# Current Electricity

### Electric Current

The motion of charges constitue electric current. The electric current is defined as the ratio of flow of charges. If a charge 1 flow through any cross-section of the conductor in time than,

$\text{electric} \, \, current \, \, i = \dfrac{q}{t} \cdots \text{equation 1a}$

If small charge dq flows in time dt, then,

$\text{Current} \,\, = \dfrac{dq}{dt} \cdots \text{equation 1b}$

The unit of electric current in M.K.S. and S.I. system is coulomb/second or Ampere. Ampere is the fundamental unit in S.I. system. The electric current has direction, but it is a scalar quantity, since it does not obey the laws of vector addition. Conventionally the direction of current is taken along the direction of flow of positive charges and opposite to the direction of flow of negative charges.

The electric current in metals is due to flow of free electrons.

### Current Density

The current density at any point inside a conductor is defined as a vector quantity whose magnitude is equal to current through infinitesimal area at that point, the area being normal to the direction of flow of current and whose direction is along the direction of current at that point.

Current Density

If $A_n$ is small area at point P, normal to current I, then:

$\text{Current Density} \, \, J = \dfrac{I}{A_n} \cdots \text{equation} \, \, 2a$

If the plane of the small area A is not normal to current, but makes an angle $\theta$ is not normal to current, but makes an angle $\theta$ with the normal to current, then:

$J = \dfrac{I}{A_n} = \dfrac{I}{A \, \, cos \theta} \cdots \text{equation} \, \, 2b$

The unit of current density is $\dfrac{amp}{m^2}$

From equation (2b) we have,

$I = J A cos \theta = \overrightarrow{J} \, \, \overrightarrow{A} \cdots \text{equation} \, \, 3$

### Drift Velocity

When no potential difference is applied across a conductor, the free electrons are in thermal equilibrium with the rest of the conductor and are in random motion. That is the average velocity vector of free electrons is zero and consequently this motion does not constitute a net transport of charge across any section of the conductor and hence there is no current in the conductor.

If a potential difference is applied across a conductor, the electrons gain some average velocity in the direction of positive potential. This average velocity is superimposed over the random velocity of electrons and is called the drift velocity.

The speed of random motion is determined by temperature by,

$\dfrac{1}{2} m v^2 = \dfrac{3}{2} K T$

i.e.

$v = \sqrt{ ( \dfrac{3 K T}{m} ) }$

Where k is Boltzmann’s constant = $1.38 \times 10^{-23}$ Joule/Kelvin.

Its order is $10^5$ m/sec, while the drift velocity $v_d$ is determined by potential difference (V) applied across the conductor. The order of drift velocity is $10^{-4} m / s$

Relation between drift velocity and potential difference: If V is the potential difference applied across a conductor of length l, then electric field strength $E = \dfrac{V}{l}$

Force of electron F = e E

Force of electron

If m is the mass of electron, acceleration produced,

$a = \dfrac{F}{m} = \dfrac{eE}{m} \cdots \text{Equation 1}$

Average random velocity of free electrons, u=0. If ‘v’ is the velocity just before the start of next collision the n,

$Drift \, \, Velocity \, \, v_d = \dfrac{u + v}{2} = \dfrac{0 + v}{2} = \dfrac{v}{2}$

If $\tau$ is time between successive collisions, or relaxation time, then $v = a \tau = 0 + a \tau = a \tau$

Using (1), we get,

$v = \dfrac{e E}{m} \tau$

Therefore,

$Drift \, \, velocity \, \, v_d = \dfrac{e E \tau}{2 m}$

Clearly drift velocity is directly proportional to the electric field strength E.

As $E = \dfrac{V}{i}$

$v_d = \dfrac{e ( \dfrac{V}{I} ) \tau}{2m} = \dfrac{e v \tau}{2 m l}$

The relaxation time changes with change of temperature. Actually it decreases with rise of temperature. Thus at a given temperature drift velocity $v_d$ is directly proportional to the potential difference, inversely proportional to length and is independent of cross-sectional area.

