Electricity One-Shot Study Notes

Introduction to Electricity

• Electricity is a fundamental chapter in physics, crucial for examinations ranging from half-yearlies to board exams. Success in solving its numerical problems and understanding its theory depends on a strong grasp of conceptual foundations like charge, current, potential difference, and resistance.

Electrical Charge

• Definition: Charge is a fundamental property of matter due to which matter particles can be positive or negative in nature.

• Types of Charge:

  • Positive Charge.

  • Negative Charge.

• Electron Charge: An electron is a negatively charged particle. The magnitude of charge on a single electron is approximately $1.6 \times 10^{-19}\,C$.

• S.I. Unit: The unit of charge is the Coulomb ($C$).

• Properties of Charge:

  • Like Charges: Same types of charges (e.g., positive-positive) repel each other.

  • Opposite Charges: Opposite types (e.g., positive-negative) attract each other. A common reality check is that while you might be attracted to someone different, attraction must be mutual, just like in physics.

  • Additivity of Charge: The total charge on a body is the algebraic sum of all individual charges. For a body with charges of $2\,C$, $3\,C$, and $4\,C$, the overall charge is $y = 2 + 3 + 4 = 9\,C$.

  • Conservation of Charge: Charge can neither be created nor destroyed; it can only be transferred.

  • Quantization of Charge: Charge does not exist in arbitrary amounts; it exists in discrete multiples of the elementary charge of an electron ($e$). This is comparable to buying sweets; you can buy 1 kg or 2 kg, but you cannot buy a specific arbitrary decimal like $22.33\,g$ without being scoffed at by the shopkeeper.

• Quantization Formula: $Q = ne$, where:

  • $Q$ is the total charge.

  • $n$ is the number of electrons (must be an integer like 1, 2, 3, etc.).

  • $e$ is the charge of one electron ($1.6 \times 10^{-19}\,C$).

Numerical Practice: Charge and Electrons

• Problem: A body has a charge of $4.8 \times 10^{-19}\,C$. Calculate the number of electrons and determine if they were gained or lost.

  • Calculation: Using $Q = ne$, we have $n = \frac{Q}{e}$. Substituting the values: $n = \frac{4.8 \times 10^{-19}}{1.6 \times 10^{-19}} = 3$.

  • Result: There are 3 electrons. Since the body has a positive charge, it means it has lost its negative particles (electrons). If it had gained electrons, it would have been negatively charged.

Electric Current

• Definition: Electric current is the rate of flow of charge through a cross-section of a conductor. In physics, the term "rate" always implies division by time.

• Formula: $I = \frac{Q}{t}$, where:

  • $I$ is the current.

  • $Q$ is the charge.

  • $t$ is the time.

• An alternate version of this formula is $Q = It$ (often remembered by the mnemonic "Kyon Aayi Thi" - QIThi).

• S.I. Unit: The unit is the Ampere ($A$). It can also be expressed as Coulomb per second ($C/s$).

• Small Units of Current:

  • Milliampere ($mA$): $1\,mA = 10^{-3}\,A$.

  • Microampere ($\mu A$): $1\,\mu A = 10^{-6}\,A$.

• Direction of Current flow: Current flows from the Positive terminal to the Negative terminal. Electrons, however, move in the opposite direction, from the Negative terminal to the Positive terminal.

• Defining 1 Ampere: One Ampere is the current flowing through a conductor when one Coulomb of charge flows through it in one second ($1\,A = \frac{1\,C}{1\,s}$).

Numerical Practice: Current and Charge

• Problem 1: A current of $0.5\,A$ is drawn by a filament for 10 minutes. Find the amount of charge.

  • Calculation: First, convert time to seconds ($10 \times 60 = 600\,s$). Using $Q = It$, charge $Q = 0.5 \times 600 = 300\,C$.

• Problem 2: A current of $10\,A$ flows for 2 minutes. Find the charge and number of electrons.

  • Calculation: Time is $120\,s$. Charge $Q = 10 \times 120 = 1200\,C$. To find the number of electrons, use $n = \frac{Q}{e} = \frac{1200}{1.6 \times 10^{-19}}$. This results in $n = 750 \times 10^{19}$ electrons.

