4. Electric Current in Counductors by Sanjay Pandey

8/14/2019 4. Electric Current in Counductors by Sanjay Pandey

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SANJAY PANDEYMail add:[email protected]

Contact No.: 09415416895, 09721573337, 094537630584. Electric Current in Conductors

1. Electric current"Electric current is defined as the amount of electric charge passing through a cross section of a

conductor in unit time."

In other words

"The rate of flow of electric charge through a cross section of a conductor is called

Electric Current".

Mathematically

= /

= /Q is charge, t is time

= = lim 0( /) = /Electric current is a scalar quantity.

Unit: AMPERE.1 = 1 / 1

AmpereIn S.I system unit of elec tric current is am pere.

Ampere is defined as:

Current through a conductor will be 1 ampere if one coulomb of electric charge passes through

any cross section of conductor in 1 second.

1 = 1 / 1

Types of currentThere are two types of current.

1. Electronic CurrentElectronic cu rrent flows from negative to positive terminal.

2. Conventional CurrentDirection of conventional current is taken from higher potential to the lower potential.

2. Current densityAverage current density = /

is current and A is area of the conductorThe current density at a point P is
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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= lim0(/) = /If current i is uniformly distributed over an area S and is perpendicular to it, then

= /

For a finite area = .

Where

= density of current (vector)

= area (vector)3. Drift speed

A conductor contains free electrons moving randomly in a lattice of positive ions. Electrons

collide with positive ions and their direction changes randomly. In such a random movement, from any area

equal numbers of electrons go in opposite directions and due to that no net charge moves and there is no

current. But when there is an electric field inside the conductor a force acts on each electron in the direction

opposite to the field. The electrons get biased in their random motion in favor of the force. As a result

electrons drift slowly in the direction opposite to the field.

If be the average time between successive collisions, the distance drifted during this period is

=1

2()2 =

1

2(/)( )

The drift speed is

= =1

2(/) =

the average time between successive collisions, is constant for a given material at a giventemperature. It is called the Relaxation time

4. Relaxation TimeSuppose electron suffered its last collis ion time ago, then relaxation time


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=

5. Relation between current density and drift speed

= / = 6. Ohm's law

It states that the current density in a conductor is directly proportional to electric field across the conductor.

=

is field and is electrical conductivity of the material.

Proof:We know that

= / =

And =1

2(/)

= 12

(/) = 2

2

Or = (proved)The resistivity of the material is defined as

= 1

and =

=1

=1

.

=

or = (other form of Ohms law)


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where =

is called the resistance of the given conductor. The quantity 1/R is called the conductance.

Unit of resistivity is ohm-meter (or -m). The unit of conductivity is ( )1 written as / .

Resistivity of a material = 1/ Another form of Ohm's law

= = ( = length of the conductor) =

= =

1/ is called conductance7. Temperature dependence of resistivity

As temperature of a resistor increases its resistance increases. The relation can be expressed as

() = (0)[1 + ( 0 )] is called temperature coefficient of resistivity.

8. Thermistors: Measure small changes in temperatures9. Superconductors:For these materi als resistivity suddenly drops to zero below a certain temperature. For Mercury

it is 4.2 K. For the super conducting material if an emf is applied the current will exist for long periods of time

even for years without any further a pplication of emf.

Scientists have achieved superconductivity at 125 K so far.

10. Battery


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Battery is a device which maintains a potential difference between its two terminals A and B.

In the battery some internal mechanism exerts forces on the charges and drives the positive charges

of the battery towards one side (terminal A) and negative charges of towards another side (terminal B).

Let force on a positive charge is (a vector quantity). As positive charge accumulates on A and

negative charge on B, a potential difference develops and grows between A and B. An electric field isdeveloped in the battery material from A to B and exerts a force = on a charge . The direction of thisforce is opposite to . In steady state, the charge accumulation on A and B is such that = . No furtheraccumulation takes place.