### Ohm’s Law and Electrical Resistance

When a potential difference is applied across a conductor, a current ‘I’ is set up in the conductor. According to Ohm’s law under given physical conditions e.g. at constant temperature and pressure, the potential difference applied across a conductor is directly proportional to the current produced in it.

I.e.

$V \propto I \, \, \, \, V = r I \cdots \text{equation} \, \, 1$

When the constant ‘R’ is called the electrical resistance of the given conductor.

Conductance: The reciprocal of resistance is called the conductance. It is denoted by K.

i.e.

$K = \dfrac{1}{R}$

The unit of resistance R is volt/ampere = Ohm and that of conductance is mho or Siemen.

Physical Concept of Electrical Resistance: The free electrons of the conductor make collisions with themselves and imperfections of lattice. The electric current is opposed by these collisions. The net hindrance offered by a conductor to the flow of current is called the electrical resistance of the conductor. Naturally the electrical resistance of a conductor depends upon the size, geometry, temperature and internal structure of the conductor.

### Ohmic and Non-ohmic conductors

If voltage-current graph of a conductor is a straight line, it is said to be ohmic conductor: The examples are metallic conductors Cu, Fe, tungsten etc, provided the current is not too high; because when current becomes high, the temperature of conductors becomes sufficiently large to change the resistance of conductor; so that linearity between V and I breaks down.

If voltage current graph of a conductor is nonlinear, it is said to be non-ohmic conductor. The examples are the torch bulb, junction diode, thermistor etc.

Ohmic Conductor

In this case the resistance varies with voltage and is called the dynamic resistance. It if found by the formula $R = ( \dfrac{ \delta V}{\delta I} )$ near given voltage.

### Resistivity and Conductivity

For a given conductor of uniform cross-section A and of length l, the electrical resistance R is directly proportional to length I and inversely proportional to cross-sectional area A.

I.e.

$R \propto \dfrac{I}{A} \, \, \, or \, \, \, R = \dfrac{\rho l}{A} \cdots \text{equation} \, \, 1$

Where $\rho$ is a constant of proportionality called the specific resistance or resitivity of the metal of the conductor at given temperature.

From (1), $\rho = \dfrac{RA}{l} \cdots \text{equation} \, \, 2$

If l=1m, A = $1 m^2$ , then $\rho = R$

That is the specific resistance of the material of the conductor defined as the resistance offered by the conductor of 1 m length and 1 $m^2$ cross-sectional area, when current flow is normal to area.

The unit of resistivity is ohm X meter.

The reciprocal of resistivity is called the conductivity. The unit of conductivity is mho/meter.

Ohm’s law in alternative from may be expressed as $J = \sigma E \cdots equation \, \, 3$

Where J = Current density and E = electric field strength.

### Recasting a wire of a given mass

Usally $R = \dfrac{\rho l}{A}$ i.e. resistance is proportional to length of the conductor. But if a wire of given mass is recasted to increase its length, then area of cross-section also decrease; so this must be taken into account.

As, $R = \dfrac{\rho l}{A} \cdots equation \, \, 1$ and mass, M = volume X density = (Al d) = constant.

Therefore,

$A = \dfrac{m}{id} \cdots equation \, \, \, 2$

Substituting this (1), we get,

$R = \dfrac{\rho l}{ ( \dfrac{m}{ld} ) } = \dfrac{\rho d}{m} l^2$

As $\rho$ , d and m are constants.

$R' \propto l^2$

Thus if a wire of initial resistance R is stretched to make its length n-times, then the new resistance become $n^2 - times$ .

i.e.

$R' = n^2 R \cdots equation \, \, 3$

But if radius is given, then from (2)

$l = \dfrac{m}{Ad}$

Substituting this in (1), we get

$R = \dfrac{\rho ( \dfrac{m}{Ad} )}{A} = \dfrac{\rho m}{A^2 d}$

As $\rho$ , m and d are constants,

$R \propto \dfrac{1}{A^2}$

As A = $r^2$ , $R = \dfrac{1}{1^4}$

Thus if a wire of initial resistance R is stretched to make its radius $\dfrac{1}{n}$ –time, then the new resistance becomes $n^4$ – times.

