Electric Current Notes

Electric Current

Defined as the rate of flow of electric charge through a cross-section of a conductor.

$I = \frac{dq}{dt}$

For steady current (does not change with time):

$I = \frac{q}{t}$

where:

• $q$ is the charge that flows across the cross-sectional area.

• $t$ is the time.

Nature of Current

Current is a scalar quantity because it does not follow the laws of vector addition.

Weightage

Maximum weightage is of topic 4: Drift of Electrons and the Origin of Resistivity.
Maximum MCQ and VSA type questions were asked from topic 7: Temperature Dependence of Resistivity.
Maximum SA type questions were asked from topic 4: Drift of Electrons and the Origin of Resistivity.

Topics Covered

1. Electric Current

2. Electric Currents in Conductors

3. Ohm's Law

4. Drift of Electrons and the Origin of Resistivity

5. Limitations of Ohm's Law

6. Resistivity of Various Materials

7. Temperature Dependence of Resistivity

8. Electrical Energy, Power

9. Cells, EMF, Internal Resistance

10. Cells in Series and in Parallel

11. Kirchhoff's Rules

12. Wheatstone Bridge

Units and Dimensions

Dimensional formula of current: $[M^0L^0T^0A^1]$
SI unit of current: Ampere (A)

Practical unit:

$1 A = 1 \frac{C}{s} = 6.25 \times 10^{18} \text{ electrons/s}$

Conventional Direction of Current

The direction of current is conventionally taken to be the direction of flow of positive charges.
Since electrons are negatively charged, their direction will be opposite to that of the conventional current flow.

Current Due to Moving Particles

If $n$ particles, each having a charge $q$, cross through a given area in time $t$, then:

$I = \frac{nq}{t}$

The current is the same for all cross-sections of a conductor of non-uniform cross-section.

Current in Different Situations

• Conductors and vacuum tubes: due to motion of electrons.

• Electrolytes: due to motion of both positive and negative ions.

• Semiconductors: due to motion of electrons and holes.

• Discharge tube (containing atomic gases): due to motion of positive ions and negative electrons.

Current Density

Defined as the amount of current flowing per unit area around that point of the conductor, provided the area is held in a direction normal to the current. Denoted by symbol $J$. SI unit is $A/m^2$.

$J = \frac{I}{A}$

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

$J = \frac{I}{A \cos \theta}$

$J \cdot A = I$

Current density is a vector quantity.
Dimensional formula: $[M^0L^{-2}T^0A^1]$

Drift Velocity

Defined as the average velocity with which free electrons get drifted towards the positive end of the conductor under the influence of an external electric field.

Drift velocity of electrons is given by:

$v_d = -\frac{eE}{m} \tau$

Where:

• $e$ is the charge on the electron.

• $m$ is the mass of the electron.

• $E$ is the electric field applied.

• $\tau$ is the time of relaxation.

Negative sign shows that drift velocity of electrons is in a direction opposite to that of the external electric field.

Drift velocity depends on electric field as $v_d \propto E$. So greater the electric field, larger will be the drift velocity.

Unit of drift velocity is $m/s$ and its dimensions is $[M^0LT^{-1}]$.

Relationship between current and drift velocity:

$I = nAev_d$

Where:

• $n$ is the number density of electrons (number of electrons per unit volume of the conductor).

• $A$ is the area of cross-section of the conductor.
  Where:

Relationship between current density and drift velocity:

$J = nev_d$

Mobility

Mobility: It is defined as the magnitude of drift velocity per unit electric field. It is denoted by symbol $\mu$.

$\mu = \frac{|v_d|}{E} = \frac{qE \tau}{mE} = \frac{q \tau}{m}$

Where $q, \tau$ and $m$ are charge, relaxation time and mass of a charge carrier respectively.
The SI unit of mobility is $m^2 V^{-1} s^{-1}$ and its dimensional formula is $[M^{-1}L^0T^2A^1]$.

Ohm's Law

It states that the current (I) flowing through a conductor is directly proportional to the potential difference (V) across the ends of the conductor, provided physical conditions of the conductor such as temperature and mechanical strain are kept constant.

