Complete Current Electricity Summary

1. Electric Current

Flow of charge is called electric current and the rate of flow of charge in a circuit represents the magnitude of the electric current in that circuit.

\[I = \frac{dq}{dt}\]

Key Points about Electric Current

The direction of electric current is taken to be opposite to the direction of the flow of electrons
The direction of flow of positive charge in a conductor is taken as the direction of the electric current
Electric current is a scalar quantity
S.I. unit of electric current is coulomb/second or ampere

1.1 Drift Velocity of Free Electrons

The drift velocity is the velocity acquired by the electron in a direction opposite to applied electric field due to the acceleration of electron during the relaxation time.

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

Where:

\(v_d\) = drift velocity
\(a\) = acceleration
\(\tau\) = relaxation time
\(e\) = charge of electron
\(E\) = electric field
\(m\) = mass of electron

1.2 Relaxation Time

The average time taken between two successive collisions is called relaxation time. It is represented by \(\tau\).

2. Ohm's Law and Resistance

Ohm's Law

If the physical conditions of a conductor (such as temperature etc.) remain same, then the potential difference across the ends of the conductor is directly proportional to the current flowing in it.

\[V \propto I \quad \text{or} \quad V = IR\]

Where, R is a constant, called resistance of the conductor.

2.1 Resistance

The resistance of a conductor is directly proportional to its length \(l\) and inversely proportional to its cross-sectional area \(A\).

\[R = \frac{\rho l}{A}\]

Where:

\(R\) = resistance
\(\rho\) = resistivity of the material
\(l\) = length of conductor
\(A\) = cross-sectional area

Key Points:

The SI unit of resistance is ohm (\(\Omega\))
Resistivity (\(\rho\)) depends on the material of the conductor
Resistance depends on temperature
For most conductors, resistance increases with temperature

3. Circuits with Resistors

3.1 Resistors in Series

Two resistances are said to be in series if the same current passes through both resistors.

Series Combination: \(R_1\) — \(R_2\) — ... — \(R_N\)

\[R = R_1 + R_2 + \ldots + R_N = \sum_{i=1}^{N} R_i\]

Properties of Series Combination:

At every instant of time all circuit elements connected in a series combination carry the same current
The equivalent resistance of a series combination of resistances is always larger than any of the individual resistances
The voltage across each resistor is proportional to its resistance

3.2 Resistors in Parallel

Two resistors are said to be in parallel combination if the current splits but the voltage drop across each resistor is the same.

Parallel Combination: All resistors connected between the same two points

\[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_N} = \sum_{i=1}^{N} \frac{1}{R_i}\]

Properties of Parallel Combination:

At every instant of time the potential difference across all circuit elements connected in parallel is the same
The total resistance of any parallel combination of resistors must be less than any one of the individual resistors
The current through each resistor is inversely proportional to its resistance

3.3 Electromotive Force (EMF) and Internal Resistance

When a cell produces a current \(I\), the terminal potential difference is less than the EMF due to internal resistance.

\[V = E - Ir\]

Where:

\(V\) = terminal potential difference
\(E\) = EMF of the source
\(r\) = internal resistance of the source
\(I\) = current

Series Combination of Cells

\[E = E_1 + E_2 + E_3 + \ldots + E_N\] \[r = r_1 + r_2 + r_3 + \ldots + r_N\]

Parallel Combination of Cells

\[\frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \ldots + \frac{1}{r_N}\]

For identical cells: \(E_{eq} = E\) and \(r_{eq} = r/N\)

4. Kirchhoff's Laws

Kirchhoff's Laws are general and can be applied to any type of circuit.

4.1 Kirchhoff's Current Law (KCL)

At any circuit junction, the sum of the currents entering the junction must be equal to the sum of the currents leaving the junction.

\[\sum I_{in} = \sum I_{out}\]

Example: For a junction with four currents:

\[I_1 + I_3 = I_2 + I_4\]

Key Point:

KCL is based on the conservation of charge.

4.2 Kirchhoff's Voltage Law (KVL)

The sum of voltage changes across all the circuit elements found when traversing one direction around any closed circuit-loop must be zero.

\[\sum V = 0\]

Key Point:

KVL is based on the conservation of energy.

Application: When applying KVL:

Choose a direction to traverse the loop
Voltage drops across resistors: \(-IR\) (when moving with current) or \(+IR\) (when moving against current)
EMF sources: \(+E\) (when moving from - to + terminal) or \(-E\) (when moving from + to - terminal)

5. Electrical Instruments

5.1 Galvanometer

A galvanometer detects the presence of current in the branch where it is connected.

An instrument that measures current is called an ammeter
An instrument that measures potential difference is called a voltmeter

5.2 Conversion of Galvanometer to Ammeter

A galvanometer may be converted into an ammeter by connecting a low resistance (called shunt) in parallel to the galvanometer.

\[S = \frac{I_g G}{I - I_g}\]

Where:

\(S\) = shunt resistance
\(G\) = resistance of the galvanometer
\(I_g\) = full scale deflection current of the galvanometer
\(I\) = maximum current to be measured by the ammeter

5.3 Conversion of Galvanometer to Voltmeter

A galvanometer may be converted into a voltmeter by connecting a high resistance in series with the galvanometer.

\[R = \frac{V}{I_g} - G\]

Where:

\(R\) = series resistance
\(V\) = maximum voltage to be measured
\(I_g\) = full scale deflection current of the galvanometer
\(G\) = resistance of the galvanometer

5.4 Wheatstone Bridge

The Wheatstone bridge provides an accurate method of determining the resistance of an unknown resistor.

Wheatstone Bridge: Four resistors arranged in a diamond pattern with a galvanometer in the middle

When the bridge is balanced (no current flows through the galvanometer):

\[\frac{R_1}{R_2} = \frac{R_4}{R_3}\]

If \(R_3\) is the unknown resistance:

\[R_3 = R_2 \times \frac{R_4}{R_1}\]

5.5 Potentiometer

A potentiometer is an instrument that allows one to compare unknown EMF with a standard EMF. It is a null device, like the Wheatstone bridge.

Advantages of Potentiometer:

Measures the true EMF of a cell (unlike a voltmeter which measures terminal potential difference)
Does not draw any current from the cell when measuring EMF
More accurate than direct measurement with a voltmeter