current electricity.docx

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INTRODUCTIONElectric current is a means by which electrical energy is transferred from one place to another for utilisation. Charges in motion constitute an electric current. However, to maintain a steady current, a source of emf is needed.

The Flow of Electric CurrentA current is any motion of charge from one region to another. The illustration below shows two bodies at different potentials. When these are connected with a wire, free electrons flow from B to A until both bodies attain the same potential, after which the current ceases to flow. Current flows if a potential difference exists throughout a conductor. This branch of physics dealing with charges in motion is called current electricity.

The amount of charge flowing through a given cross-section of a conductor per unit time constitutes electric current.

When a battery or a cell is connected across the ends of a conductor, the thus set up exerts a force on the free

electron causing them to move as shown in the figure. The arrows give the direction of the conventional current.

Current is a scalar quantity. The direction of conventional current in any circuit is the direction in which the positive charges flow. Mathematically, for a steady current across any area,

where I is the current, and q is the charge that flows across that area in time t.

The S.I. unit of current is Ampere (A).

One ampere of current is said to flow through a wire if at any cross section, one coulomb of charge flows in one second.

In solid conductors, the current carriers are the free electrons; in electrolytes, the anions and cations; and in gases, the electrons and anions.

Electromotive Force (EMF) and VoltageNo current flows in a copper wire by itself, just as water in a horizontal tube does not flow. If one end of the tube is connected to a tank with water such that there is a pressure difference between the two ends of the horizontal tube, water flows out of the other end at a steady rate. The rate at which water flows out depends on the pressure difference, for a given tube. If the flow

I I T - J E E | M E D I C A L | F O U N D A T I O N S

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rate (current) is to be kept constant, the water flowing out for instance has to be put back into the jar to maintain the pressure head. This requires work to be done by an external agency. The above analogy brings out several features of electrical current flow. An electric current flows across a conductor only if there is an electric potential difference between its two ends. To maintain a steady current flow, one needs an agency, which does work on the charges. This agency is called the electromotive force or emf.

In the case of water flow, the agency is the pump 'P' which does work at a steady rate in putting the water back into the tank. Just as for a given tube, the current of fluid flowing out depends on its viscosity, the electrical current flowing for a given potential difference depends on the electrical resistance of the conductor.

The above circuit diagram shows the flow of charges. A steady electric current 'I' flows through the resistance R, from A to B. That is, positive charges flow from higher potential (A) to lower potential (B).

The potential drop from A and B is V. The source P of emf does work on these charges, as they come through at B because it has to take the positive charges from lower to higher potential. The charge is transferred from one end of the source of emf to the other and 'qV' work is done on the charge. The source of emf by doing work on the electric charges, maintains a potential difference V between its terminals.

Resistance and ResistivityIt is found experimentally that the current I flowing through a conductor is directly proportional to the potential difference V across its ends, provided the physical conditions (temperature, mechanical strain, etc.,) remain constant. That is

or V = IR

where R is the resistance of the conductor.

This equation, called Ohm's law, is named after G.S. Ohm who first discovered it in 1828. The unit of resistance is Ohm (1W = 1VA-1). Ohm's law is only empirical. These conductors (e.g.; metals) which obey Ohm's law are called Ohmic conductors.

If the material is not Ohmic, the relation between V and I is not linear, so that R is not constant. Still, R=V/I serves as the definition of resistance.

ResistivityThe resistance of a resistor depends on the nature of the material, its geometrical features (length and cross-sectional area) and on the temperature and pressure. It is useful to separate out the dependence on the geometrical factor.

Consider a rectangular slab of some conducting material, of length l and cross-sectional area A. If the length is doubled (fig. (a)) for a fixed current I, the potential drop across the slab also doubles. (This is because the potential difference is the electric field times the distance, and for a fixed current, the electric field that drives it, stays the same).

Therefore, the resistance doubles on doubling the length. That is


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Now, suppose that the slab is cut lengthwise into two parallel slabs (fig.(b)) each of area A/2. If the current flowing through the full slab is I for a given voltage V, then the current flowing through each half -slab is I/2. Therefore, the resistance of each half - slab is twice that of original full slab. But these two slabs differ only in the cross - sectional area. Thus,

Combining (1) and (2), we get

where r, the constant of proportionality, is called resistivity. It depends only on the nature of the material and its physical conditions. The unit of r is ohm m (Wm).

