ICSE Class 10 Physics Chapter 08 Current Electricity

Electricity And Magnetism

Syllabus

(i) Ohm's law, concepts of e.m.f., potential difference, resistance; resistances in series and parallel; internal resistance.

Scope of syllabus: Concepts of p.d. (V), current (I) and resistance (R) and charge (Q). Ohm's law: statement, V = IR; SI units; experimental verification; graph of V vs I and resistance from slope; ohmic and non-ohmic resistors, factors affecting resistance (including specific resistance) and internal resistance, super conductors, electromotive force (e.m.f.); combination of resistances in series and parallel and derivations of expressions for equivalent resistance. Simple numerical problems using the above relations. (Simple network of resistors).

(ii) Electrical power and energy.

Scope of syllabus: Electrical energy; examples of heater, motor, lamp, loudspeaker etc., electrical power, measurement of electrical energy W = QV = UI = I2t from definition of p.d., Combining with Ohm's law W = Vit = I2Rt = (V2/R)t and electrical power P = (W/t) = VI = I2R = V2/R. Units: S.I. and commercial; power rating of common appliances, household consumption of electric energy; calculation of total energy consumed by electrical appliances; W = Pt (kilowatt - hour = kWh), (simple numerical problems).

A - Concept Of Charge, Current, Potential, Potential Difference, And Resistance; Ohm's Law

8.1 Concept Of Charge

In class IX, we have read that when two bodies (such as glass and silk or ebonite and fur) are rubbed together, there is a transfer of electrons from one body to the other. The body which gains electrons becomes negatively charged and the body which loses electrons becomes positively charged. Thus there are two kind of charges: (1) positive charge, and (2) negative charge. Two like charges repel each other, while the two unlike charge attract each other. The charge on a body is denoted by the symbol q (Q).

Unit of charge: The S.I. unit of charge is coulomb (symbol C). The smaller units of charge are milli-coulomb (mC), micro-coulomb (\(\mu\)C) and nano-coulomb (nC) where

1 mC = 10-3 C, 1 \(\mu\)C = 10-6 C, and

1 nC = 10-9 C

The quantity of charge on a body is determined by the number of electrons in deficit (if the body is positively charged) or the number of electrons in excess (if the body is negatively charged). Thus, charge on a body

q = \(\pm\) ne where n is the number of electrons in deficit for + sign and in excess for - sign while e is the charge on an electron. The charge on an electron is -1.6 x 10-19 C, so 1 C charge means a deficit of \(\frac{1}{6\times 10^{-19}}\) = 6.25 x 1018 electrons.

8.2 Concept Of Current

The charge in motion constitutes electric current. Current is defined as the rate of flow of charge. In other words, the current flowing in a conductor is the amount of charge flowing per second through it. If a charge Q flows through the cross section of a conductor in time t, then current I through it is given as:

\[I = \frac{Q}{t}\]

Note: The current in a circuit is measured by an ammeter by connecting it in series in that circuit, taking care that the +ve marked terminal of ammeter is towards the positive terminal of the source of current.

Unit of current: The S.I. unit of charge is coulomb and therefore current is measured in coulomb per second which has been given the name ampere (symbol A) in honour of the French physicist Andre Ampere. We define one ampere as the current which flows when one coulomb of charge passes in one second. i.e.,

\[1 \text{ A} = \frac{1 \text{ C}}{1 \text{ s}} = 1 \text{ C s}^{-1}\]

To express a weak current, the smaller units of current are milli-ampere (mA) and micro-ampere (\(\mu\)A) which are related to ampere (A) as follows:

1 mA = 10-3A and 1 \(\mu\)A = 10-6 A

Flow of current: In metals, the moving charges are free electrons which constitute the current, while in electrolytes and ionised gases, both the positively charged ions (cations) and the negatively charged ions (anions) are the moving charges which constitute current.

If n electrons pass through the cross section of a conductor in time t, then total charge passed through the conductor is given as

\[Q = n \times e\]

and the current in conductor is

\[I = \frac{Q}{t} = \frac{ne}{t}\]

Since 1 C charge is carried by 6.25 x 1018 electrons, so if 1 A current flows through a conductor, it implies that 6.25 x 1018 electrons pass in 1 second across the cross section of conductor.

Note: Current is a scalar quantity. By stating the direction of current, we mean that the direction of motion of electrons is opposite to it.

8.3 Concept Of Potential And Potential Difference (P.D.)

Potential: We have read that the like charges repel, while the unlike charges attract each other. If a charge A is brought near another like charge B, some work has to be done in moving the charge A against the repulsive force on it due to the other charge B. Similarly if a charge A is brought near another unlike charge B, work is done by the attractive force on the charge A due to the unlike charge B. Thus some work is always involved in moving a charge in the vicinity of another charge. Therefore, the potential at a point in a region of charges, is measured in terms of the work done in moving a test charge from a point of zero potential to that point. Since force between the two charges at infinite separation is zero, therefore we consider the potential to be zero when the test charge is at infinite separation from the other charge. Then we define potential as under:

The potential at a point is defined as the amount of work done in bringing a unit positive charge from infinity to that point.