I.e.

$R' = n^4 R$

### Variation of Resistance with temperature

The resistance of a conductor varies with temperature figure represents the graph of variation of resistance of pure metal with temperature.

Varation of resistance with temperature

Mathematically the dependence of (R) on temperature (t) is expressed as:

$R_i = R_0 ( 1 + \alpha t + \beta t^2 ) \cdots equation \, \, 1$

Where $\alpha > \beta$ are temperature coefficents of resistance. Their values vary from metal to metal. If temperature t is not sufficiently large as in most practical cases, then equation (1) may be expressed as:

$R_t = R_0 ( 1 + \alpha t )$

The constant $\alpha$ is called the temperature coefficient of resistance of the material $\alpha$ is positive for metals and negative for semi-conductors and electrolytes. If $R_1 \, \, and \, \, R_2$ are the resistances of the same specimen at temperature $t_1 \, \, and \, \, t_2 \, \, _0 C$ , then

$R_2 = R_1 [ 1 + \alpha ( t_2 - t_1 ) ]$

### Thermistor

It is a heat sensitive device made of a semi-conductor. The temperature coefficient of thermistor is negative but us unusually large.

The voltage current graph of thermistor is unusual as shown in figure.

Thermistor

Thermistor is used:

(i) In resistance thermistors to measure low temperature of the order of 10 K.

(ii) To safeguard electronic circuits against current jumps, because initially thermistor has high resistance when cold and its resistance decrease appreciably when it warms up.

### Color code of carbon Resistances

It is indicate resistance and its percentage reliability. The color bands are formed from left to right. The first three bands give the value of resistance. The first and second band indicate the first and second significant digit while the third band gives the number of zeros which follow the first two digits, often called multiplier. The fourth band represents its tolerance. Absence of any fourth band means a tolerance of 20%.

 Memory Letter Color Band ‘1’ Band “2’ Band ‘3’ (multiplier) Band ‘4’ B Black 0 0 $10^0$ Gold 5% B Brown 1 1 $10^1$ Silver 10% R Red 2 2 $10^2$ No Color 10% O Orange 3 3 $10^3$ Y Yellow 4 4 $10^4$ G Green 5 5 $10^5$ B Blue 6 6 $10^6$ V Violet 7 7 $10^7$ G Grey 8 8 $10^8$ W White 9 9 $10^9$

The memory letters may be remembered by the following sentence:

Color code

“B.B.ROY 9of) Great Britain (has) Very Good Wife.”

### Sources of EMF

A source of emf is a device that drives charge carriers from one point to another. The sources of emf may be chemical, thermal, electromagnetic, piezoelectric etc. Some sources of emf are thermocouple, piezoelectric crystal, photocell and electric cell. In a thermocouple heat energy is converted into electrical energy; in a photocell light energy is converted into electrical energy; in an electric cell (usually called a cell), chemical energy is converted into electric energy.

These cells are divided into two categories:

(i) Primary cells: A primary cell consists of an electrolyte, two electrodes and a depolarizing agent. The Lechlanche cell, Daniel cell, dry cell are examples of primary cells. They are used where no continuous current is required. They have high internal resistance and their material used up. Moreover they cannot be recharged.

(ii) Secondary cells: In secondary cell the chemical reaction is reversible; so they can be recharged. Its internal resistance is smaller than a primary cell. They used where continuous supply of current is required.

When a secondary cell is giving current in a circuit; the chemical energy is convened into electrical energy and during the process its emf and density of electrolyte falls; while during charging process; the electrical energy is convened into chemical energy and its emf and density of electrolyte increase and finally becomes steady.

Remarks: When a secondary cell is charged, its positive terminal is connected to positive terminal of the charging supply.

### Electromotive Force and Potential Difference

The E.M.F of a source of potential difference is the potential difference between the terminals of the source when no current is drawn from it i.e. when the source is in the open (or infinite resistance) circuit. Alternatively the e.m.f. of a cell is defined as the work done in moving per unit positive test charge in the entire closed circuit including the solution of the cell.

I.e.

