VI
Power in the resistance is Pr
By ohm's law, V=IR
Pr=VI
Ps+Pr=0
67
definition

Seebeck effect

Two metallic strips made of different metals are joined at the ends to form a loop. If the junctions are kept at different temperatures, there is an electric current in the loop. This effect is called the Seebeck effect and the emf developed is called the Seebeck emf or thermo-emf.
68
definition

Neutral and inversion temperature

The temperature of the hot junction at which the thermo-emf is maximum is called the neutral temperature and the temperature at which the emf changes its sign (current reverses) is called the inversion temperature.
69
definition

Working of a thermocouple

The working principle of thermocouple is based on three effects - Seebeck Effect, Peltier Effect and Thomson Effect.It comprises of two dissimilar metals, A and B, joined together to form two junctions, p and q, which are maintained at the temperatures T1 and T2 respectively which generates the Peltier emf within the circuit and it is the function of the temperatures of two junctions.The total emf or the current flowing through the circuit can be measured easily by the suitable device.Now, the temperature of the reference junctions is already known, while the temperature of measuring junction is unknown. The output obtained from the thermocouple circuit is calibrated directly against the unknown temperature. Thus the voltage or current output obtained from thermocouple circuit gives the value of unknown temperature directly.
70
law

Law of intermediate temperatures

Let ϵθ1,θ2
denote the thermo-emf of a given thermocouple when the temperatures of the junctions are maintained at θ1 and θ2. Then,
ϵθ1,θ2=ϵθ1,θ3+ϵθ3,θ2
.
This is known as the law of intermediate temperatures
71
law

Law of intermediate metal

Suppose ϵAB
=thermo-emf between metals A and B
ϵBC=thermo-emf between metals B and C
ϵAC=thermo-emf between metals A and C.
Also, suppose the temperatures of the cold junctions are the same in the three cases and the temperatures of the hot junctions are also the same in the three cases. Then,
ϵAB=ϵACϵBC
.
This law is known as the law of intermediate metal
72
definition

Peltier effect

The Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors. When a current is made to flow through a junction between two conductors, A and B, heat may be generated or removed at the junction. The Peltier heat generated at the junction per unit time,\dot { Q }  , is equal to
Q˙=(ΠAΠB)I

73
definition

Thomson effect

If a metallic wire has a nonuniform temperature and a current is passed through it, heat may be absorbed or produced in different sections of the wire. This heat is over and above the Joule heat i2Rt
and is called Thomson heat. The effect itself is called the Thomson effect.
74
definition

Voltage rating of a bulb

The voltage rating of an incandescent bulb is the voltage that bulb is designed to operate in the circuit measured at the base of the bulb and in which published data watts, amps, lumens, colour temperature and life hours are measured.
75
definition

Power rating of a bulb

The power rating on a light bulb indicates how much power it would dissipate when it is hooked up to the standard household voltage of 120V.
76
definition

Difference between breaking voltage and voltage rating of a bulb

Voltage rating of a bulb is the voltage at which the bulb operates.

If an excess voltage is applied to the bulb, a dangerously high electric field is induced which may damage the bulb. The minimum value of voltage at which breakdown occurs is called the breakdown voltage of the bulb. 
77
example

Example of power rating of a combination of bulbs

Example:
Two bulbs one of 200
volts, 60 watts & the other of 200 volts, 100 watts are connected in series to a 200 volt supply. Find the power consumed.Solution:
Here, two bulbs one of 200 volts, 60 watts & the other of 200 volts, 100 watts are connected in series to a 200 volt supply.
For bulb 1,
I=PV=60200=620A

Hence,
R=VI=200×206=40006Ω

Similarly, for bulb 2,
I=PV=100200=12A

Hence,
R=VI=200×21=400Ω
When bulbs are connected in series their resistances will added
 up
R=(40006)+400

R=64006

Hence, current  is
I=VR=200×66400=1264A

Hence, P=VI=200×1264

P=37.5W

78
definition

Bulb brightness and power dissipated

An unchanged capacitor is connected in circuit as shown in figure Power ratings of bulbs are given in diagram At t=0
switch is closed :
We know that Po=V2oR.
Now, at t=0, since voltage across C=0, power consumed by B1 will be Po.
Power consumed by B2 and B3 will be Po4 each.
 Hence, total power consumed at t=0 will be 2(Po4)+Po=3Po2.
 After long time, power through B1 will be 0.
Hence total power will be P02.
Brightness of B2 will be constant but that of B1
will decrease with time.
79
definition

Relationship between current and voltage of a capacitor

The relationship between a capacitors voltage and current define its capacitance and its power. We have Q=CV.
Then $$\dfrac{dQ}{dt}=C \dfrac{dV}{dt} \Rightarrow i_c=
C \dfrac{dV_c}{dt}$$
80
definition

Voltage across a capacitor doesn't change suddenly

i=CdVdt

An instantaneous change in the voltage across a capacitor would require taking limits.
limt0i=Climt0dVdt

i.e. the rate of change of the voltage (dv/dt) be infinite, and hence the current would have to be infinite which is not possible.
81
definition

Behaviour of capacitor on closing the switch

Voltage across a capacitor does not change suddenly. Hence, for an uncharged capacitor, just after the switch is closed voltage remains zero. This is analogous to a short circuit. Hence, an initially uncharged capacitor behaves as a short circuit just after closing the switch.
Example:
Find the current in the circuit shown in the attached figure just after the closing of the switch. The capacitor is initially uncharged.
Solution:
Just after the closing of the switch, capacitor behaves as a short circuit. Hence, the current is given by:
I=VsR
82
example

Capacitor as an open circuit

In the steady state, the energy stored in the capacitor is :In steady state current flow in capacitor branch is zero.

