Revision: Current Electricity >> Current Electricity Physics Science (English Medium) Class 12 CBSE

A continuous and closed path of an electric current is called an electric circuit.

Substances whose resistance decreases tremendously with decreasing temperature and reaches nearly zero near absolute zero are called superconductors; e.g., lead, tin, etc.

 Semiconductors: Substances whose resistance decreases with the increase in temperature are named as semiconductors. E.g. manganin, constantan etc.

An electric current is measured by the amount of electric charge moving per unit time at any point in the circuit.

The magnitude of an electric current is the number of electric charges flowing through a conductor in one second.

Electromotive force: When no current is drawn from a cell, when the cell is in open circuit, the potential difference between the terminals of the cell is called its electromotive force (or e.m.f.).

The movement of the positive charge is called conventional current.

The unit of electric current is ampere (A). When 1 C of charge flows through a conductor in 1 s, it called 1 ampere (A) current.
I = `Q/t`

The resistivity of a material is the resistance of a wire of that material of unit length and unit area of cross-section.

Current is defined as the rate of flow of charge.

The reciprocal of resistance is called conductllnce. It is denoted by the letter G. 

One coulomb is the amount of electric charge transferred by a current of one ampere in one second.

One ohm is the resistance of a component when the potential difference of one volt applied across the component drives a current of one ampere through it.

The temperature coefficient is defined as the ratio of the increase in resistivity per degree rise in temperature to its resistivity at T₀.

Current density is a vector quantity, often known as an area vector or cross-sectional area vector, whose value is equal to the electric current flowing per unit area.

J = `"I"/"A"`

S.I unit is A/m².

In an electrical circuit, electric power is defined as the rate at which electrical energy is supplied by the source.

Electric power (P) is the rate at which electrical energy is transferred or consumed in an electrical circuit.

Internal Resistance is the resistance which is present within the battery that resists the current flow when connected to a circuit.

The potential gradient of a potentiometer wire is defined as the change in electric potential (voltage) per unit length of the wire.

Mathematically,

Potential Gradient = `V/L`

A potentiometer is a manually adjustable, variable resistor with three terminals. Two terminals are connected to the ends of a resistive element, and the third terminal is connected to an adjustable wiper. The position of the wiper sets the resistive divider ratio.

The reciprocal of specific resistance is known as conductivity.

Specific resistance of a material is the resistance of a wire of that material of unit length and unit area of cross section.

The charge flowing per second in an electric circuit is the measure of electric current in that circuit.

Mathematically,

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

• 1 ampere = 1 coulomb/second ⇒ 1A = 1Cs^-1
• 1 ampere = 6.25 x 10¹⁸ electrons per second
  

The average distance moved by a free electron between two successive collisions is called 'mean free path' of the electron.

The average time-interval between two successive collisions is called the 'relaxation time' of the electron.

If a small change ΔV in the potential difference across a part of a non-ohmic circuit causes a change ΔI in electric current, then the ratio ΔV/ΔI is called the 'dynamic resistance' of that part of the circuit.

Mathematically.
\[\frac {ΔV}{ΔI}\]

The ratio of the intensity of the electric field E at any point within the conductor and the current-density j at that point is called ‘specific resistance' or ‘electrical resistivity' of the conductor and is represented by ρ.

Mathematically,
ρ = \[\frac {E}{j}\]

Dimensions = [M L³ T^-3 A^-2]

The reciprocal of specific resistance is called 'specific conductance' and is represented by σ.

σ = \[\frac {1}{ρ}\]

SI unit = (ohm-metre)^-1 ⇒ (Ω-m)^-1
Dimension = [M^-1 L^-3 T³ A²]

Current density is defined as the current flowing through unit cross-sectional area drawn through that point perpendicular to the direction of flow of current.

Mathematically,
j = \[\frac {I}{A}\]

SI unit = ampere/metre² (A m^-2), Dimensions = [A L^-2].

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

OR

The average velocity with which free electrons drift opposite to the direction of the applied electric field.

The rate at which electric energy is transferred into other forms of energy is called ‘electric power’ P.