$E = \dfrac{W}{q_0}$

When the terminals of a cell are connected to an external resistance, the cell is said to be in closed circuit. The potential difference across the terminals of a cell in closed circuit is called the potential difference across the external resistance. Alternatively the p.d. across the external circuit is the work done in carrying per unit positive test charge from one terminal to another in external circuit.

I.e.

$V = \dfrac{W_{ext}}{q_0}$

In general E > V : but E may be less than V, if a opposite current flows in the cell e.g. when the cell is being charged.

Internal Resistance of Cell (r): The internal resistance of a cell is the resistance offered by the solution of the cell between its electrodes. It is denoted by r.

The internal resistance of a cell:

(i) Varies directly as concentration of the solution of the cell.

(ii) Varies directly as the separation between electrodes i.e. length of solution between electrodes.

(iii) Varies inversely as the area of immersed electrodes.

(iv) is independent of the material of electrodes.

Terminal p.d. of a cell: If a cell of emf E internal resistance is giving current I in external resistance R, then its terminal p.d. V = E — Ir = IR and internal resistance of a cell

$r = ( \dfrac{E}{V} - I ) R$

### Combination of Resistances

There are two arrangements for connecting a number of resistances.

1. Series combination : In this arrangement the resistances are connected end to end in succession (Fig.). In this combination,

Series arrangement

(i) The current in each resistor is same.

(ii) The total p.d. V across the combination is equal to the sum of the p.d. across individual resistance.

I.e.

$V = v_1 + V_2 + V_3$

(iii) The equivalent or effective resistance (r) of the combination is equal to the sum of individual resistance.

I.e.

$R = R_1 + R_2 + R_3$

2. Parallel combination:  In this arrangement one each end of each resistor is connected at one point and the other end of each to other point. Then these two points are connected across a source p.d. v.

Parallel Combination

In this arrangement:

(i) The p.d. (v) across each resistor is the same.

(ii) The current is different in different resistances such that the total current flowing in the combination is shard by the individual resistances.

i.e.

$I = i_i + i_2 + i_3$

(iii) The equivalent or effective resistance (R) of the combination is given by:

$\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$

Or effective conductance K is the sum of inductance resistors.

i.e.

$K = K_1 + K_2 + K_3$

### Kirchhoff’s Laws

Kirchhoff in 1882 gave two laws which are used to solve the complicated problems.

1. First law: It states that “The algebraic sum of currents meeting at any junction is zero.”

I.e.

$\sum I= 0$

First law

Conventionally the currents towards the junction are taken as positive; while those directed away from the junction are taken as negative. Accordingly $i_1 \, \, and \, \, \, i_2$ are positive while $i_3 \, \, \, and\, \, \, i_4$ are negative for junction O.

$i_1 + i_2 - i_3 - i_4 =0$

2. Second law (or Loop Law): It states that “The algebraic sum of potential differences across each element of a closed circuit (or loop) is zero.

I.e.

$\sum V = 0$

Conventionally the potential fall is taken as negative; while potential rise is taken as negative. In a resistor the current flows from higher to lower potential; therefore potential difference across a resistor is taken as negative if we proceed in the direction of current and is taken as positive if we proceed opposite to direction of current.

Second law

If there is source of emf E, there is rise of potential from negative to positive terminal and hence is taken positive; while there is fall of potential from positive to negative terminal and hence is taken negative. For example we consider a simple circuit and proceed along path abcda. Let ‘I’ be the current in circuit. Then,

$- I R - I r + E = 0$

That is,

$E = I R + I r$

This law is based on conservation of energy.

### Wheatstone’s Bridge

The Wheatstone’s bridge is shown in figure P, Q, R and S are four resistances, G is galvanometer and E is a battery. The Wheatstone’s bridge is said to be balanced when no current flows in galvanometer i.e when potential of B = potential of D.

WheatStone Bridge

Condition of balance, $\dfrac{P}{Q} =\dfrac{R}{S}$

Remarks:

(i) The sensitivity of Wheatstone’s bridge is maximum when all the four resistances become equal.

I.e.