I=E1R1+R2+r1

VCE2=IR1

VC=E2+R1E1R1+R2+r1
Energy stored =12cv22
                         =12c[E2+R1E1R1+R2+r1]2

83
example

RC circuits at the time of closing of switch

Only switch S1
closed in Figure. What is the charge on the 3μF capacitor in μC?Without power supply V=12 we can not solve.
When S1 is closed only , the capacitance 3μF is charged to voltage V=12.
Thus, the charge on 3μF is Q=3V=3×12=36μC

84
example

RC circuits at steady state

In the given circuit, the steady state voltage drop across the capacitor C isIn steady state C=0

V=I[r1+r2]
VC=Ir1
         =Vr1r1+r2

85
formula

Discharging RC circuit

The differential equation of RC circuit is
dqq=1CRdt
86
formula

Charging of an RC circuit

Rdqdt=ϵCqC
where ϵ
is the emf of the cell
87
definition

Time constant in an RC circuit

The time constant of an RC circuit is the time required to charge the capacitor, through the resistor, by 63.2 percent of the difference between the initial value and final value or discharge the capacitor to 36.8 percent.
τ=RC
where τ
is the time constant and R and C are the values of resistance and capacitance respectively 
88
formula

General solution of RC circuit for a capacitor

General solution of RC circuit for a capacitor after time t is given by :
q(t)=q()(q()q(0))etRC

where:
q(t): Charge on the capacitor after time t
q(0): Initial charge on the capacitor
q(): Charge on the capacitor at steady state
RC:
Time constant
Note:
The solution remains valid for other parameters of capacitors like voltage, current, etc.
89
definition

General solution of RC circuit for a resistor

General solution of RC circuit for a resistor after time t is given by :
i(t)=i()(i()i(0))etRC

where:
i(t): Current in the resistor after time t
i(0): Initial current in the resistor
i(): Current in the resistor at steady state
RC:
Time constant
Note:
The solution remains valid for other parameters of resistors like voltage.
90
formula

Charge on a capacitor in a charging RC circuit

Charge on a capacitor in a charging circuit is given by the following equation.
q=q0(1etτ)

where q0 is the initial charge of the capacitor and q is the charge at a time t
.
91
formula

Voltage in a charging/discharging RC circuit

Charging  V(t)=V0(1et/τ)
             Vr(t)=V0(et/τ)
 Discharging  Vc(t)=V0(et/τ)           Vr(t)=V0(et/τ)
where τ=RC
is the time constant.
92
formula

Current in a charging/discharging RC circuit

Charging: i=VRet/τ

Discharging i=V0Ret/τ
where V0
is the initial voltage.
93
definition

Non-ideal capacitor

If the dielectric material between the plates of a capacitor has finite resistivity as opposed to infinite resistivity in case of an ideal capacitor then there will be a small amount of current flowing between the plates of the capacitor.Lead resistance and plate effects also exist in non-ideal capacitor.
94
example

Energy stored in a capacitor in RC circuit

Example:
Consider the circuit shown in the attached figure. Find the energy stored in the capacitor after time t. The initial charge on the capacitor is 0.
Solution:
Given, q(0)=0

At steady state, capacitor will be fully charged and hence will have a potential of Vs across it.
Hence, charge stored in C in steady state is q()=CVs
Charge stored in the capacitor after time t is given by:
q(t)=q()[q()q(0)]etRC
q(t)=CVs(1etRC)
Energy stored in capacitor after time t is given by:
E(t)=q2(t)2C
E(t)=CVs(1etRC)2C

95
formula

Heat generated in resistor in RC circuit

A capacitor C
is charged to V0 volts, a resistor R is connected across it. The heat produced in the resistor is CV20/2
96
formula

Time to reach specific percentage of charge in a RC circuit

qq0=1etτ

ln(1qq0)=tτ
t=τln(11qq0)
97
definition

Sudden change in voltage s forced across a capacitor

If a source of voltage is suddenly applied to an uncharged capacitor, the capacitor will draw current from that source, until the capacitors voltage equals that of the source. Once the capacitor voltage reached this charged state, its current decays to zero. 
98
example

Solving complicated RC circuit excited by DC

Example:
Consider the circuit shown in the figure. The capacitor is initially uncharged. Find the current across R2
after time t.
Solution:
To find the time-constant, the emf source is short-circuited. The equivalent resistance is then given by:
Req=(R1+R3)||R2
Req=500011Ω
Time-constant is then given by: τ=ReqC=122s
Now, voltage across the capacitor initially is V(0)=0 and remains the same at the time of switching.
This is the same as the voltage across R2.
By ohm's law, i2(0)=0
In steady state, the capacitor is fully charged and acts as open circuit. Then current across R2 can be found using KCL.
i2()=20R1+R2+R3=1275A  
Using general solution,
i2(t)=i2()[i2()i2(0)]etτ
i2(t)=1275(1e22t)A

Note:
This approach is useful when the resistors and capacitors network can be separated into two groups.
99
example

Charge on a capacitor in a charging/discharging circuit

Initially capacitor was uncharged and switch was open. Switch is closed at t=0. Ammeter and voltmeter are ideal. [All units are in SI]After long time, current through the circuit will be zero since capacitor will behave as open circuit. Therefore voltmeter will read 10V.
Just after closing the switch, voltage across capacitor will be 0V. Hence current will be i=20+104+2=5A
.
Reading of voltmeter just after closing the switch will be 102i=0V.
After long time, entire voltage will appear across the capacitor, hence, Q=CV=2X30=60C

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