Electric Potential and Potential Difference

• Definition: Electric Potential Difference ($V$) between two points in an electric circuit is the work done ($W$) to move a unit positive charge ($Q$) from one point to another.

• Analogy: Imagine picking up a charge at one spot and tossing it to another, like a "Disco Dancer" move; the effort/work required for that move is the potential difference.

• Formula: $V = \frac{W}{Q}$.

• Derived Formula for Work: $W = V \times Q$.

• S.I. Unit: The unit for Potential Difference is the Volt ($V$). The unit for work is Joule ($J$), making the volt equal to Joule per Coulomb ($J/C$).

• Defining 1 Volt: One Volt is the potential difference when 1 Joule of work is done to move a charge of 1 Coulomb from one point to another.

• Why Current Flows: Current flows due to a potential difference between the terminals of a battery. Electrons flow from Low Potential (Negative) to High Potential (Positive), while current flows from High Potential (Positive) to Low Potential (Negative).

Electric Circuit and Symbols

• Circuit Diagram: A schematic drawing of an electrical circuit where components are shown using standard symbols.

• Standard Symbols:

  • Electrical Cell: A single long line (Positive) and a short thick line (Negative).

  • Battery: A combination of multiple cells.

  • Switch (Plug Key): Represented as brackets. A dot inside ($\cdot$) means the switch is closed (current flows). Empty brackets mean the switch is open (current stops).

  • Ammeter: Represented by 'A'. It measures current and is always connected in series.

  • Voltmeter: Represented by 'V'. It measures potential difference and is always connected in parallel.

  • Resistor: Represented by a zigzag line.

  • Variable Resistor (Rheostat): A resistor with an arrow through it or over it, indicating its value can be changed.

Ohm's Law

• Definition: Ohm's Law states that the potential difference ($V$) across the ends of a conductor is directly proportional to the current ($I$) flowing through it, provided that the temperature remains constant.

• Mathematical Expression: $V \propto I \rightarrow V = IR$.

• Constant ($R$): Here, $R$ is the Resistance, which remains constant at a fixed temperature.

• Resistance: It is the property of a conductor to oppose the flow of electric current. It is like traffic on a road preventing a car from moving fast.

• S.I. Unit of Resistance: The Ohm ($\Omega$).

• Experimental Verification: By adding batteries in series (increasing voltage), it is observed that the current measured in the ammeter increases proportionally. Doubling the voltage doubles the current.

Graphs and Resistance (V-I Graph)

• V-I Graph: A plot of Voltage ($V$) vs Current ($I$) results in a straight line passing through the origin. The slope of this line represents the Resistance ($R$).

• Comparing Slopes: In a $V-I$ graph, a line with a steeper angle (greater slope) indicates higher resistance.

• The "Inverse" Graph Trick: In an $I-V$ graph (Current on y-axis, Voltage on x-axis), the rule reverses: the line with the smaller angle has the highest resistance. If $IV$ is given, $\text{smaller angle} = \text{more power/resistance}$.

Factors Affecting Resistance

• Length ($L$): Resistance is directly proportional to the length of the wire ($R \propto L$). Longer wires have more traffic (resistance).

• Area of Cross-section ($A$): Resistance is inversely proportional to the area ($R \propto \frac{1}{A}$). A wider road has less traffic (resistance).

• Nature of Material: Measured by Resistivity ($\rho$).

• Temperature: For metals and alloys, resistance increases with temperature. For semiconductors like Silicon or Germanium, resistance decreases as temperature increases.

• Combined Formula: $R = \rho \frac{L}{A}$.

Resistivity

• Resistivity ($\rho$): A property of the material itself (e.g., copper has a different resistivity than iron). It does not change with length or area.

• S.I. Unit: The unit of resistivity is the Ohm-meter ($\Omega \cdot m$).

• Properties:

  • Insulators (like plastic) have very high resistivity.

  • Metals (conductors) have very low resistivity.

  • Alloys generally have higher resistivity than their constituent pure metals and are used in heating elements.

• The Stretching Case: If a wire of length $L$ is stretched to length $3L$, its area decreases by 3 ($A/3$) because volume remains constant. Consequently, the resistance increases by 9 times ($R_{new} = \rho \frac{3L}{A/3} = 9R$).