$P = Q = R = S$

(ii) If battery and galvanometer are interchanged, the balanced position of bridge remains unchanged while its sensitivity changes.

(iii) When Wheatstone’s bridge is balanced, the resistance in arm BD may be ignored while calculating the equivalent resistance of bridge between A and C.

(iv) To calculate the resistance between terminals B and D, the resistance of G is counted. In this case P and R are in series, Q and S are in series, while all the three arms (arm containing P, R), G, and (arm containing Q, G) are in parallel.

### Combination of Cells

There are three possible arrangement of a number of cells.

1. Series Arrangement: In this arrangement the positive terminal of one cell is connected to negative terminal of the other in succession. Figure represents n cells, each of e.m.f. E and internal resistance r connected in series and an external resistance R is connected across the combination

The net e.m.f = nE

Net internal resistance = nr

Total resistance of circuit = R + nr

Therefore,

$Current \, \, I = \dfrac{Net EMF}{Net \, \, Resistance}$ $= \dfrac{nE}{R + nr}$

Series Arrangement

If R >> nr, then $I = n \dfrac{E}{R} = n time \, \, current \, \, due \, \, to \, \, one \, \, cell$

Obviously for maximum current, the cells should be connected in series, when net external resistance >> net internal resistance.

2. Parallel Arrangement: In this arrangement the positive terminals of all cells are connected to one point and negative terminals to the other point. Figure represents m cells, each of e.m.f E and internal resistance r, connected in parallel and an external resistance R is connected across the combination.

Net e.m.f + E

Parallel Arrangement

Net internal resistance $R_{int} = \dfrac{r}{m}$

Total resistances of circuit = $R + \dfrac{r}{m}$

Therefore,

$Current I = \dfrac{E}{R} + ( \dfrac{r}{m} )$

Obviously for maximum current, the cells should be connected in parallel when net internal resistance >> net external resistance.

3. Mixed Grouping: In this arrangement the total number N of cells is divided into m groups in parallel and in each group n cells are connected in series. Fig. represents the mixed grouping of N = mn cells, having m rows of cells connected in parallel, each row containing n cells in series. The e.m.f. of each cell is E and internal resistance of each cell is r. The combination is connected to external resistance R.

Mixed Grouping

Net e.m.f = nE

Net internal resistamce $r_{int} = \dfrac{nr}{m}$

Therefore,

Net resistance of circuit = $R + \dfrac{nr}{m}$

Therefore,

$Current \, \, = \dfrac{n E}{R + ( \dfrac{nr}{m} ) } = \dfrac{m n r}{m R + n r}$

For maximum current:

$R = \dfrac{n r}{m}$ $r_{ext} = r_{int}$

Thus for maximum current, the cells should be connected in mixed grouping when external resistance R = net internal resistance.

I.e.

$R_{ext} = R_{int}$

Cells of different emf’s in parallel: If two cells of different emfs $E_1$ and $E_2$ and of different internal resistances $r_1$ and $r_2$ are connected in parallel, then the net effective emf,

$E = \dfrac{ \dfrac{E_1}{r_1} + \dfrac{E_2}{r_2}}{ \dfrac{1}{r_1} + \dfrac{1}{r_2}}$ and net internal resistance.

$r_{int} = \dfrac{r_1 r_2}{r_1 + r_2}$

### Potentiometer

Potentiometer is an ideal device to measure the p.d. between two points. It consists of a long resistance wire AB of uniform cross-section with a steady direct current set up in it by means of a main battery B. this maintains a uniform potential gradient along the length of the wire. If $V_{AB}$ is potential differences across wire AB and $r_{AB}$ is its total resistance, then potential gradient,

$K = \dfrac{V_{AB}}{R_{AB}} = I \rho = I \dfrac{R_{AB}}{L}$

Where $\rho= \dfrac{R_{AB}}{L}$ = resistance per unit length of potentiometer wire.

Potentiometer

If E is the e.m.f of source balanced between points A and C, then e.m.f of source,

E = K X length of AC = Kl

The potentiometer is said to act as ideal voltmeter since it is based on cancellation of currents.

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