When two or more resistances connected between two points are replaced by a single resistance such that there is no change in the current of the circuit and the potential difference between those two points, the single resistance is called the equivalent resistance.

1 kilowatt-hour, or 1 unit, is the quantity of electric-energy which is dissipated in 1 hour in a circuit when the electric power in the circuit is 1 kilowatt.

The ratio of the potential difference to the current is called the ‘electric resistance’ R of the conductor.

Mathematically.
R = \[\frac {V}{I}\]

1 ohm = 1 volt/ampere ⇒ 1Ω = 1VA^-1
Dimensions =  [M L² T^-3A^-2]

The mobility of a free electron is numerically equal to the magnitude of drift velocity imparted by a uniform electric field of strength 1 V-m^-1.

SI unit: m²v^-1s^-1.

OR

The mobility m defined as the magnitude of the drift velocity per unit electric field:

μ = \[\frac {v_d}{E}\] = \[\frac {eτ}{m}\]

The potential difference between two points in an electric circuit is defined as the work done in carrying a unit charge from one point to the other.

It is an important instrument for measuring the emf of a cell or the potential difference between two points of an electric circuit.

An electric cell is a source of electrical energy which maintains a continuous flow of charge in a circuit.

The work done by the cell in forcing a unit positive charge to flow in the whole circuit (including the cell) is called the ‘electromotive force' (emf) of the cell.

Mathematically,
E = \[\frac {dW}{dq}\]

OR

The potential difference between the positive and negative terminals of a cell in an open circuit (when no current flows).

Metre bridge is a sensitive device based on the principle of Wheatstone's bridge, for the determination of the resistance of a conductor (wire).

If in the flow of 1 C of charge in a circuit, the work done by the cell be 1 J, then the emf of the cell is 1 V.

The terminal potential difference of a cell is equal to the work done for the flow of a unit charge in the external circuit only.

Mathematically,
V = \[\frac {W_{ext}}{q}\]

Electric Power P = \[\frac {W}{t}\] = VI = \[\frac {V^2}{R}\] = I²R

1 kW-h = 3.6 x 10⁶ W-s = 3.6 × 10⁶ J

Units = \[\frac {watt × hour}{1000}\]

v_d = \[-\frac{e\mathbf{E}}{m}\tau\]

P = \[\frac {W}{t}|] = V I

Power in a Resistor:
P = I²R and P = \[\frac {V^2}{R}\]

I = \[\frac{nE}{nr+R}\]

\[\frac{1}{r_{eq}}=\frac{1}{r_1}+\frac{1}{r_2}+\ldots\]

I = \[\frac{E}{\left(\frac{r}{n}+R\right)}=\frac{nE}{r+nR}\]

r = R\[\left[\frac{E}{V}-1\right]\]

Cells in Series:

r_eq​ = r₁​ + r₂​ + …

I = \[\frac{mnE}{nr+mR}\]

According to Ohm’s law, the current flowing in a conductor is directly proportional to the potential difference across its ends, provided the physical conditions and temperature of the conductor remain constant.
No, it is not always true. E.g., Diode valve, junction diode, etc., do not obey Ohm’s law.

Statement: Ohm’s Law

"The electric current flowing through a conductor is directly proportional to the potential difference across its ends, provided the temperature and other physical conditions of the conductor remain constant."

Mathematically,

I ∝ V or V = I R

where:

• V = Potential difference (in volts)
• I = Current (in amperes)
• R = Resistance of the conductor (in ohms, Ω)

Explanation:

When two conductors at different electric potentials are joined by a metallic wire, electrons flow from the conductor at a lower potential (excess electrons) to the one at a higher potential (deficit of electrons). This movement of electrons results in an electric current.

• The current continues to flow until both conductors reach the same potential.
• For continuous current flow, a constant potential difference must be maintained across the ends of the conductor (e.g., using a battery or power supply).

Derivation / Mathematical Proof:

From Ohm’s Law:

I ∝ V ⇒ \[\frac {V}{I}\] = constant

This constant is defined as the resistance (R) of the conductor. Therefore,

V = I R   ---(1)

This is the mathematical form of Ohm’s Law.