It is denoted by the symbol V. It is a scalar quantity.

Note: The potential is positive at a point in the vicinity of a positive charge since work has to be done on the positive test charge against the repulsive force due to the positive charge it from infinity, while it is negative at a point in the vicinity of a negative charge since work is done on the test charge by the attractive force itself.

In Fig. 8.1, a test charge Q is brought from infinity to a point P in the vicinity of a positively charged body.

If W joule of work is done in bringing the test charge Q coulomb from infinity to the point P, then electric potential V at the point P is given as

\[V = \frac{W}{Q}\]

Obviously the work needed to move a charge Q from infinity to a point P where electric potential is V, will be

\[W = QV\]

Unit Of Electric Potential

From eqn. (8.5),

Unit of potential V = \(\frac{\text{unit of work W}}{\text{unit of charge Q}}\)

The S.I. unit of work is joule and that of charge is coulomb, so the S.I. unit of potential is joule/coulomb (or J C-1) which has been given the name volt (symbol V) in the honour of the scientist Volta. Hence, the potential at a point is said to be 1 volt when 1 joule of work is done in bringing 1 coulomb charge from infinity to that point. Thus,

\[1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} = 1 \text{ J C}^{-1}\]

Potential Difference (P.D.)

In practice when we consider the flow of current between two points A and B of an electric circuit, we consider only the flow of charge between the two points A and B, therefore it is not necessary to know the exact potential at each point A and B. It is sufficient to know the potential difference (abbreviated as p.d.) between the two points A and B. Using the definition of potential at a point, the potential difference between two points can be defined as follows:

The potential difference (p.d.) between two points is equal to the work done in moving a unit positive charge from one point to the other.

It is a scalar quantity.

If W joule of work is done in moving a test charge Q coulomb from a point A to the point B, the potential difference between the two points A and B is

\[V_A - V_B = \frac{W}{Q}\]

Note: The potential difference between two points in an electric circuit is measured by a voltmeter which is connected across those points in parallel with the circuit, taking care that the +ve marked terminal of voltmeter is connected to the higher potential point.

Unit Of Potential Difference

Potential difference (p.d.) is also expressed in volt (V). The potential difference between two points is said to be 1 volt if the work done in moving 1 coulomb charge from one point to other is 1 joule, i.e.,

\[1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} = 1 \text{ J C}^{-1}\]

8.4 Concept Of Resistance

When current flows through a conductor (say, a metallic wire), the wire offers some obstruction to the flow of current i.e., it offers some resistance. Thus,

The obstruction offered to the flow of current by the conductor (or wire) is called its resistance.

Cause of resistance: A metal has a large number of electrons and an equal number of positive ions (i.e., the atoms which have given out electrons). The positive ions do not move, while the electrons move almost freely inside the metal.

These electrons are called the free electrons. They move at random, colliding among themselves and with the positive ions as shown in Fig. 8.2 (a).

When the ends of the metal wire are connected to a cell (or a source of current), i.e., when a potential difference is applied across the ends of the metal wire, the electrons inside it (due to acceleration) start moving from the end at negative potential towards the end at positive potential [Fig. 8.2 (b)] during which their speed increases. But during the movement, they collide with the fixed positive ions and lose some of their kinetic energy due to which their speed decreases. After the collision, they are again accelerated towards the positive potential due to the existing potential difference so their speed again increases and then again in collision with the positive ions, their speed decreases. This process continues. As a result, the electrons do not move in bulk with a continuously increasing speed, but there is a drift of electrons towards the positive terminal. Thus a metal wire offers some resistance to the flow of electrons through it.

The resistance of a conductor depends on the number of collisions suffered by the electrons with the positive ions while moving from one end to the other end.

The relationship between potential difference and current in a conductor. This relationship is stated in form of a law known as Ohm's law.

Statement Of Ohm's Law

According to Ohm's law, the current flowing in a conductor is directly proportional to the potential difference applied across its ends provided the physical conditions and the temperature of conductor remain constant.

The direct proportionality between the current and potential difference implies that if the potential difference across the ends of a conductor is doubled, the current flowing in it also gets doubled.

If a current I flows in a conductor when the potential difference across its ends is V, then according to Ohm's law

\[I \propto V\]

or

\[\frac{V}{I} = \text{constant}\]

In eqn. (8.9), we put the constant equal to R, the resistance of conductor. Then,

\[V = IR\]

Here the resistance R is a constant at a given temperature for the given conductor.

If I = 1, then V = R

Thus the resistance of a conductor is numerically equal to the potential difference across its ends when unit current flows through it.

