Comprehensive Study Notes on Electricity and Electric Current and Circuits

Electric Current and Circuit

• Electricity is a controllable and convenient form of energy used in homes, schools, hospitals, and industries.

• Flowing water constitutes a water current in rivers. Similarly, if electric charge flows through a conductor, such as a metallic wire, it constitutes an electric current.

• In a torch, the cells or a battery provide the flow of charges to make the bulb glow.

• A switch serves as a conducting link between the cell and the bulb.

• A continuous and closed path of an electric current is defined as an electric circuit. If the circuit is broken or the switch is turned off, the current stops and the bulb does not glow.

• Electric current is expressed by the amount of charge flowing through a particular area in unit time. It is the rate of flow of electric charges.

• In circuits using metallic wires, electrons constitute the flow of charges.

• At the time electricity was first observed, electrons were unknown. Therefore, electric current was considered the flow of positive charges.

• Conventionally, the direction of electric current is taken as opposite to the direction of the flow of electrons (negative charges). The direction of current is the direction of flow of positive charges.

• If a net charge $Q$ flows across any cross-section of a conductor in time $t$, then the current $I$ is:     $I = \frac{Q}{t}$

• The SI unit of electric charge is the coulomb ($C$).

• One coulomb is equivalent to the charge contained in nearly $6 \times 10^{18}$ electrons.

• An electron possesses a negative charge of $1.6 \times 10^{-19},C$.

• The SI unit of electric current is the ampere ($A$), named after the French scientist Andre-Marie Ampere ($1775\text{--}1836$).

• One ampere is defined as the flow of one coulomb of charge per second:     $1,A = \frac{1,C}{1,s}$

• Small quantities of current are expressed in milliampere ($1,mA = 10^{-3},A$) or microampere ($1,\mu A = 10^{-6},A$).

• An instrument called an ammeter measures electric current in a circuit. It is always connected in series.

• Electric current flows from the positive terminal of the cell to the negative terminal through the bulb and ammeter.

Example 11.1

• A current of $0.5,A$ is drawn by a filament of an electric bulb for $10$ minutes.

• Given: $I = 0.5,A$; $t = 10,min = 600,s$.

• Calculation: $Q = It = 0.5,A \times 600,s = 300,C$.

Electric Potential and Potential Difference

• Charges do not flow in a copper wire by themselves. Just as water flows in a tube only if there is a pressure difference, electrons move in a conductor only if there is a difference of electric pressure, called the potential difference.

• Electric potential difference is produced by a battery consisting of one or more electric cells. Chemical action within the cell generates this difference across its terminals even when no current is drawn.

• When the cell is connected to a circuit, the potential difference sets charges in motion.

• To maintain current, the cell must expend chemical energy stored within it.

• The electric potential difference ($V$) between two points is the work done ($W$) to move a unit charge ($Q$) from one point to the other:     $V = \frac{W}{Q}$

• The SI unit of electric potential difference is the volt ($V$), named after the Italian physicist Alessandro Volta ($1745\text{--}1827$).

• One volt is the potential difference between two points when $1$ joule of work is done to move a charge of $1$ coulomb:     $1,V = \frac{1,J}{1,C}$     $1,V = 1,J\,C^{-1}$

• The potential difference is measured by an instrument called a voltmeter, which is always connected in parallel across the points of measurement.

Example 11.2

• Work done in moving a charge of $2,C$ across two points with a potential difference of $12,V$:

• Calculation: $W = VQ = 12,V \times 2,C = 24,J$.

Circuit Diagram Symbols

• Commonly used electrical components and their symbols:

  • An electric cell: A long vertical line (positive) and a shorter, thicker vertical line (negative).

  • A battery: A combination of cells.

  • Plug key or switch (open): ( )

  • Plug key or switch (closed): (•)

  • A wire joint: A T-junction with a dot.

  • Wires crossing without joining: A line jumping over another line.

  • Electric bulb: A M-shaped filament inside a circle or a loop symbol.

  • A resistor of resistance $R$: A zigzag line.

  • Variable resistance or rheostat: A zigzag line with an arrow through it or pointing to it.

  • Ammeter: A circle containing 'A' with $+$ and $-$ signs.

  • Voltmeter: A circle containing 'V' with $+$ and $-$ signs.

Ohm’s Law

• In $1827$, German physicist Georg Simon Ohm ($1787\text{--}1854$) discovered the relationship between current ($I$) and potential difference ($V$).

• The potential difference ($V$) across the ends of a metallic wire is directly proportional to the current ($I$) flowing through it, provided its temperature remains constant.     $V \propto I$     $\frac{V}{I} = \text{constant} = R$     $V = IR$

• $R$ is the resistance, a constant for a given metallic wire at a given temperature. It is the property of a conductor to resist the flow of charges.

• The SI unit of resistance is ohm, represented by the Greek letter $\Omega$.

• According to Ohm's law: $R = \frac{V}{I}$.

• $1$ ohm is the resistance of a conductor such that when a potential difference of $1,V$ is applied, a current of $1,A$ flows through it:     $1,\Omega = \frac{1,V}{1,A}$

• Current is inversely proportional to resistance: $I = \frac{V}{R}$. If resistance is doubled, current is halved.

• A rheostat is a device used to change the resistance in a circuit to regulate current without changing the voltage source.

• Different components offer different levels of resistance because electrons are restrained by the attraction of atoms within the conductor.

• Good conductors have low resistance; resistors have appreciable resistance; poor conductors have higher resistance; insulators offer highest resistance.

Factors on Which the Resistance of a Conductor Depends

• Resistance ($R$) of a uniform metallic conductor depends on:

  1. Its length ($l$): Resistance is directly proportional to length: $R \propto l$.

  2. Its area of cross-section ($A$): Resistance is inversely proportional to cross-sectional area: $R \propto \frac{1}{A}$.

