Revision: Current Electricity CUET (UG) Current Electricity

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 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. 

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

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.

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.

Conductivity, denoted by σσ, measures how easily current flows through a material. From the microscopic model, conductivity depends on free electron density and relaxation time.

The average velocity acquired by free electrons in a conductor under the influence of an electric field is called the drift velocity.

The average time interval between two successive collisions of a free electron with the ions of the metallic lattice is called the relaxation time and is denoted by τ.

Current density is the current flowing per unit cross-sectional area of the conductor. The source material connects current density with the drift motion of electrons.

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.

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

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

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

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

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.

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

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 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

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

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.

\[V=E-Ir\]

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).

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 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`

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.

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

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

Where:

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

SI unit of current = ampere (A).

Using the average time between collisions ττ, the source derives the drift velocity as: 

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.

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

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

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

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.

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

Based on 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.

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.

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

• 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.

• 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.

• 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

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

• 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:
  \[I=\frac{nE}{nr+R}\]

• 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.