Material Nature Resistivity at room temp in ohm-metre

Copper Metal 1.7x10- 8

Iron Metal 9.68x10- 8

Manganin Alloy 48x10- 8

Nichrome Alloy 100x10- 8

Pure Silicon Semiconductor 2.5x 103

Pure Germanium Semiconductor 0.6

Glass Insulator 1010 to 1014

Mica Insulator 1011 to 1015

The reciprocal of resistance of a conductor is called its conductance (G). Therefore, G = 1/R. The S.I. unit of conductance is mho or siemen (S). The inverse of resistivity of a conductor is called its conductivity (s);

The S.I. unit of conductivity is siemen metre-1 (sm-1) or mho m-1 or (Wm)-1.

Good electrical conductors such as metals are usually good conductors of heat also. Insulators such as ceramic and plastic materials are also poor conductors of heat.

In metals, the free electrons that carry charge in electrical conduction also provide the principal mechanism for heat conduction.


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SemiconductorsSemiconductors have resistivities intermediate between those of metals and insulators. They are important because of the way they are affected by temperature and small amount of impurities.

Colour Code for ResistorsA resistor is a component in a circuit whose function is to provide a specified value of resistance. The commonly used resistors are the carbon resistors made from powdered carbon mixed with binding material and baked into a small tube with wire attached to each end. These have resistance values from a fraction of an ohm to several million ohms.

Carbon resistors are marked with a set of co-axial coloured rings. An example is shown in the figure below. These rings indicate the value of their resistance (the table gives the colour code) and the percentage accuracy.

Equation ColourNumbe

rIn order to remember the colour code, the following

sentence may be helpful.

(Black) B (Brown) B(Red) R

(Orange)O(Yellow) Y(Green) G(Blue) B (Violet) V (Grey) G (White) W

0 1 2 3 4 5 6 7 8 9

BB Roy Great Britain Very Good Wife

The first two colour bands (i.e., A and B) from the end indicate the first two significant figures of the resistance in ohms. The third colour band i.e., C indicates the decimal multiplier and the last band D stands for the tolerance in percent about the indicated value. If this last band is gold, the tolerance is 5%, if silver it is 10%. If it is absent, the tolerance is 20%.

Example: If ABCD represent colours yellow, violet, brown and gold respectively, the value of the resistance will be

Origin of ResistivityIn electrostatic situations, the electric field is zero everywhere within the conductor, and there is no current. This does not mean that all charges within the conductor are at rest. In metals such as copper or aluminium, some of the electrons are free to move within the conducting material. These free electrons move randomly in all directions with a speed of the order of 106 ms-1. But, the electrons do not escape from the material as they are attracted to the positive


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ions of the material. Since the motion of the electrons is random, there is no net flow in any direction and hence the current is zero.

When the same metal is in an electric field, the electrons are

subjected to a steady force given by If the electrons were moving in vacuum, the steady force would cause a steady

acceleration in the direction of an the electrons speed would have gone up. But in a conductor, the electrons undergo frequent collisions with the ions of the material. Hence, the direction of the electrons undergoes a random change. See figure below, the net effect is that in addition to the random motion of the electrons, there is also a very slow net motion or drift of the electrons. This drift or flow with a

constant velocity (in a direction opposite to ) is called the drift velocity VD.

To find the relationship between current and drift velocity, consider a conductor of length l and area of cross-section A. If V is the potential difference across the ends of the conductor, then the strength of the electric field is

The acceleration acquired by each electron due to the electric field is

where is the coulomb's force experienced by each electron and m is the mass of the electron.

The drift velocity of the electrons is given by

where t, the relaxation time, is the average time that an electron spends between two collisions. It is of the order of 10-14 s.

Now the volume of the conductor is equal to Al and if n is the number of free electrons per unit volume, then, the total number of free electrons in the conductor will be equal to n Al.

Hence, the total charge q = - n Ale

The time taken by free electrons to cross the conductor is

where we have substituted for t and q.