  3. The nature of its material.

• Combining these proportionalities:     $R = \rho\frac{l}{A}$

• $\rho$ (rho) is the electrical resistivity, a characteristic property of the material.

• The SI unit of resistivity is $\Omega\,m$.

• Metals and alloys have very low resistivity ($10^{-8}$ to $10^{-6},\Omega\,m$) and are good conductors.

• Insulators (rubber, glass) have high resistivity ($10^{12}$ to $10^{17},\Omega\,m$).

• Both resistance and resistivity vary with temperature.

• Resistivity of an alloy is generally higher than its constituent metals. Alloys do not oxidize (burn) readily at high temperatures, making them suitable for heating devices like toasters and electric irons.

• Tungsten is used for bulb filaments due to its high melting point ($3380^\circ\text{C}$).

• Copper and aluminum are used for electrical transmission lines due to low resistivity.

Table 11.2: Electrical Resistivity at $20^\circ\text{C}$

• Conductors:

  • Silver: $1.60 \times 10^{-8},\Omega\,m$

  • Copper: $1.62 \times 10^{-8},\Omega\,m$

  • Aluminum: $2.63 \times 10^{-8},\Omega\,m$

  • Tungsten: $5.20 \times 10^{-8},\Omega\,m$

  • Iron: $10.0 \times 10^{-8},\Omega\,m$

  • Mercury: $94.0 \times 10^{-8},\Omega\,m$

  • Manganese: $1.84 \times 10^{-6},\Omega\,m$

• Alloys:

  • Constantan (Cu and Ni): $49 \times 10^{-6},\Omega\,m$

  • Manganin (Cu, Mn, Ni): $44 \times 10^{-6},\Omega\,m$

  • Nichrome (Ni, Cr, Mn, Fe): $100 \times 10^{-6},\Omega\,m$

• Insulators:

  • Glass: $10^{10}\text{--}10^{14},\Omega\,m$

  • Hard rubber: $10^{13}\text{--}10^{16},\Omega\,m$

  • Diamond: $10^{12}\text{--}10^{13},\Omega\,m$

  • Paper (dry): $10^{12},\Omega\,m$

Resistance of a System of Resistors

Resistors in Series

• When resistors are joined end to end, they are in series.

• The current ($I$) remains the same in every part of a series circuit.

• The total potential difference ($V$) across the combination is the sum of individual potential differences ($V_1, V_2, V_3$):     $V = V_1 + V_2 + V_3$

• Using Ohm's law ($V = IR$):     $IR_s = IR_1 + IR_2 + IR_3$     $R_s = R_1 + R_2 + R_3$

• The equivalent resistance ($R_s$) of a series combination is the sum of individual resistances and is greater than any individual resistance.

Resistors in Parallel

• When resistors are connected between the same two points, they are in parallel.

• The potential difference ($V$) is the same across each resistor.

• The total current ($I$) is the sum of separate branch currents:     $I = I_1 + I_2 + I_3$

• Using Ohm's law ($I = \frac{V}{R}$):     $\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$     $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

• The reciprocal of the equivalent resistance ($R_p$) equals the sum of the reciprocals of the individual resistances.

• Parallel circuits are advantageous because:

  1. They divide current, allowing gadgets with different resistances to function correctly.

  2. If one component fails, the circuit is not broken for other components.

  3. The total resistance is decreased.

Heating Effect of Electric Current

• Chemical energy in a cell is used to maintain current. Part of this energy performs useful work, while the rest is dissipated as heat.

• If a circuit is purely resistive, the energy is dissipated entirely as heat. This is the heating effect of electric current.

• Power input to the circuit ($P$):     $P = V\frac{Q}{t} = VI$

• Energy supplied in time $t$ is $P \times t = VIt$.

• Heat produced ($H$) for a steady current $I$:     $H = VIt$

• Applying Ohm's law ($V = IR$) results in Joule’s law of heating:     $H = I^2Rt$

• Joule's Law of Heating states that heat produced is:

  1. Directly proportional to the square of the current ($I^2$).

  2. Directly proportional to the resistance ($R$).

  3. Directly proportional to the time ($t$) for which current flows.

Practical Applications

• Devices like electric laundry irons, toasters, ovens, kettles, and heaters use Joule's heating.

• Electric bulbs use heating to produce light. The filament must have a high melting point (like Tungsten at $3380^\circ\text{C}$) and be thermally isolated.

• Bulbs are filled with inactive Nitrogen and Argon to prolong filament life.

• A fuse is a safety device placed in series with a circuit. It is made of a metal or alloy (AI, Cu, Fe, Pb) with an appropriate melting point. If current exceeds a limit, the fuse melts and breaks the circuit.

• Domestic fuses are rated at $1,A, 2,A, 3,A, 5,A, 10,A$, etc.

Electric Power

• Power is the rate of doing work or the rate of consumption of energy.

• Electric power ($P$) is given by:     $P = VI$     $P = I^2R = \frac{V^2}{R}$

• The SI unit of electric power is the watt ($W$).

• One watt is the power consumed by a device carrying $1,A$ of current at $1,V$ potential difference:     $1,W = 1,V \times 1,A = 1,V,A$

• The unit watt is small; in practice, kilowatt ($1,kW = 1000,W$) is used.

• Electrical energy is power multiplied by time ($W\,h$ or $kW\,h$).

• One watt hour ($1,W\,h$) is the energy consumed when $1,W$ of power is used for $1$ hour.

• The commercial unit of energy is kilowatt hour ($kW\,h$), also called a 'unit'.     $1,kW\,h = 1000,W \times 3600,s = 3.6 \times 10^6,J$

• Electrons themselves are not consumed; we pay for the energy required to move them through gadgets.