Special Case:

If the current I = 1 A, then:

V = R

This implies that the resistance of a conductor is numerically equal to the potential difference across it when 1 ampere of current flows through it.

Conclusion:

Ohm's Law provides a fundamental relationship between voltage, current, and resistance in an electric circuit. It is widely used in the design and analysis of electrical and electronic systems.

Let `I_3` and `I_4`  be the currents in resistors Q and S respectively . Let `I_g` be the current through galvanometer. For balanced condition, 

`I_g = 0`

Applying junction law at ‘b’ we get

`I_1 = I_3 + I_g`

`because I_g = 0 , I_1 = I_3`    ....(i)

Applying junction law at ‘d’, we get

`I_2 + I_g = I_4`

`because I_g = 0 , I_2 = I_4`    ....(ii)

Applying loop law in the loop abda, we get

`-I_1·P - I_g·Q + -I_2·R = 0`

⇒ `-I_1P + I_2R = 0`  (`because I_g = 0`)

⇒ `I_1P = I_2R`

⇒ `P/R = I_2/I_1`               ....(iii)

Applying loop law in the loop bcdb, we get

`-I_3·Q + I_4·S + I_g·6 = 0`

⇒ `-I_3·Q + I_4·S + 0 = 0  (because I_g =0)`

⇒ `-I_3Q = I_4S`

⇒ `Q/S = I_4/I_3`

⇒ `Q/S = I_2/I_1`             ...(iv) [using eq.(i) and (ii)]

From eq. (iii) and (iv), `P/ R = Q/s`

⇒ `P/Q = R/S`

This is the balanced condition. 

Statement

The variation of current with voltage is the macroscopic form of Ohm’s law. When the situation is considered at a point, the law is known as Ohm’s law in microscopic (vector) form.

Explanation/Proof

From, V = \[\frac{m}{ne^2\tau}\frac{l}{A}I\]

or

\[\frac{V}{l}=\left(\frac{m}{ne^{2}\tau}\right)\left(\frac{I}{A}\right)\]

But,

\[\frac {V}{l}\] = E, \[\frac {m}{n e^2 τ}\] = ρ and \[\frac {I}{A}\] = j,

\[\therefore\] E = ρ j

Also, ρ = \[\frac {1}{σ}\]​

Hence,

E = \[\frac {1}{σ}\]j or j = σ E

In vector notation,

\[\vec j\] = σ\[\vec E\]

Conclusion

Therefore, for an isotropic substance,

\[\vec j\] ∝ \[\vec E\]

and Ohm’s law in vector form states that the current density is directly proportional to the applied electric field strength, and the ratio of current density to electric field is a constant σ, independent of the electric field producing the current.

Statement

When a Wheatstone bridge is balanced, that is, when there is no deflection in the galvanometer, the ratio of the resistances of any two adjacent arms is equal to the ratio of the resistances of the remaining two adjacent arms, i.e.,
\[\frac {P}{Q}\] = \[\frac {R}{S}\]

Explanation/Proof

In a Wheatstone bridge, four resistances P, Q, R, and S are connected to form the four arms of a parallelogram. A galvanometer is connected across one diagonal and a cell across the other diagonal.

When the key is pressed, the current entering the junction is divided into two parts: current I₁​ flows through arm AB, and current I₂​ flows through arm AD. The resistances are adjusted such that there is no current through the galvanometer, indicating that the bridge is balanced.

Since there is no current in the diagonal BD, the same current I₁ flows through arms AB and BC, and the same current I₂​ flows through arms AD and DC.

Applying Kirchhoff’s second law to loop ABDA,

I₁P − I₂R = 0 ⇒ I₁P = I₂R   ---(i)

Applying Kirchhoff’s second law to loop BCDB,

I₁Q − I₂S = 0 ⇒ I₁Q = I₂S   ---(ii)

Dividing equation (i) by equation (ii),

\[\frac {P}{Q}\] = \[\frac {R}{S}\]

Conclusion​

Thus, when a Wheatstone bridge is balanced, the ratio of the resistances in one pair of adjacent arms is equal to the ratio of the resistances in the other pair of adjacent arms. This principle is used to determine an unknown resistance accurately.