For a given conductor, I a Vd


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A small value of drift velocity 10-5m/sec produces a large amount of current, as there are a large number of free electrons in a conductor. The drift velocity of the electrons Vd is (using E = V / l)

Also I = - neAVd

On substituting for Vd in the above expression we get,

or

From this, the resistance R can be identified as

where,

R : resistance of the conductor, m : mass of the electrons, l : length of the conductor, n : density of free electrons in the conductor. e : electronic charge. A : area of cross-section, : relaxation time

From this, the relaxation time r for a metal can be estimated, using the observed values of r. For copper at room temperature, r = 1.7 x 102 mW m. The number density of electrons is ~ 8.5 x 1022m-3 (using the density of copper). Substituting these, along with the known values of m and e, we get t = 2 x 10-7 s, which agrees with values obtained by other methods.

MobilityIn metals, the mobile charge carriers are the electrons, in an ionized gas, they are electrons and positive ions; in electrolytes, these are both positive and negative ions. In a semiconductor such as Ge or Si, conduction is partly duer to electrons and partly due to holes which are sites of missing electrons. Holes act as positive charges.

The mobility m is defined as the magnitude of the drift velocity per unit electric field, i.e.,

Mobility is positive for both electrons and holes.

The electrical conductivity of a superconductor can be expressed as

= neme + pemh

Here, me, mh are electron and hole mobilities and n,p are electron and hole concentrations.


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The S.I unit of mobility is m2 / Vs.

Temperature Dependence of Resistivity

The resistivity of a metallic conductor nearly always increases with increasing temperature. As temperature increases, the ions

of the conductor vibrate with greater amplitude, making it more likely that a moving electron will collide with an ion. This

impedes the drift of electrons and hence the current. Over a small temperature range, the resistivity of a metal can be

represented by a linear relation

where o is the resistivity at a reference temperature To and (T) is the resistivity at temperature T. is called the temperature

coefficient of resistivity and has dimensions of (oC)-1.

However, the temperature dependence of at low temperatures is non-linear as shown in figure given below.

Fig (a) - Resistivity T of copper as a function of temperature T

In metallic alloys, the resistivity is very large, but has a weak temperature dependence, as seen in below figure.


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Fig(b) - Resistivity of nichrome as a function of absolute temperature T

Alloys have a residual resistivity even at absolute zero, but a pure metal has a vanishingly small resistivity. This can be used

to check the purity of metals. The resistivity of a semiconductor decreases rapidly with increasing temperature as shown in fig

(c).

Fig (c) - Temperature dependence of resistivity for a typical semiconductor

This means that is negative. The resistivity of an insulator too decreases exponentially with increase in temperature.

These observations may be understood qualitatively using the equation for .

Since m and e are constants,

In metals, the number of free electrons, n does not change with temperature. But, as temperature increases, the atoms/ions vibrate with increasing amplitude. Therefore, the collisions of electrons with them become more frequent, resulting in a decrease in t. This means an increase in with increase in temperature.

In both insulators and semiconducotors, t remains almost constant, but the number of free charge carriers increases with

temperature. At any temperature T, the number of carriers is given by

n(T) = n0 exp (–Eg/kB T)

where Eg is the energy gap between the conduction and valence bonds. From this, we can get the temperature dependence

of to be

(t) = 0 exp (Eg/kB T)

In semiconductors, Eg ~ 1 eV, \ is not very high.


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In insulators, Eg >> 1 eV; \ is very high.

Also, this last equation shows that for semiconductors and insulators, resistivity increases with decreasing temperature.

Limitations of Ohm's Law

Ohm's law is not a fundamental law of nature. There are a number of commonly used circuit elements which do not obey this

law. They have one or more of the following properties:

1. V depends on I non-linearly.

2. The relationship between V and I depends on the sign of V for the same absolute value of V.

3. The relation between V and I is non-unique, that is, for the same current I, there is more than one value of voltage V.

The graph shows the (V-I) graph of a vacuum diode.

A vacuum tube is used to convert high voltage alternating current to direct current.

The behavior of semiconductor diodes is somewhat different but still strongly asymmetric. It exhibits properties 1 and 2 that

are listed above. For a diode of either type, a positive potential difference V will cause a current to flow in the positive direction,

but a potential difference of the other sign will cause little or no current. Hence, a diode acts like a one-way valve in a circuit,

and is then called a rectifier.


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The graph shows the characteristics of a device known as a thyristor, which consists of four alternate layers of P and n-type

semiconductors. We find that V is not directly proportional to I. All the properties 1 to 3 are seen and the region (PQ) is still

more interesting. Here the current carried by the device increases as the voltage decreases. Other examples of non-ohmic

devices are electrolytes, thermistors.