$V \propto I$

$V = RI$

Where the constant of proportionality $R$ is called resistance of the conductor.

The graph between potential difference (V) and current (I) through a metallic conductor is a straight line passing through the origin.

$R = \frac{V}{I} = \frac{1}{\tan \theta} = \frac{1}{\text{slope of } I-V}$

Resistance

The resistance of a conductor is the obstruction posed by the conductor to the flow of current through it.
The SI unit of resistance is ohm ($\Omega$) and its dimensional formula is $[ML^2T^{-3}A^{-2}]$.

The resistance of a conductor is

$R = \frac{ml}{ne^2 \tau A} = \rho \frac{l}{A}$

Where:

• $m$ is the mass of the electron.

• $e$ is the charge of the electron.

• $n$ is the number density of electrons.

• $\tau$ is the relaxation time.

• $l$ is the length of the conductor.

• $A$ is its area of cross-section.

• $\rho$ is the specific resistance or resistivity of the conductor.

Resistivity

Resistivity: The specific resistance offered by the conductor of unit length and unit cross-section area. It is denoted by $\rho$.

The SI unit of resistivity is $\Omega \cdot m$ and its dimensional formula is $[ML^3T^{-3}A^{-2}]$.
If the conductor is in the form of wire of length $l$ and a radius $r$, then its resistance is

$R = \rho \frac{l}{\pi r^2}$

• If a conductor has mass $m$, volume $V$ and density $d$, then its resistance $R$ is

$R = \rho \frac{l}{A} = \rho \frac{l^2}{Al} = \rho \frac{l^2}{V} = \rho \frac{l^2 d}{m}$

• If length of a given metallic wire of resistance $R$ is stretched to $n$ times, its resistance becomes $n^2R$ but its resistivity remains unchanged.

• If radius of the given metallic wire of resistance $R$ becomes $n$ times, its resistance becomes $\frac{R}{n^4}$.

• If the area of cross-section of the given metallic wire of resistance $R$ becomes $n$ times, then its resistance becomes $\frac{R}{n^2}$.

Conductivity

Conductivity: The reciprocal of resistivity is known as conductivity or specific conductance. It is denoted by symbol $\sigma$.

$\sigma = \frac{1}{\rho} = \frac{ne^2 \tau}{m} = ne \mu = \frac{nev_d}{E}$

The SI unit of conductivity is $\Omega^{-1} m^{-1}$ or $S \cdot m^{-1}$ or mho $m^{-1}$ and its dimensional formula is $[M^{-1}L^{-3}T^3A^2]$.

Relationship Between $J, \sigma \text{ and } E$

$J = \sigma E$

It is a microscopic form of Ohm's law.

Ohmic and Non-Ohmic Conductors

Ohmic Conductors

Those conductors which obey Ohm's law are known as ohmic conductors, e.g., metals.
For ohmic conductors, the graph between current and potential difference is a straight line passing through the origin.

Non-Ohmic Conductors

Those conductors which do not obey Ohm's law are known as non-ohmic conductors, e.g., diode valve, junction diode.
For non-ohmic conductors, the graph between the current (I) and potential difference (V) has one or more of the following characteristics:

• The relation between $V$ and $I$ is non-linear.

• The relation between $V$ and $I$ depends on the sign of $V$.

• The relation between $V$ and $I$ is not unique, i.e., there is more than one value of $V$ for the same current $I$.

Effect of Temperature on Resistance and Resistivity

• The resistance of a metallic conductor increases with increase in temperature.
  The resistance of a conductor at temperature $t \, ^\circ C$ is given by

$R<em>t = R</em>0 (1 + \alpha t)$

Where:

• $R_t$ is the resistance at $t \, ^\circ C$.

• $R_0$ is the resistance at $0 \, ^\circ C$.

• $\alpha$ is the characteristics constants of the material of the conductor.

