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Resistor Color Codes
electric circuit | education
Learn how to identify different resistors with their colour code
Printer Friendly Version The humble resistor is a key component of many renewable energy systems. They are designed to resist electricity and can therefore be used to supply the correct voltage to the various components in an electric circuit. Because resistors are so small - too small to print numerical resistance values on they are instead marked with four, five, or (less commonly) six coloured bands. Each colour represents a different number and so the value (in Ohms) of the resistor can easily be calculated.
Resistor Colour Code Chart
The last coloured band is usually used to indicate the tolerance - a measure of how accurate the stated resistance is, i.e. 1%, 2%, 5%, or 10%. In the case of six band resistors, is used to show the temperature coefficient - a measure of how much the resistance of a resistor is changed by changes in temperature.
Remembering the Resistor Colour Code Values
There are many mnemonic phrases to help you to remember the order of the colours: Black Brown Red Orange Yellow Green Blue Violet Gray White ...which correspond to the values 0-9. For example: Bad Beer Rots Our Young Guts But Vodka Goes Well.
Electric current
is a means by which electrical energy is transferred from one place to another for utilisation. Charges in motion constitute an electric current. We know from electrostatics that when an isolated conductor is placed in an electric field, the charges in the conductor rearrange themselves so that the interior has no net electric field. This constitutes of a transient current, that is a current of short duration. It ceases the moment the net electric field in the conductor becomes zero. However , to maintain a steady current, a source of emf is needed. This and other aspects of current electricity, such as resistance, are discussed in this chapter A 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 the free electron causing them to move as shown in the figure. The arrows give the direction of the conventional current. thus set up exerts a force on
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. No 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 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.
Ohm's Law
It 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. Departures from Ohm's law are discussed in a later section.
Resistivity
The 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
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 Copper Iron Manganin Nichrome Pure Silicon Pure Germanium Glass Mica Nature Metal Metal Alloy Alloy Semiconductor Semiconductor Insulator Insulator Resistivity at room temp in ohm-metre 1.7x10- 8 9.68x10- 8 48x10- 8 100x10- 8 2.5x 103 0.6 1010 to 1014 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.
Semiconductors
Semiconductors 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 Resistors
A 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 Colour (Black) B (Brown) B (Red) R (Orange)O (Yellow) Y (Green) G (Blue) B (Violet) V (Grey) G (White) W
Numbe r 0 1 2 3 4 5 6 7 8 9
In order to remember the colour code, the following sentence may be helpful.
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
In 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 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 crosssection 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 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 t : 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 1022 m-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.
Mobility
In 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 s = neme + pemh Here, me, mh are electron and hole mobilities and n,p are electron and hole concentrations.
The S.I unit of mobility is m2 / Vs. Mobilities of some materials, at room temperature, in cm2/vs Material Diamond Silicon Germanium InSb GaAs Electrons 1800 1350 3600 800 8000 Holes 1200 480 1800 450 300
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 ro is the resistivity at a reference temperature To and r (T) is the resistivity at temperature T. a is called the temperature coefficient of resistivity and has dimensions of (oC)-1. However, the temperature dependence of r at low temperatures is non-linear as shown in figure given below.
Fig (a) - Resistivity rT 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.
Fig(b) - Resistivity rT
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 a is negative. The resistivity of an insulator too decreases exponentially with increase in temperature. These observations may be understood qualitatively using the equation for r.
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 r 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 r to be r(t) = r0 exp (Eg/kB T)
In semiconductors, Eg ~ 1 eV, \ r is not very high. In insulators, Eg >> 1 eV; \ r is very high. Also, this last equation shows that for semiconductors and insulators, resistivity increases with decreasing temperature.
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.
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 T C, a
phase transition occurs and the resistivity drops suddenly to zero. And electric current established in a super conducting ring 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 Tc superconductors 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 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.
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, 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 R 2 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.
ELECTRICAL CIRCIUT AND KIRCHOFF RULE
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 Resistance
When 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
'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.
Therefore, I1 + I4 = I2 + I3 + I5 Hence, I1 + I4 - I2 - I3 - I5 = 0 or SI = 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.
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.
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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.
Voltmeter and Ammeter
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 (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 R A) 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, the 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.
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) 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 l 1 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
where s is the resistance per unit length of the wire and l 1 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 a and b be the lengths on the respective end under the metallic strips, then we have
Solving the equations (1) and (2) for a and b, we have
Hence the values of a, b 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., r, 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
From the above figure, by means of a battery B and rheostat R h, 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 r 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.
SUMMARY
• •
Current through a given area of a conductor is the net charge passing per unit time through the area. To maintain a steady current, we must have a closed circuit in which an external agency moves electric charge
from lower to higher potential energy. The work done per unit charge by the source in taking the charge from lower to higher potential energy (i.e., from one terminal of the source to the other) is called the electromotive force, or emf, of the source. Note that the emf is not a force; it is the voltage difference between the two terminals of a source in open circuit.