A more fundamental breakdown of Ohm's Law occurs in certain alloys at very low temperatures ( <4K)

It is observed that the resistance of a uniform wire made of the alloy of length 2l is more than twice the resistance of the wire

of length l. This can be understood in terms of the wave character of electrons, which leads to interference effects. This

increases the resistance beyond what would be if electrons flowed like water.

Variations of current versus voltage for GaAs

Strong non-linearity occurs in a material of GaAs. Here, after a certain voltage, the current decreases as voltage increases.

That is, the effective resistance is negative.

Superconductivity

Some materials, including several metals and alloys, exhibit a phenomenon called superconductivity. As the temperature

decreases, the resistivity at first decreases smoothly like that of any metal. But then at a certain critical temperature TC, a

phase transition occurs and the resistivity drops suddenly to zero. And electric current established in a super conducting ring


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continues indefinitely without the presence of any driving field. Superconductivity was discovered in 1911 by the Dutch

physicist Heike Kamerlingle Onnes.

Superconductors display the Meissner effect which is discussed below.

Meissner Effect

In 1933, Meissner and Ochsenfeld found that if a superconducting material is cooled to below its transition temperature, in a magnetic field, then the lines of induction B are pushed out. This is shown in the figure below.

Meissner effect in a superconductor

The Meissner effect shows that a bulk superconductor behaves as if inside it, B is zero.

High Tc Superconductivity

This table shows that some materials, mainly oxides, have a fairly high critical temperature Tc. The 125K bulk super

conducting oxide was discovered in 1988. High Tcsuperconductors have promising commercial applications in thin film

devices, levitated vehicles and long distance power transmission.

In MRI (magnetic resonance imaging) instruments, superconducting magnets made with these high Tc materials are being

used. These can be cooled with liquid nitrogen. Earlier, liquid helium was required which was much more expensive.

Resistors in Series and in Parallel

When resistors, can be connected in such a way that the same current flows in them, then they are said to be connected in

series. The resistors are said to be connected in parallel if the potential difference is the same across each resistor.


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For any combination of resistors in a circuit, we can always find a single resistor that can replace the combination and result in

the same total current and potential difference.

For example, a string of light bulbs can be replaced by a single, chosen light bulb that can draw the same current and have the

same potential difference between its terminals as the original string of bulbs. The resistance of this single resistor is called the

equivalent resistance of the combination.

Resistances in Series

If the resistances are connected end-to-end, the same current flows through each resistance, as there is no alternative path. Then

where I is the current, V is the potential difference of the battery and RS is the equivalent resistance of the combination.

Now,

V = V1+V2+V3

IRS = I1R1 + IR2 + IR3

Therefore,

RS = R1+ R2 + R3

Resistance in Parallel

If the resistance are in parallel, the potential difference across each is the same, but the current is not. Then,


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I = I1 + I2 + I3

Since charge is not accumulated at a point,

where Rp is the equivalent resistance of the combination. Therefore,

For the special case of only two resistors in parallel, the expression for the equivalent resistance takes on a particularly simple

form, i.e.,

Next, consider the following two networks where the resistances are connected in series - parallel combinations.

The first network can be simplified by replacing the parallel combination of R2 and R3 with its equivalent resistance. This is then in series with R1.

In the second network, the combination of R2 and R3 in series forms a simple parallel combination with R1.

But, not all networks can be reduced to simple series - parallel combinations, and special methods are required.

Electric Circuits and Kirchhoff's Rules


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In many electrical circuits, Ohm's law cannot be applied. This happens when there is more than one source of emf in the

circuit or when resistors are connected in a complicated manner. To solve such complex circuits, Gustav Robert Kirchhoff

developed two laws based on charge neutrality in a metal.

Here, we first discuss the internal resistance of electrical circuits and then go on to Kirchhoff's rules.

Internal ResistanceWhen current is drawn from a cell, ions move within the cell from one electrode to another. The resistance offered by the electrodes and electrolytes to these, measure the internal resistance of the cell.The internal resistance of a cell depends on the distance between the plates, the nature of the electrolytes, the concentration of electrolytes, the nature of the electrodes and the area of the plates.

It is usual practice to represent internal resistance of a cell like a series resistor, external to the cell as shown.

Consider the circuit below.