• Over a limited range of temperatures, that is not too large. The resistivity of a metallic conductor is approximately given by

$\rho<em>t = \rho</em>0 (1 + \alpha t)$

• Where $\alpha$ is the temperature coefficient of resistivity. Its unit is $K^{-1}$ or $\, ^\circ C^{-1}$.

• In the temperature range in which resistivity increases linearly with temperature, the temperature coefficient of resistivity $\alpha$ is defined as the fractional increase in resistivity per unit increase in temperature.

• For metals, $\alpha$ is positive i.e., resistance increases with rise in temperature.

• For insulators and semiconductors, $\alpha$ is negative i.e., resistance decreases with rise in temperature.

Cells, EMF, Internal Resistance

Electrochemical cell: It is a device which converts chemical energy into electrical energy and maintains the flow of charge in a circuit.

Electromotive Force (EMF) of a Cell

It is defined as the potential difference between the two terminals of a cell in an open circuit i.e., when no current flows through the cell. It is denoted by symbol $\varepsilon$.

The SI unit of EMF is Joule/Coulomb or Volt and its dimensional formula is $[ML^2T^{-3}A^{-1}]$.
The EMF of a cell depends upon the nature of electrodes, nature and the concentration of electrolyte used in the cell and its temperature.

Terminal Potential Difference

It is defined as the potential difference between two terminals of a cell in a closed circuit i.e., when current is flowing through the cell.

Internal Resistance of a Cell

It is defined as the resistance offered by the electrolyte and electrodes of a cell when the current flows through it.
The internal resistance of a cell depends upon the following factors:

• Distance between the electrodes

• The nature of the electrolyte

• The nature of electrodes

• Area of the electrodes, immersed in the electrolyte.

Relationship Between $\varepsilon, V \text{ and } r$

When a cell of EMF $\varepsilon$ and internal resistance $r$ is connected to an external resistance $R$, the voltage across $R$ is

$V = RI$

$\varepsilon = I(R + r)$

$I = \frac{\varepsilon}{R + r}$

$\varepsilon = IR + Ir$

$V = \varepsilon - Ir$

$r = R \left(\frac{\varepsilon}{V} - 1\right)$

• During discharging of a cell, terminal potential difference < EMF of a cell - voltage drop across the internal resistance of a cell, i.e., terminal potential difference across it is less than EMF of the cell. The direction of current inside the cell is from negative terminal to positive terminal.

• During charging of a cell, terminal potential difference > EMF of a cell + voltage drop across internal resistance of a cell, i.e., terminal potential difference becomes greater than the EMF of the cell. The direction of current inside the cell is from positive terminal to negative terminal.

Grouping of Cells

Cells can be grouped in the following three ways:

Series Grouping

If $n$ identical cells each of EMF $\varepsilon$ and internal resistance $r$ are connected to the external resistor of resistance $R$, they are said to be connected in series grouping.

$\varepsilon_{eq} = n \varepsilon$

$r_{eq} = nr$

Current in the circuit,

$I = \frac{n \varepsilon}{R + nr}$

Special Cases

• If R << nr, then $I = \frac{n \varepsilon}{nr} = \frac{\varepsilon}{r}$.

• If R >> nr, then $I = \frac{n \varepsilon}{R}$.

Parallel Grouping

If $m$ identical cells each of EMF $\varepsilon$ and internal resistance $r$ are connected to the external resistor of resistance $R$, they are said to be connected in parallel grouping.

$\varepsilon_{eq} = \varepsilon$

$r_{eq} = \frac{r}{m}$

The current in the circuit

$I = \frac{\varepsilon}{R + \frac{r}{m}}$

Special Cases

• If \frac{r}{m} << R, then $I = \frac{\varepsilon}{R}$.

• If \frac{r}{m} >> R, then $I = \frac{\varepsilon}{\frac{r}{m}} = \frac{m \varepsilon}{r}$.

Mixed Grouping

If the cells are connected as shown in figure they are said to be connected in mixed grouping. Let there be $n$ cells in series in one row and $m$ such rows of cells in parallel. Suppose all the cells are identical. Let each cell be of EMF $\varepsilon$ and internal resistance $r$.