Combinations of Resistors

Series Combination

• In series, resistors are connected end-to-end in a single line.

• Current ($I$): Remains the same through all resistors.

• Voltage ($V$): Splits across the resistors ($V = V_1 + V_2 + V_3$).

• Equivalent Resistance: $R_s = R_1 + R_2 + R_3$.

• Resistance is always maximum in a series combination.

Parallel Combination

• In parallel, resistors are connected across the same two points, branching out.

• Current ($I$): Splits across branches ($I = I_1 + I_2 + I_3$).

• Voltage ($V$): Remains the same across all branches.

• Equivalent Resistance: $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$.

• Resistance is always minimum in a parallel combination.

• Calculation Shortcut (Two Resistors): $R_p = \frac{R_1 \times R_2}{R_1 + R_2}$.

• Calculation Shortcut (Identical Resistors): If $n$ identical resistors of value $R$ are in parallel, $R_{eq} = \frac{R}{n}$. For example, two $10\,\Omega$ resistors in parallel equal $5\,\Omega$.

Domestic Circuits

• In households, parallel circuits are used rather than series for several reasons:

  1. Lower net resistance.

  2. Every appliance receives the full voltage (e.g., $220\,V$).

  3. Independent Functioning: If one appliance breaks or is turned off, the others continue to work. In series, one break stops everything.

  4. Ease of adding new devices.

Joule's Law of Heating

• Definition: The heat produced in a conductor is directly proportional to the square of the current ($I^2$), the resistance ($R$), and the time ($t$) for which the current flows.

• Formula: $H = I^2 Rt$.

• Derivation Summary: Since Heat is energy, it is equivalent to work ($W = VQ$). Substituting $Q = It$ gives $W = VIt$. Applying Ohm's Law ($V = IR$) yields $H = I^2 Rt$.

Applications of Heating Effect

• Electric Bulb:

  • Filament is made of Tungsten because of its high melting point and high resistance.

  • It glows hot to produce light without melting.

  • Bulbs are filled with inactive gases like Nitrogen and Argon to prevent the filament from burning up/oxidizing.

• Electric Fuse:

  • A safety device connected in series.

  • It has a low melting point and high resistance.

  • Made of an alloy of Lead and Tin.

  • If excessive current flows, the fuse wire melts, breaking the circuit and protecting appliances.

• Electric Heater / Iron:

  • Heating element made of Nichrome (an alloy).

  • Has high resistivity and high melting point.

  • Does not oxidize (burn) even at very high temperatures.

Electric Power

• Definition: Electrical Power is the rate at which electrical energy is consumed or dissipated.

• Formula 1: $P = V \times I$.

• Formula 2: $P = I^2 R$.

• Formula 3: $P = \frac{V^2}{R}$.

• S.I. Unit: The unit is the Watt ($W$). $1\,W = 1\,V \times 1\,A$.

• Kilowatt ($kW$): $1\,kW = 1000\,W$.

• Power Rating Strategy: If a bulb is rated at $220\,V$ and $100\,W$, and the voltage is halved to $110\,V$, the power becomes one-fourth ($25\,W$) because $P \propto V^2$.

Electrical Energy and Commercial Unit

• Formula: $\text{Energy} = \text{Power} \times \text{Time}$.

• Commercial Unit: The unit used for electricity bills is the kilowatt-hour ($kWh$), commonly called a "Unit".

• Relationship to Joules: $1\,kWh = 3.6 \times 10^6\,J$.

• Electricity Billing Calculation:

  1. Find the power of each appliance in $kW$ (divide Watts by 1000).

  2. Multiply by the number of hours used per day.

  3. Multiply by the number of days (usually 30 for a month).

  4. Multiply total units ($kWh$) by the cost per unit.

Circuit Solving Procedures

• Step 1: Calculate the total/net resistance ($R_{net}$) of the entire circuit using series and parallel rules.

• Step 2: Calculate the total current ($I$) flowing from the battery using $I_{net} = \frac{V_{battery}}{R_{net}}$.

• Step 3: Solve for specific branch currents or individual resistor voltages using the properties of series (same current) and parallel (same voltage).