Let e, r be the emf and internal resistance of a cell and R - the external resistance. A high resistance voltmeter V is connected as shown.

When K is opened (i.e., open circuit) emf, the voltmeter reads the emf (e) of the cell as no current flows through the circuit.

When K is closed (i.e., closed circuit), a current 'I' flows in the circuit. Hence, we have


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'Ir' is the potential difference across the internal resistance r.

But, V = IR

Therefore, the external voltage V is less than the emf of the cell, e. It is as though an internal resistance r is in series with the

external resistance R, and this determines the current in the circuit for a given source of emf.

Also,

Kirchhoff's Rules

Consider the following two circuits. Neither can be solved by series-parallel combinations.

Many practical resistor networks cannot be reduced to simple, series-parallel combinations. The above circuit shows a DC power supply with EMF E1 charging a battery with a smaller EMF E2 and feeding current to a light bulb with resistance R. Here we cannot identify the resistances in series or in parallel. So, German physicist Gustow Robert Kirchoff developed a technique. He introduced two terms. One is 'junction' and the other is 'loop'.

Before going on to Kirchoff's rules, we need to introduce two terms - junction and loop.

In the above circuits a, b, c, d are junctions but not e, f.

Some possible loops are acdba, acdefa, abdefa and abcdefa.

Kirchhoff's Junction Rule

The algebraic sum of the currents at a junction in a closed circuit is zero.


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Therefore, I1 + I4 = I2 + I3 + I5

Hence, I1 + I4 - I2 - I3 - I5 = 0

or I = 0

(Sum of currents entering a junction = Sum of currents leaving the junction)

This rule is based on the fact that charge cannot be accumulated at any point in a conductor in a steady situation.

Kirchhoff's Loop Rule

The algebraic sum of the potential differences in any loop including those associated with emfs and those of resistive elements

must be equal to zero.

This rule is based on energy conservation, i.e., the net change in the energy of a charge after completing the closed path is

zero. Otherwise, one can continuously gain energy by circulating charge in a particular direction.

Sign Convention in Applying Kirchhoff's Rules

The emf of a cell is positive when one moves in the direction of increasing potential (i.e., negative pole to positive pole) through the cell and is negative when one moves from positive to negative.

The product of resistance and current, i.e., the IR term, in any arm of the circuit is taken negative if one moves in a closed path, in the same direction of the assumed current; and positive if in the opposite direction.


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Steps to solve circuits by Kirchhoff's laws:

Assume unknown currents in a given circuit and show their directions by arrows.

Choose any closed loop and find the algebraic sum of voltage drops plus the algebraic sum of the emfs in that closed

loop and equate it to zero.

Write equations for as many closed loops as the number of unknown quantities. Solve the equations to find the

unknown quantities.

If the value of assumed current is negative, it means that the actual direction of the current is opposite to that of the

assumed direction.

Measurement of Voltages, Currents and Resistances

These devices measure the voltage and current respectively in a circuit. The basic component of both is the moving coil

galvanometer which produces a deflection proportional to the electric current through it.

Ammeter

An ammeter is connected in series with the circuit element whose current is to be measured, so that there is only a negligible change in the circuit resistance and hence circuit current.

Let the galvanometer resistance be G and the current for full-scale deflection be Ig. To measure larger currents, a suitable low

resistance S (called shunt) is connected in parallel with the galvanometer.

The value of S is chosen by the maximum current I that we want to measure. This means that though the circuit current is I,

only a current Ig should be through the galvanometer. The remaining current

I – Ig = Is should flow through the shunt. Equating potential differences across the shunt and galvanometer, we get


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(I – Ig) S = Ig G

The resistance of the ammeter (i.e., shunted galvanometer) is

RA < S

So, the shunt not only extends the range of current (from Ig to I), it extends the range of current (from G to RA) of the ammeter.

Voltmeter

A voltmeter is connected in parallel with the circuit element across which potential difference is to be measured. It should have a very high resistance as not to alter the circuit resistance, and hence circuit current.

The galvanometer can measure voltages upto IGG. For larger potential differences, a suitable high resistance R (called

multiplier) is connected in series.