Unit Of Resistance

From relation (8.10),

Unit of R = \(\frac{\text{unit of V}}{\text{unit of I}}\)

The S.I. unit of potential difference is volt and that of current is ampere, so the unit of resistance is volt/ampere (or V A-1) which is named after the scientist Ohm as ohm. It is denoted by the symbol \(\Omega\) (omega). One ohm is defined as below:

The resistance of a conductor is said to be 1 ohm if 1 ampere current flows through it when a potential difference of 1 volt is applied across the ends of the conductor, i.e.,

\[1 \text{ ohm} = \frac{1 \text{ volt}}{1 \text{ ampere}}\]

High resistances are measured in units: kilo-ohm (k\(\Omega\)) and mega-ohm (M\(\Omega\)), where

1 kilo-ohm (or 1 k\(\Omega\)) = 103 \(\Omega\)

and 1 mega-ohm (or 1 M\(\Omega\)) = 106 \(\Omega\)

Conductance: The reciprocal of resistance is called conductance. i.e.,

\[\text{Conductance} = \frac{1}{\text{Resistance}}\]

Its unit is (ohm)-1 or siemen (symbol \(\Omega\)-1).

Note: Now a days we do not write mho for (ohm)-1.

I-V Graph

For a metallic conductor, the ratio V/I is constant for all values of V and I. If a graph is plotted for current I against potential difference V, we get a straight line passing through origin as shown in Fig. 8.3.

Slope of I-V graph: The slope of I-V graph is \(\frac{\Delta I}{\Delta V}\) which is the reciprocal of resistance of the conductor, i.e.,

\[\text{Slope} = \frac{\Delta I}{\Delta V} = \frac{1}{\text{resistance of the conductor}}\]

Limitation of Ohm's law: Ohm's law is obeyed only when the temperature of conductor remains constant.

8.6 Experimental Verification Of Ohm's Law

Electric Circuit

To verify Ohm's law, the electric circuit used is shown in Fig. 8.4.

The rheostat Rh, key K, ammeter A and resistance wire R are connected in series with the battery B, taking care that the + ve marked terminal of ammeter A is towards the positive terminal of the battery. The voltmeter V is then connected in parallel across the resistance wire R keeping its + ve marked terminal towards the positive terminal of the battery.

The battery (B) sends current in the circuit. The current in the circuit is controlled by the rheostat (Rh) and the ammeter (A) measures the current. The key K is used to make and break the circuit. The wire R (say, a nichrome wire) is the unknown resistance. The voltmeter (V) measures the potential difference across the ends of the resistance wire R.

Procedure

As the key K is closed, current flows in the circuit. The rheostat Rh is adjusted to get the minimum (non-zero) reading in the ammeter A and voltmeter V. The ammeter reading I and the voltmeter reading V are noted. The sliding terminal of rheostat is then moved to increase the current gradually and each time the value of current I flowing in the circuit and the potential difference V across the resistance wire R are recorded by noting the readings of the ammeter A and voltmeter V respectively. In this way, different sets of the values of I and V are recorded in the table given below. Then for each set of values of I and V, the ratio V/I is calculated.

V-I Graph

A graph is plotted for V against I by taking V on Y-axis and I on X-axis which is found to be a straight line as shown in Fig. 8.5. This verifies the Ohm's law.

Slope Of V-I Graph

To find the slope of straight line obtained on V-I graph, take two points P and Q on the straight line. From the points P and Q, draw normals PA and QB on the Y-axis; and PC and QD on the X-axis. Read the potential VA at A and VB at B and find the difference VA - VB = \(\Delta\)V. Similarly read the current IC at D, and ID at C and find the difference IC - ID = \(\Delta\)I. Then find the slope = \(\Delta\)V/\(\Delta\)I.

The slope of the straight line on V-I graph i.e., (\(\Delta\)V/\(\Delta\)I) gives the resistance R of the conductor (or wire), i.e.,

\[R = \frac{\Delta V}{\Delta I} = \text{slope of V vs I graph}\]

Obviously, greater is the slope of V-I graph, greater is the resistance of conductor.

8.7 Ohmic And Non-Ohmic Resistors

Ohmic resistors: The conductors which obey the Ohm's law are called the ohmic resistors (or linear resistances). Examples are: all metallic conductors (such as silver, aluminium, copper, iron, etc.), nichrome, copper sulphate solution with copper electrodes, and dil. sulphuric acid, etc. at a constant temperature.

For such resistors, a graph plotted for the potential difference V against current I is a straight line passing through the origin as shown in Fig. 8.5 and the resistance R is same irrespective of the value of V or I (i.e., the ratio V/I is constant for all values of V or I).

Non-ohmic resistors: The conductors which do not obey the Ohm's law are called the non-ohmic resistors (or non-linear resistances). Examples are: LED, solar cell, junction diode, transistor, filament of a bulb, etc.

For these conductors, the graph plotted for the potential difference V against current I is not a straight line, but it is a curve. Fig. 8.6 shows a V-I graph in case of a junction diode.

Note: For the Ohmic resistor, it is necessary that the straight line on V-I graph passes through the origin; but for a non-ohmic resistor, it is not necessary that the curve on V-I graph must pass through the origin.

The resistance of such a conductor (i.e., the ratio V/I) is different for different values of V or I. The resistance at a particular value of V or I is obtained by finding the slope of the tangent drawn at the corresponding point on the V-I graph. The value \(\Delta\)V/\(\Delta\)I is called the dynamic resistance since its value is different for the different values of V or I.

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