If a charge is moved from one terminal (say B) to the other terminal say A, the work done by thebattery force is = where is distance between A and B.

The work done by the battery force per unit charge is

E= / = /

This Eis called the emf of the battery. Pleas e note that emf is not a force it is / .

If nothing is e xternally connected b etween A and B, then

= = Or = = (because = ) = potential difference between the terminalsAs = / = / = / =

Therefore = Thus, the emf of a battery equals the potential difference between its terminals when the terminals

are not connected externally.

11. Energy transfer in an electric circuitWhen an electric charge = goes through the circuit having resistance R the electric potential

energy decreases by

= = ()() = 2This loss in electric potential energy appears as increased thermal energy of the resistor. Thus a

current for a time through a resistance increases the thermal energy by 2 = = / = = 12. Effect of internal resistance of a batteryInternal resistance of a battery is due mainly to the resistance of the electrolyte between

electrodes. It is denoted by and for ideal battery = 0.


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Since the internal resistance of the battery is in series with the load the equivalent resistance of the

circuit is = + . The current is thus reduced owing to the internal resistance, = / ( + ), fromwhat it would be in i ts absence.

The potential difference across the load, equivalent to that across the battery, is less than the full emf of thebattery because of the voltage drop across t he internal resistance.

= + = =

=

( = internal resistance)

The so-called terminal voltage of a battery is lower than the emf when it is discharging because of the voltagedrop across the internal resistance. If, on the other hand, the battery is being charged by an external source

such as a recharger, the current will be forced through the battery in the opposite direction; the terminal

voltage will then be higher than the emf by the amount of the voltage difference across its internal resistance.


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Polarity of a ResistorAssign a positive (+) sign for the voltage to the terminal of the element where the current enters and

negative () sign to the terminal of the element where the current leaves it.13. Kirchhoff's laws:

Kirchhoffs current law (KCL) or the junction lawThe sum of all currents directed towards a point in a circuit is equal to the sum of all the

currents directed away from the point.

Kirchhoffs voltage law (KVL) or the loop lawThe algebraic sum of all the potential differences along a closed loop in a circuit is zero.

Strategy for multi-loop circuits:1. Sketch circuit.2. Replace r esistor combinations with their equivalents.3. Label the positive direction of current in each branch of the circuit.4. Apply the junction rule.5. Apply the loop rule.


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Writing The KVL Equations(a) Pick a starting point on the loop you want to write KVL for.(b) Imagine walking around the loop - clockwise o r counterclockwise.(c) When you enter an element there will be a voltage defined across that element. One end will be

positive and the o ther negative.

(d)

Pick the sign of the voltage definition on the end of the element that you enter. Conversely, you couldchoose the sign of the end you leave, except that you have to be consistent all the way around theloop.

(e) Write down the voltage across the element using the sign you got in the previous step.(f) Keep doing that u ntil you have gone completely around the loop returning to your starting point.(g) Set your result equal to zero.

Now, let's write KVL for each of the three loops in adjoining Fig.

For the firs t loop (Battery, Element 1, Element 2) + 1 + 2 = 0

For the second loop (Element 2, Element 3, Element 4). Note, you have to be careful with this onebecause you might not expect the voltage across Element 3 to be defined the way it is.

2 3 + 4 = 0

For the third loop (Battery, Element 1, Element 3, Element 4) + 1 3 + 4 = 0

So, we get three equations - right?

Actually, that's not right, because we do not get three independent equations. There are only two

independent equations we can write.

That's not immediately obvious, so write the three equations as shown below. We'll put a horizontal

line between the first two and the third equation.

+ 1 + 2 = 0

2 3 + 4 = 0...

+ 1 3 + 4 = 0Can you see that you can add the first two equations to get the third? (Actually , there is a 2 and a

+2 , and those are the only things that cancel out when you add.) The third equation can be obtained fromthe first two equations, so it is not an independent equation. When you have the first two equations you can

get the third from them!


