Revision: Current Electricity >> Electric Resistance and Ohm's Law Physics (Theory) ISC (Science) ISC Class 12 CISCE

Current is defined as the rate of flow of charge.

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

The movement of the positive charge is called conventional current.

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.

The unit of electric current is ampere (A). When one coulomb charge flows through an electric circuit in one second, then the electric current flowing through the circuit is said to be an ampere. 

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.

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.).

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

The current density of a conductor is defined as the amount of current passing per unit area of the conductor held perpendicular to the flow of charge.

\[J=\frac{I}{A}\]

• SI unit is A m⁻²
• Dimensional formula = [M⁰ L⁻² T⁰ A¹]

A variable resistor has a resistance that can be varied. It is used to vary the amount of current flowing in a circuit.

A fixed resistor has a resistance of a fixed value. Common types of fixed resistors include carbon film resistors and wire-wound resistors.

Resistance is the obstacle that the wire presents to the current flow.

The resistance of a conductor is defined as the ratio of the potential difference V across the conductor to the current I flowing through it.

• S.I. unit of resistance is ohm (Ω)
• Dimensional formula: [M L² T⁻³ A⁻²]

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.

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

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

At constant temperature and other physical conditions, the current flowing through a conductor is directly proportional to the potential difference across its ends.

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.

Mobility is the magnitude of drift velocity per unit electric field.

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

S.I. Unit of resistivity is ohm-metre, i.e., Ω·m.

\[\rho=R\left(\frac{A}{l}\right)\]

Electromotive force (e) is the energy provided by a cell or battery per coulomb of charge passing through it.

\[e=\frac{E}{Q}\]

It is measured in volt (V).

The resistance offered by the electrolyte inside the cell, to the flow of current, is called the internal resistance of the cell.

An instrument used to measure the potential difference between two points in an electrical circuit, always connected in parallel with the component across which the voltage drop is to be measured, is called a voltmeter.

The condition of the Wheatstone bridge under which the galvanometer shows zero (null) deflection, i.e., I_g = 0, is called the balance condition of the bridge.

An arrangement of four resistors used to measure the resistance of one of them in terms of the other three, invented by Samuel Hunter Christie in 1833 and made famous by Sir Charles Wheatstone, is called a Wheatstone bridge.

A device, based on the Wheatstone bridge principle, which is used to measure the resistance of an unknown wire (conductor) with good accuracy is called a meter bridge (slide wire bridge).

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.

An ideal apparatus of infinite resistance, based on the null deflection method, which is used to measure unknown potential differences accurately without drawing any current from the circuit, is called a potentiometer.

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 = \[\frac {W_{ext}}{q}\]

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.

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]

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]

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.
\[\frac {ΔV}{ΔI}\]

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

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].

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

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 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.

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.

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

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

Where:

• I = electric current
• Q = charge flowing through the conductor
• t = time taken

SI unit of current = ampere (A).

V ∝ I

V = IR

Other useful forms: I = \[\frac {V}{R}\] or R = \[\frac {V}{I}\]

μ = \[\frac {∣v_d​∣}{E}\]​

where:

Balance condition (when I_g = 0):

Based on Wheatstone bridge principle:

R = S\[\left(\frac{l_1}{100-l_1}\right)\]

where R = unknown resistance, S = known resistance, l₁​ = distance of null point from the first end.

r = \[\left(\frac{l_1-l_2}{l_2}\right)\]R

\[\frac{E_1}{E_2}=\frac{l_1}{l_2}\]

\[\frac{E_1+E_2}{E_1-E_2}=\frac{l_1+l_2}{l_1-l_2}\]

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

1 kW-h = 3.6 x 10⁶ W-s = 3.6 × 10⁶ J

Units = \[\frac {watt × hour}{1000}\]

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

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.

Statement

At any junction in an electric circuit, the sum of currents entering the junction is equal to the sum of currents leaving the junction. 

Derivation

When the current in a circuit is steady, charge does not accumulate at any junction. Therefore, the amount of charge entering the junction per second must be equal to the amount of charge leaving the junction per second. 

If currents I₁​ and I₂ enter a junction and currents I₃​ and I₄​ leave it, then

or

Hence,

Conclusion

Kirchhoff's First Law is a direct consequence of the conservation of charge. 

Statement

In any closed loop of an electric circuit, the algebraic sum of all changes in potential is zero. 

Derivation

Consider a charge moving around a closed loop. After completing one full loop, the charge returns to its starting point. Since electric potential depends only on position, the net change in potential over a complete loop must be zero. 

Therefore, in a closed loop,

If a loop contains cells and resistors, then the total emf supplied by the sources is equal to the total potential drop across the resistors. Thus,

Conclusion

Kirchhoff's Second Law is a direct consequence of the conservation of energy.

At any junction, the sum of currents entering = the sum of currents leaving.

Example: I₁ + I₃ = I₂ + I₄​. Based on conservation of charge.

The algebraic sum of potential differences in a closed loop is zero.

Based on conservation of energy.

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. 

 V ∝ L ⇒ V = xL

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.

• 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.

• Free electrons in a metal move randomly; without a potential difference, there is no net flow of current.
• When a potential difference is applied, electrons drift towards the positive terminal, but collide with fixed positive ions, losing energy.
• These collisions cause resistance, and the number of collisions determines the amount of resistance in the conductor.

• 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.

• Kirchhoff's laws are used for complex circuits. 
• Kirchhoff's First Law: Total current entering a junction = total current leaving a junction. 
• Kirchhoff's Second Law: Total potential rise in a closed loop = total potential drop in the loop. 
• KCL is based on conservation of charge. 
• KVL is based on conservation of energy. 
• Mathematical forms are ∑I = 0 and ∑V = 0. 
• The correct sign convention is essential in numericals. 

• 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.

• 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.

• 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.

• 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.

• 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.

• 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