$\varepsilon_{eq} = n \varepsilon$

$r_{eq} = \frac{nr}{m}$

$\therefore I = \frac{n \varepsilon}{R + \frac{nr}{m}}$

In the case of mixed grouping of cells, current in the circuit will be maximum, when

$R = \frac{nr}{m}$
i.e., external resistance = total internal resistance of all cells

Kirchhoff's Laws

Kirchhoff in 1842 put forward the following two laws to solve the complicated circuits. These two laws are stated as follows:

Kirchhoff's First Law / Junction Rule / Current Law

It states that the algebraic sum of the currents meeting at a junction is zero.
Kirchhoff's first law supports the law of conservation of charge.
According to sign convention the current flowing towards a junction is taken as positive and the current flowing away from the junction is taken as negative.

Kirchhoff's Second Law / Loop Rule / Voltage Law

It states that in a closed loop, the algebraic sum of the EMFs is equal to the algebraic sum of the products of the resistance and the respective currents flowing through them.

$\sum \varepsilon = \sum IR$

Kirchhoff's second law supports the law of conservation of energy.
According to sign convention while traversing a closed loop (in clockwise or anti-clockwise direction), if negative pole of the cell is encountered first then its EMF is positive, otherwise negative. The product of resistance and current in an arm of the circuit is taken positive if the direction of current in that arm is in the same sense as one moves in a closed loop and is taken negative if the direction of current in that arm is opposite to the sense as one moves in the closed loop.

Wheatstone Bridge

It is an arrangement of four resistances $P$, $Q$, $R$ and $S$ connected as shown in the figure. Their values are so adjusted that the galvanometer $G$ shows no deflection. The bridge is then said to be balanced. When this happens, the points $B$ and $D$ are at the same potential and it can be shown that

$\frac{P}{Q} = \frac{R}{S}$

This is called the balancing condition. If any three resistances are known, the fourth can be found.

Joule's Law of Heating

According to Joule's heating effect of current, the amount of heat produced ($H$) in a conductor of resistance $R$, carrying current $I$ for time $t$ is

$H = I^2Rt \text{ (in Joule)}$

$H = \frac{I^2Rt}{J} \text{ (in calorie)}$

Where $J$ is Joule's mechanical equivalent of heat ($J = 4.2 \text{ cal}$).

Electric Power

It is defined as the rate at which work is done by the source of EMF in maintaining the current in the electric circuit.

$P = \frac{\text{Electric work done}}{\text{time taken}} = VI = I^2R = \frac{V^2}{R}$

SI unit of power is Watt ($W$).
The practical unit of power is kilowatt ($kW$) and horse power ($hp$).

Dimension of power: $[ML^2T^{-3}A^0]$.
Power dissipated in connecting wires, which is wasted is $P = I^2 \cdot R<em>c = \frac{P^2}{V^2} \cdot R</em>c$
If $P<em>1, P</em>2, P_3, …$ are the powers of electric appliances in series with source of rated voltage $V$, the effective power consumed is

$\frac{1}{P<em>s} = \frac{1}{P</em>1} + \frac{1}{P<em>2} + \frac{1}{P</em>3} + …$

If $P<em>1, P</em>2, P_3$ are the powers of electric appliances in parallel with a source of rated voltage $V$, the effective power consumed is

$P<em>p = P</em>1 + P<em>2 + P</em>3 + …$

Electric Energy

It is defined as the total electric work done or energy supplied by the source of EMF in maintaining the current in an electric circuit for a given time.

Electric energy = electric power x time = $P \times t$.
The SI unit of electrical energy is Joule ($J$).
The commercial unit of electric energy is kilowatt-hour ($kWh$).

$1 kWh = 1000 Wh = 3.6 \times 10^6 J = \text{one unit of electricity consumed}$

The number of units of electricity consumed is:

$= \frac{\text{total wattage} \times \text{time in hour}}{1000}$

Wheatstone Bridge Application

It calculates the unknown resistance by balancing two legs of the bridge circuit.

$\frac{P}{R} = \frac{Q}{S}$