Statement

In an electric circuit, the 'algebraic' sum of the currents meeting at any junction in the circuit is zero, that is, ∑ I = 0.

Proof

When applying this law, currents entering the junction are taken as positive, while currents leaving the junction are taken as negative.

Consider a junction O where five conductors meet, carrying currents I₁, I₂, I₃, I_4​ and I₅.
Let I₁ and I₂ enter the junction, and I₃, I_4​ and I_5​ leave the junction.

According to Kirchhoff’s first law,

∑ I = 0

That is,

I₁ + I₂ − I₃ − I₄ − I₅ = 0

or,

I₁ + I₂ = I₃ + I₄ + I₅​

Thus, the sum of incoming currents is equal to the sum of outgoing currents.

Conclusion

Kirchhoff’s first law, also called Kirchhoff’s current law (KCL), states that when a steady current flows in a circuit, no charge accumulates at any junction. Hence, the law is a direct consequence of the principle of conservation of electric charge.

Statement

In any closed loop of a circuit, the algebraic sum of the products of current and the resistance in each part of the loop is equal to the algebraic sum of the emfs in that loop, that is,
∑ IR = ∑ E

Proof

While applying Kirchhoff’s second law, the following sign conventions are used:

1. When we traverse a resistance in the direction of current, the product I R is taken as positive.
2. The emf is taken as positive when we traverse from the negative to the positive electrode of the cell through the electrolyte.

Consider the circuit shown, containing two cells of emfs E₁​ and E₂​ and three resistances R₁, R_2​, and R₃​. Let the currents in R_1​ and R₂ be I₁​ and I₂, respectively. Applying Kirchhoff’s first law at junction A, the current through R₃​ is I₁ + I₂.

The circuit has two closed loops.

For loop 1, applying Kirchhoff’s second law:

I₁R₁ − I₂R₂ = E₁ − E₂​

For loop 2, applying Kirchhoff’s second law:

I₂R₂ + (I₁ + I₂)R₃ = E₂

Conclusion​

Kirchhoff’s second law, also called Kirchhoff’s Voltage Law (KVL), is a direct statement of the conservation of energy. Using the equations obtained from this law, the currents in different parts of the circuit can be determined.

• Electricity is a convenient and controllable form of energy widely used in homes, industries, schools, and hospitals.
• Electric current is produced when electric charges flow through a conductor, and it flows only through a closed, continuous electric circuit.
• A switch completes or breaks the circuit; when the circuit is broken, current stops flowing, and devices like bulbs do not glow.
• Electric current is the rate of flow of charge, given by the relation I = Q / t, where Q is charge and t is time.
• In metallic wires, electrons are the charge carriers, but by convention, current flows from the positive to the negative terminal, in the opposite direction to electron flow.

• Electrical power represents the rate at which electrical energy is supplied by the source in an electric circuit.
• The S.I. unit of electrical power is a watt (W), and larger units such as kilowatt, megawatt, and gigawatt are used for measuring higher power.

• Specific resistance is a characteristic property of a substance and differs among metals, semiconductors, and insulators.
• Specific resistance depends on temperature: it increases with temperature for metals and decreases with temperature for semiconductors, while it remains nearly constant for some alloys.
• Specific resistance does not depend on the shape and size of the conductor and remains unchanged when a wire is stretched or doubled.

• Carbon resistors use colour codes to indicate resistance value; the first two bands give significant figures and the third band gives the multiplying power of 10.
• The fourth colour band indicates the resistor tolerance: gold (±5%), silver (±10%), and no band (±20%).
• The colour sequence Black to White represents digits 0 to 9, and the same colours in the third band represent multipliers 10⁰ to 10⁹.

• Series combination: Same current flows through all resistances, and the equivalent resistance is
  R = R₁ + R₂ + R₃​
• Series property: In a series, the equivalent resistance is greater than the largest individual resistance, and the voltage divides in the ratio of resistances.
• Parallel combination: Same potential difference exists across all resistances and the equivalent resistance satisfies
  ​\[\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\]
• Parallel property: In parallel, the equivalent resistance is less than the smallest individual resistance, and current divides inversely with resistance.
• Practical use: Household electrical appliances are connected in parallel, so each works independently at the same voltage.