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What this means is that you have to be careful when you write KVL. You can write too many

equations, and in being careful you might not write enough. Fortunately, if you look at a circuit you can

almost always see how many independent loops there are by inspection.

14. Combination of resistors in series

For resistors in series, the current through each resistor is identical.

= + + +...15. Combination of resistors in parallel

For resistors in parallel, the voltage drop across each resistor is identical

= 1/ + 1/ + 1/+...16. Division of current in resistors joined in parallel

/ = / = /( + )

17. Batteries connected in series = ( + )/( + )

Where = external resistance = +

, are internal resistances of two batteries18. Batteries connected in parallel

= = +21( + )

where , are emfs of of b atteries , and , are internal resistances. = = /( + )

So = /( + )


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19. Wheatstone bridgeIt is an arrangement of four resistances, and one of them can be measured if the other are

known resistances.

Relation Among Resistances In Balanced ConditionR/R = R/R

From figure:

R & R are connected in series. Reason: (only one path for the flow of current)R & R are connected in series.R & R are connected in parallel. Reason: (two paths for the flow of current)R & R are connected in parallel.

If the there is no deflection in the galvanometer, then

/ = /4 4 = /

20. AmmeterUsed to measure current in a circuit. A small resistance is connected in parallel to the coil

measuring current in an ammeter to reduce the overall resistance of ammeter.

21. VoltmeterA resistor with a large resistance is connected in series with the coil.

When a volt meter is connected in parallel to the point between which the potential is to be

measured, i f a large resistance is connected, the equivalent resistance is less than the sm all resistance.22. Charging of the capacitorIt takes time to charge a capacitor and it takes time to discharge one.

This time is dependent on the sizes of the capacitor and the resistor in the circuit.

The case for charging a capacitor is described first, then discharging a capacitor.


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Figure shows charging of a capaci tor. Just before the switch is closed the charge on the capacitor is zero.

When the switch is closed (at time t= 0), the charging starts.

By KVL

= 0

= 0

=

=

=

=

0

0

=

1 =

= (1 / ) is charge on the capacitor, is time, = emf of the battery, = capacitance, is resistance of

battery and connecting wires,

has units of time and is termed time constant.In one tim e constant (= )

= 1 1 = 0.63


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Thus, 63% of the maximum charge is deposited in one time constant.


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23. Discharging

Just before the switch is closed (at t= 0) the charge on the capacitor is 0 , and the current is zero. At the tim e

t after the switch is closed, the charge on the capacitor is q, the current is i.


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By KVL,

= 0

Here =

=

= 1

dqqqQ 0 = 1 0

ln 0 =

= 0/ Where, is charge remaining on the capacitor and 0 is the initial charge. The constant is the timeconstant. A t = , the remaining charge is = 0

1

= 0.370 . Thus in one time constant 0.63% dischargingis complete.


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24. Atmospheric electricity

At about 50 km above the earths surface, the air becomes highly conducting and thus there is

perfectly conducting surface having potential of 400 kV with respect to earth and current (positive charge)

comes down from this surface to earth.

25. Thunderstorms and lightning bring negative charge to earth .Water vapour condenses to form small water droplets and tiny ice particles. A parcel of air

(cloud) with these droplets and ice particles forms a thunderstorm. A matured thunderstorm is formed with

its lower end at a height of 3-4 km above the earths surface and the upper end at about 6-7 km above the

earths surface. Negative charge is at the lower end and positive charge is at the upper end of this

thunderstorm. This negative charge creates a potential difference of 20 to 100 MV between these clouds and

the earth. This cause dielectric breakdown of air and air becomes conducting.

There are number of thunderstorms every day throughout the earth. They charge the

atmospheric battery by supplying negative charge to the earth and positive charge to the upper atmosphere.

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4. Electric Current in Counductors by Sanjay Pandey