Unit 16: Electricity

Electric Current: Amount of charge passing through any cross section of a conductor per unit time is called electric current.

If \(Q\) charge flows in time \(t\), then current \(I\) can be expressed as:
\[I = \frac{Q}{t}\]
Unit: In SI the unit of current is Ampere \((A)\).

Ampere: One ampere is defined as the flow of one coulomb of electric charge per second, which can be written as;
\[1A = 1Cs^{-1}\]

Conventional Current: Before the discovery of electrons, electric current was considered to be due to the flow of positive charges from the positive terminal to the negative terminal.

Electronic Current: After the discovery of electrons, electric current was explained as the movement of electrons from the negative terminal to the positive terminal.

Measurement of Electric Current: An ammeter is a device used to measure the current flowing through a circuit. It must be connected in series with the component whose current is to be measured.

Electromotive Force (e.m.f.): Electromotive force is the electric potential energy supplied by a battery to a unit positive charge when it flows through the closed circuit.
\[e.m.f = \frac{Energy}{Charge}\]
\[E = \frac{W}{Q}\]
where \(E\) is the \(e.m.f.\), \(W\) is the energy supplied, and \(Q\) is the charge.

Potential Difference: Electric potential \((V)\) is the amount of electric potential energy \((W)\) per unit charge \((Q)\) at a specific point in an electric field.
\[V = \frac{W}{Q}\]
The potential difference is the difference in electric potential between two points in an electric field. Its SI unit is volt \((V)\), where:
\[1V = 1JC^{-1}\]

How does a battery create a potential difference? A battery converts chemical energy into electrical energy. A chemical reaction inside the battery causes a buildup of electrons at the negative terminal and a shortage of electrons at the positive terminal. This imbalance creates a potential difference between the terminals, which causes charges to flow through a connected circuit.

Role of e.m.f. in maintaining the flow of current: The e.m.f. supplies the energy needed to keep the charges moving. After the charges reach the negative terminal, the battery does work to move them back to the positive terminal. This continuous supply of energy maintains the flow of electric current in the circuit.

Ohm's Law: The current flowing through a conductor is directly proportional to the potential difference \(V\) across its two ends of a conductor, provided the physical state (dimension, temperature, etc.) of the conductor remains same. Mathematically
\[I \propto V\]
\[V \propto I\]
\[V = IR\]
where \(R\) is the constant of proportionality, and is the resistance of the conductors.

The SI unit of resistance is the ohm \((\Omega)\).
\[1\Omega = 1V A^{-1}\]

Effect of Resistance on Current: Resistance is the opposition to the flow of electric current. Higher resistance allows less current to flow for a given potential difference, while lower resistance allows more current to flow.

Applications of Ohm's Law: Ohm's law is used in practical electrical circuits to:
i. Calculate the current when voltage and resistance are known.
ii. Determine the voltage when current and resistance are known.
iii. Find the resistance when voltage and current are known.
iv. Design and analyze electrical circuits by selecting suitable resistor values.

Series Combination of Resistors: When resistors are connected in such a way that there is only one path for the current to flow, then it is called series combination of resistors.

Explanation: Consider three resistors \(R_{1}, R_{2}\) and \(R_{3}\) connected in series. When this combination is connected to a battery of \(V\) volts, it draws current \(I\) from the battery.

Let \(R_{e}\) be a single resistor representing the total resistance of the circuit. This resistor is called the equivalent resistor, and its resistance is called the equivalent resistance.

The total voltage in the circuit will be the sum of voltage across the three resistors.
\[V = V_{1} + V_{2} + V_{3}\]
Applying Ohm's law:
\[V_{1} = IR_{1}, V_{2} = IR_{2} \text{ and } V_{3} = IR_{3}\]
\[I R_{e} = I R_{1} + I R_{2} + I R_{3}\]
\[I R_{e} = I(R_{1} + R_{2} + R_{3})\]
\[R_{e} = R_{1} + R_{2} + R_{3}\]
Thus, equivalent resistance is equal to the sum of individual resistances.

Parallel Combination of Resistors: When there are multiple paths for current flow in a circuit, the combination of resistances is referred to as parallel combination.

Explanation: Consider three resistors \(R_{1}, R_{2}\) and \(R_{3}\) connected in parallel. When these are connected to a battery of \(V\) volts, it draws a current \(I\) from the battery.

Let \(R_{e}\) be a single resistor that represents the total resistance of the parallel combination. When connected to the same battery of \(V\) volts, it draws the same current \(I\) from the battery as the entire parallel combination. Therefore, \(R_{e}\) is called the equivalent resistor, and its resistance is called the equivalent resistance.

The total current in the circuit will be the sum of currents flowing through the resistors:
\[I = I_{1} + I_{2} + I_{3}\]
Applying Ohm's law:
\[V = I_{1}R_{1} \Rightarrow I_{1} = \frac{V}{R_{1}}\]
\[V = I_{2}R_{2} \Rightarrow I_{2} = \frac{V}{R_{2}}\]
\[V = I_{3}R_{3} \Rightarrow I_{3} = \frac{V}{R_{3}}\]
Putting values:
\[\frac{V}{R_{e}} = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}\]
\[\frac{V}{R_{e}} = V\left[\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}\right]\]
\[\frac{1}{R_{e}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}\]
Thus, the reciprocal of equivalent resistance is equal to the sum of reciprocals of individual resistances.

Resistivity: Electrical resistivity \((\rho)\) is a property of a material that indicates how strongly it resists the flow of electric current. It is also defined as the resistance of a cubic metre of a material.

The relation between resistance and resistivity is:
\[R = \rho \frac{L}{A}\]
where \(R\) is the resistance, \(L\) is the length of the conductor, \(A\) is its cross-sectional area, and \(\rho\) is the resistivity.

The SI unit of resistivity is ohm-metre \((\Omega m)\).

Dependence on Material: The value of resistivity depends entirely on the nature of the material. Materials with low resistivity are good conductors, whereas materials with high resistivity are poor conductors (insulators).

Role of Temperature:
• Conductors: The resistance and resistivity increase with an increase in temperature because the increased vibration of atoms makes it more difficult for electrons to flow.
• Semiconductors: The resistance and resistivity decrease as the temperature increases because more free charge carriers are produced, increasing conductivity.