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

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)

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 \[\mathrm{ms^{-1}N^{-1}C}\]

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 \[v_d=\frac{eE\tau}{m}\]

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.

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

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.

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 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 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 = \[3.6\times10^6\mathrm{~J}\])

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

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

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]

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]

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

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.

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.

\[V=E-Ir\]

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

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 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).

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

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.

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.

Junction Law or Current Law:

It states that the sum of the currents flowing into a junction is equal to the sum of the currents flowing out of the junction.

At Junction A: Incoming current = outgoing current

I₁ + I₂ = I₃ + I₄ or I₁ + I₂ − I₃ − I₄ = 0

∑I = 0

Loop Law or Potential Law:

Kirchhoff’s second law states that the algebraic sum of changes in potential around any closed loop is zero.

• Kirchhoff’s second law can be expressed as ∑V = 0

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:
  \[I=\frac{E}{R+r_{net}}=\frac{E}{R+\frac{r}{n}}=\frac{nE}{nR+r}\]