•
Ohm's law; the electric current I flowing through a substance is proportional to the voltage V across its ends. I or V = RI. where R is called the resistance of the substance. The unit of resistance is ohm: 1W = 1VA-1.
i.e., V
•
The resistance R of a conductor depends on its length l and constant cross-sectional area A through the
relation.
where r, called resistivity is a property of the material and depends on temperature and pressure.
•
8
Electrical resistivity of substances varies over a very wide range. Metals have low resistivity, in the range of 10-
Wm to 10-6Wm. Insulators like glass and rubber have 1022 to 1024 times greater resistivity. Semiconductors like Si
and Ge lie roughly in the middle range of resistivity on a logarithmic scale.
•
In most substances, the carriers or current are electrons; in some cases e.g., ionic crystals and electrolytic
liquids, positive and negative ions carry the electric current.
•
j = nqv
Current density j gives the amount of charge flowing per second per unit area normal to the flow.
where n is the number density (number per unit volume) of charge carriers each of charge q, and v is the drift velocity of the charge carriers. For electrons q = -e. If j is normal to a cross-sectional area A and is constant over the area, the magnitude of the current I through the area is (nev A).
•
Using E = V/l, I = nev A and ohm's law, one obtains.
The proportionality between the force eE on the electrons in a metal due to the external field E and the drift velocity v d (not acceleration) can be understood, if we assume that the electrons suffer collisions with ions in the metal, which deflect them randomly. If such collisions occur on an average at a time interval t, vd = at = eEt/m where a is the acceleration of the electron. This gives
•
In the temperature range in which resistivity increases linearly with temperature, the temperature coefficient of
resistivity a is defined as the fractional increase in resistivity per unit increase in temperature.
•
Ohm's law is obeyed by many substances, but it is not a fundamental law of nature. It fails if
(a) V depends on I non-linearly. (b) The relation between V and I depends on the sign of V for the same absolute value of V. (c) The relation between V and I is non-unique. An example of (a) is when increases with I (even if temperature is kept fixed). A rectifier combines features (a) and (b). A thyristor shows all the features (a), (b) and (c).
•
When a source of emf e is connected to an external resistance R, the voltage Vext across R is given by
where r is the internal resistance of the source.
•
(a) Total resistance R of n resistors connected in series is given by
R = R1 + R2 + …Rn (b) Total resistance R of n resistors connected in parallel is given by
•
Kirchhoff's Rule
(a) First Rule (Junction Rule): At any junction of circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it. (b) Second Rule (Loop Rule): The algebraic sum of changes in potential around any closed loop must be zero.
•
A voltmeter consists of a galvanometer (or resistance RG) in series with a high resistance R. It is put in parallel
to the circuit element across which the voltage is to be measured. Because of its high (R + R @ R). It draws a very small current and thus does not disturb the circuit. If the full scale deflection of the galvanometer occurs for a current Io and the voltmeter is to have range Vo, we have
•
An ammeter consists of a galvanometer (or resistance RG) and low resistance R in parallel. The effective
resistance of the ammeter is
Because of its very low resistance, the ammeter placed in series in a circuit does not materially change the current in the circuit to be measured. If the full-scale deflection of the galvanometer occurs for current I o and the ammeter is to have a range Imax, we have
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The potentiometer is a device to compare potentials. Since the method involves a condition of no current flow,
the device can be used to compare emfs of two sources.
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The Wheatstone bridge is an arrangement of four resistances - P, Q, R, S as shown in the text. The null-point
condition is given by
using which the value of one resistance can be determined, knowing the other three resistances.
PROBLEMS AND SOLUTIONS
01.
In the case of hydrogen atom, an electron moves in an orbit of radius 5x10-11 m with a speed of 2.2 x 106 ms-1. Calculate the equivalent current. Given: charge on an electron = 1.6 x 10-19C.
Suggested solution:
Radius of the orbit, r = 5 x 10-11m Circumference = 2pr Speed of electron,
Number of revolutions in 1 second,
Charge flowing in one second, q = nr Current,
02.
How many electrons flow through the filament of a 120 V, 60 W electric lamp in one second? Given: Electric power is the product of voltage and current.
Suggested solution:
Now,
or
03.
What is the number of free electrons in a piece of silver of cross-section 1.0 x 10-4 m2 and length 1m? Atomic weight of silver = 108, density of silver = 105 x 102 kg m-3. Assume that there is one free electron per atom.
Suggested solution:
Area, A = 1.0 x 10-4 m2 Length, l = 1 m Atomic weight = 108 Density, d = 105 x 102 kg m-3 Volume of given piece = A x l = 1.0 x 10-4 m2 x 1m = 10-4 m3 Mass = Volume x density = 10-4 m3 x 105 x 102 kg m-3 = 1.05 kg Number of atoms in 108 kg silver = 6.023 x 1026
Since there is only free electron per atom, therefore the number of free electrons is 5.856 x 1024