• Series combination: The net power consumed decreases; for identical bulbs,
  P_consumed = \[\frac {P}{n}\]​
  and it is directly proportional to bulb resistance and inversely proportional to rated power.
• Parallel combination: The net power consumed increases; for identical bulbs,
  P_consumed = n P
  and it is inversely proportional to bulb resistance and directly proportional to rated power.

• Ohm’s law does not hold when temperature changes due to current flow, causing resistance to vary (e.g., filament bulb).
• In some materials, current starts flowing only after a minimum applied voltage, so the V–I graph is not linear.
• Devices like diodes, thermistors, and vacuum tubes are non-ohmic because their resistance is not constant

• Ohm’s law is not valid for all materials; in some devices, voltage is not proportional to current.
• In certain materials (like diodes), reversing the voltage does not produce equal current in the opposite direction.
• Some materials show non-unique V–I characteristics, meaning more than one voltage value may correspond to the same current.

• Metals: Resistivity increases with a rise in temperature due to increased electron collisions.
• Temperature coefficient: For metals, resistance varies with temperature as
  R_t = R₀(1 + αt),
  and for most metals, α ≈ \[\frac {1}{273}\] per °C, so R ∝ T (approximately).
• Alloys: The resistivity of alloys changes very little with temperature and remains relatively high.
• Semiconductors: Resistivity decreases with an increase in temperature due to an increase in charge carriers.
• Electrolytes: Resistivity decreases with a rise in temperature because ions move more freely.

• Null-deflection method: At balance, no current flows through the galvanometer, making the measurement independent of the cell's internal resistance.
• Uniform wire requirement: The potentiometer wire must have a uniform cross-section and material so that the potential drop along the wire is uniform.
• True emf measurement: The emf is measured in open circuit, ensuring the true value of the emf is obtained without energy loss in the cell.
• Sensitivity dependence: The sensitivity of a potentiometer increases as the potential gradient decreases, using a long wire and low current.
• Experimental precautions: Current should not flow for a long time to avoid heating of the wire, and touch the jockey lightly to prevent wire damage.

• Principle: The metre bridge works on the Wheatstone bridge principle, and balance is obtained at the null point where the galvanometer shows no deflection.
• Null point condition: At the null point, points B and D are at the same potential and
  \[\frac {P}{Q}\] = \[\frac {R}{S}\]
• Finding unknown resistance: If the wire is divided into lengths l and 100 − l, the unknown resistance is
  S = R\[\frac {(100−l)}{l}\].
• Reducing errors: Errors are reduced by interchanging the known and unknown resistances and taking the mean value.
• Precautions: Keep the null point near the middle, avoid heating the wire, and press the jockey lightly without rubbing.

• Battery and the need for a combination of cells
  A single cell cannot give a strong current, so two or more cells are combined to form a battery to obtain a suitable current or emf.
• A series combination of cells
  In a series combination, emfs and internal resistances add up. It is useful when the external resistance is much larger than the internal resistance.
• Parallel combination of cells
  In parallel combination, the emf remains the same as one cell, but the internal resistance decreases. It is useful when the external resistance is small.
• Mixed grouping of cells
  In mixed grouping, cells are connected in series and parallel to obtain both a large current and a suitable emf.
  I_max = \[\frac {nE}{2R}\]
• Condition for maximum current in mixed grouping
  Maximum current flows when the battery's internal resistance equals the external resistance. i.e. \[\frac {nr}{m}\] = R

• Purpose of a rheostat: A rheostat is used to control the current in an electric circuit by changing resistance.
• Construction: It has a Nichrome wire wound on a china-clay cylinder with a sliding contact.
• As a current controller: When connected through A–C or B–C, moving the sliding contact changes the current in the circuit.
• As a potential divider: When connected across A and B, and the circuit is taken from A–C (or B–C), the rheostat provides a variable fraction of the applied potential difference.
• Working principle: Sliding the contact changes the wire's effective length, thereby changing its resistance.