The value of R is chosen according to the maximum voltage V that we want to measure. But the galvanometer by itself can

only handle a voltage of IgG. The remaining potential difference (V - IgG) should be across the multiplier R. The current through

it is Ig. Therefore, equating voltage drops, we get

V = Ig G + Ig R

The resistance of the voltmeter (i.e., a galvanometer in series with a high resistance) is

RV = G + R

Since R is high, multiplier increases the resistance of the voltmeter, and of course, extends the voltage range (from IgG to V).

Wheatstone Bridge

This is used to measure an unknown resistance accurately. It consists of 4 resistors (2 fixed known resistances P and Q, a known variable resistance R and the unknown resistance X) connected as shown in the figure.


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Wheatstone's network

A source of emf is connected across one pair of opposite junctions (A and C), and a galvanometer G across the other opposite

pair (B and D). The key K1 is closed first and then K2. The value of R is varied till the galvanometer shows no deflection, i.e.,

Ig = 0. Then, the bridge is said to be balanced.

The wheatstone bridge principle states that under balanced conditions, the products of the resistances in the opposite arms are equal, i.e.,

We will show this using Kirchoff's rules.

Applying the Kirchhoff's law to loop 1, we have

– I1P – IgRg + (I – I1)R = 0 ...(1)

Similarly for loop 2, we have

– (I1–Ig)Q +(I – I1+ Ig)X + IgRg = 0 ...(2)

(where Rg is the resistance of the galvanometer)

In the balanced condition, putting Ig = 0, we have

–I1P + (I – I1)R = 0 …(1)

and – (I1)Q + (I – I1)X = 0 …(2)

Simplifying the two equations, we get

I1P = (I – I1)R …(1)


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I1Q = (I – I1)X …(2)

Dividing the above two equations, we get

Resistor Q is called the standard arm of the bridge, and resistor P and R are called the ratio arms.

Metre Bridge

This is the simplest form of wheatstone bridge and is specially useful for comparing resistances more accurately. The

construction of the metre bridge is as shown in the below figure.

It consists of one metre resistance wire clamped between two metallic strips bent at right angles and it has two points for

connection. There are two gaps; in one of them a known resistance whose value is to be determined is connected. The

galvanometer is connected with the help of jockey across BD and the cells is connected across AC. After making connections,

the jockey is moved along the wire and the null point is obtained. The segment of length l1 and (100 – l1) form two resistances

of the wheatstone bridge, the other two reistances being R and S. The wire used is of uniform material and cross-section. The

resistance can be found with the help of the following relation


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where is the resistance per unit length of the wire and l1 is the length of the wire from one end where null point is obtained.

The bridge is most sensitive when null point is somewhere near the middle point of the wire. This is due to end resistances.

End Correction

Sometimes at the end points of the wire, some length is found under the metallic strips and as a result, in addition of length

l1 or (100 – l1), some additional length should be added for accurate measurements. The resistance due to this additional

length is called end resistance. If the end resistance is small, it can be determined by first introducing known resistances P

and Q in the gap and obtaining the null point reading l1, then interchanging P and Q and obtaining the null point reading l2.

Let and be the lengths on the respective end under the metallic strips, then we have

Solving the equations (1) and (2) for and , we have

Hence the values of , can be calculated and suitably accounted for when accurate measurements are required.

Potentiometer

This instrument is identical to the meter bridge except that in this case, the resistance wire is of more than a meter length. This

enables greater accuracy. A standard cell of emf e1 maintains a constant current throughout the wire. As the wire is of uniform

material and cross section, it has uniform resistance per unit length. The potential gradient, i.e., , depends upon the current in

the wire.

If an emf e1 is balanced against the length, say, l1 we have

Similarly, if another emf e2 is balanced against the length, say, l2, we have

From equations (1) and (2), we have


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From the above figure, by means of a battery B and rheostat Rh, a steady current is passed through the potentiometer wire

AC. Two cells e1 and e2 whose emf's are to be compared are put in such a way that positive terminals are connected to A and

negative terminal to the galvanometer through a two-way plug key k.

First the cell e1 is connected by connecting 1 and 3 points of key K2 and by moving the jockey K on the potentiometer wire, the

no deflection point is obtained. Let the reading be l1, then

where is the potential gradient and l1 is the length CN. After this, the points 2 and 3 of the key K2 are connected i.e., the cell

of emf e2 is put into the circuit and again the no deflection point on the wire is obtained. Let this reading be l2. Then

e2 = rl2

Different sets of observations are taken by varying the variable resistance Rk and then mean value of ratio is computed.


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