Revision: Class 12 >> Principles of Electrical Circuits and their Applications NEET (UG) Principles of Electrical Circuits and their Applications

Electric current through a conductor is said to be one ampere if charge of one coulomb flows through any cross-section of the conductor in one second.

\[1\mathrm{~ampere~(A)}=\frac{1\mathrm{~coulomb~(C)}}{1\mathrm{~second~(s)}}=1\mathrm{~C~s}^{-1}\]

It is defined as the velocity with which the free electrons are drifted towards the positive terminal under the effect of the applied electric field.

• The drift velocity of electrons is given by \[v_d=\frac{eE\tau}{m}\]

It is defined as the magnitude of the drift velocity of the charge carriers per unit electric field.

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

The S.I. unit of mobility is \[\mathrm{m^{2}s^{-1}~V^{-1}}\] or \[\mathrm{ms^{-1}N^{-1}C}\]

Electric current is defined as the amount of electric charges flowing through any cross-section of a conductor per unit time.

\[I=\frac{\text{Total charge flowing (Q)}}{\text{Time taken (t)}}\]

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

The S.I. unit of current is ampere (A)

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.

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

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

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.

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

The conductors which do not obey Ohm's law are called non-ohmic resistors (or non-linear resistances).

i.e., Current depends on voltage, but it does not vary linearly.

The conductors which obey Ohm's law are called ohmic resistors (or linear resistances).

i.e., Voltage and current vary linearly in ohmic conductors/materials.

The electrical energy consumed in a circuit is defined as the total work done in maintaining the current in the electric circuit for a given time.

Electrical Energy = \[VIt=I^2Rt=\frac{V^2t}{R}\]

S.I. unit of electric energy is joule (1 kWh = \[3.6\times10^6\mathrm{~J}\])

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.

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.

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

Electrical conductivity, or conductivity of a substance, is equal to the inverse of its resistivity.

\[\sigma=\frac{1}{\rho}\]

• SI unit: S m⁻¹
• Dimensions: [M⁻¹ L⁻³ T³ A²]

Conductance of a substance is equal to the inverse of its resistance.

\[G=\frac{1}{R}\]

• S.I. unit: ohm⁻¹ or mho or siemens (S).
• Dimensions: [M⁻¹ L⁻² T³ A²]

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]

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]

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

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

The temperature coefficient of resistivity, denoted by α, measures the fractional change in resistivity per degree change in temperature in the linear range.

• Unit: per degree Celsius or per kelvin.
• For metals, α > 0.
• For semiconductors, α < 0.

Resistivity, denoted by ρ, is the intrinsic property of a material that determines how much it resists current flow.

The resistance offered by the electrolyte of the cell when an electric current flows through it is known as internal resistance.

The emf of a cell is defined as the work done in carrying a unit positive charge through the complete circuit, including the charge flow inside the cell.

Unit: J/C (or) volt

When current is drawn through a cell or current is supplied to it, then the potential difference across its terminals is called the terminal potential difference.

\[V=E-Ir\]

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).

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.

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.

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.

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).

V ∝ I

V = IR

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

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

ρ_T ​= ρ₀​[1 + α(T − T₀​)]

Here:

• ρ_T​ = resistivity at temperature T.
• ρ₀​ = resistivity at reference temperature T₀.
• α = temperature coefficient of resistivity.

R_T ​= R₀​(1 + αΔT)

where ΔT = T − T₀​.

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.

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.

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.

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

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

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

Based on conservation of energy.

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. 

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. 

• 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.

• 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.

Resistivity and Temperature:

\[\rho_T=\rho_0[1+\alpha(T-T_0)]\]

Resistance and Temperature:

\[R_T=R_0(1+\alpha\Delta T)\]

Temperature Coefficient (α):

• Unit: °C⁻¹ (or K⁻¹)
• Metals: α > 0→ resistivity increases with temperature

Semiconductors & insulators:

α < 0 → resistivity decreases with temperature

• In series, resistors are connected one after another (in a single path).
• Current is the same through all resistors.

Equivalent resistance:
R_eq = R₁ + R₂ + R₃ + ...

For n identical resistors:
R_eq = nR

Voltage relation:
V = V₁ + V₂ + V₃

Voltage divider rule:
V₁ : V₂ : V₃ = R₁ : R₂ : R₃

R_eq > R_max

• In parallel, resistors are connected across the same two points (multiple paths).
• Voltage is the same across all resistors.

Equivalent resistance:
\[\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdots\]

For n identical resistors:
Req = R/n

Current relation:
I = I₁ + I₂ + I₃

Current divider rule:
I₁ : I₂ : I₃ = \[\frac{1}{R_{1}}:\frac{1}{R_{2}}:\frac{1}{R_{3}}\]

R_eq < R_min

• Cells are connected from the positive terminal to the negative terminal.
• Total emf is the sum of individual emfs:
  E_net = E₁ + E₂ + E₃ + ...
• Total internal resistance:
  rnet = r₁ + r₂ + r₃ + ...
• For n identical cells:
  E_net = nE
  r_net = nr
• Current in the circuit:
  \[I=\frac{E_{\mathrm{net}}}{r_{\mathrm{net}}+R}\]
• For identical cells:
  \[I=\frac{nE}{nr+R}\]

• All positive terminals are connected together, and all negative terminals are connected together.
• Total emf remains the same:
  E_net = E
• Internal resistance reduces:
  \[r_{net}=\frac{r}{n}\]
• Current in the circuit:
  I = E / (R + rnet)
• For identical cells:
  \[I=\frac{E}{R+r_{net}}=\frac{E}{R+\frac{r}{n}}=\frac{nE}{nR+r}\]

• 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.