Revision: Current Electricity CUET (UG) Current Electricity

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.).

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.

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.

Current is defined as the rate of flow of charge.

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

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

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. 

The movement of the positive charge is called conventional current.

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.

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 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.

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

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 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.

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).

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.

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`

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

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.

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

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

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

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. 

 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:
  \[I=\frac{nE}{nr+R}\]