Chapter 8 Solved Numericals: Magnetism

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Chapter Overview & SLOs

What is covered in these solved numericals? This section provides detailed step-by-step solutions for the numerical problems of Chapter 8, "Magnetism." The exercises focus on the mathematical application of electromagnetic principles to determine field strength and forces. How do we calculate magnetic field due to a long straight wire? You will find step-by-step solutions for calculating the magnetic field (B) produced by a long straight wire at a specific distance using Ampere's circuital law: - Formula: $B = \frac{μ₀I}{2πr}$ (Magnetic field = permeability of free space × current / 2π × distance) - Where: - $B$ = Magnetic field (Tesla, T) - $μ₀ = 4π × 10^{-7} \text{ Tm/A}$ (permeability of free space/air) - $I$ = Current in amperes (A) - $r$ = Perpendicular distance from wire to point (meters, m) - Key concept: The magnetic field decreases as distance from the wire increases (B ∝ 1/r) What are the important unit conversions? The notes emphasize critical unit conversions: - Distance conversion: Convert cm to m before calculation (divide by 100). Example: 10 cm = 0.1 m, 5 cm = 0.05 m, 2 cm = 0.02 m - Current: Usually given in amperes (A) - ensure it's in amperes, not milliamperes (mA = ×10⁻³) - Final unit: Magnetic field expressed in Tesla (T) How do we simplify Ampere's law calculations? You will learn the cancellation of π trick: - $B = \frac{(4π × 10^{-7}) × I}{2π × r} = \frac{4π × 10^{-7} × I}{2π × r} = \frac{2 × 10^{-7} × I}{r}$ - The π cancels out, leaving a simpler formula: $B = \frac{2 × 10^{-7} × I}{r}$ - This shortcut saves time in exams How do we calculate magnetic field inside a solenoid? You will learn to calculate the magnetic induction within a solenoid: - Formula: $B = μ₀ × n × I$ - $n = \frac{N}{L}$ (number of turns per unit length, where N = total number of turns, L = length of solenoid in meters) - $B = μ₀ × \frac{N}{L} × I$ - Important: The magnetic field inside a solenoid is uniform (constant) and parallel to the axis What is the magnetic force on current-carrying conductors? Additional problems cover: - Force on straight wire in magnetic field: $F = BIL \sinθ$ (Force = magnetic field × current × length × sine of angle between wire and field) - Maximum force when wire perpendicular to field (θ = 90°, sin90° = 1) - Zero force when wire parallel to field (θ = 0°, sin0° = 0) Why is the permeability of free space constant important? $μ₀ = 4π × 10^{-7} \text{ Tm/A}$ is a fundamental constant of nature. It relates magnetic field strength to the current producing it and is essential for all electromagnetic field calculations. What is the SI unit Tesla? The Tesla (T) is the SI unit of magnetic field: - $1 \text{ T} = 1 \frac{\text{N}}{\text{A·m}} = 1 \frac{\text{kg}}{\text{A·s²}}$ - Reference values: Earth's magnetic field ≈ $5 × 10^{-5} \text{ T}$ (0.05 mT), Small bar magnet ≈ $10^{-2} \text{ T}$ (10 mT), Strong electromagnet ≈ $1 \text{ T}$, MRI machine ≈ $1.5 - 3 \text{ T}$ These solutions emphasize the correct application of the permeability of free space constant, proper unit conversions (especially converting cm to m), and the cancellation of π to simplify expressions. They are strictly designed to help students master the mathematical requirements of the FBISE 2026 annual examination.

• How do we calculate magnetic field intensity from a straight wire? Calculate the magnetic field intensity (B) at a given distance from a long straight wire using Ampere's circuital law: B = μ₀I / 2πr, where μ₀ = 4π × 10⁻⁷ Tm/A, I is current in amperes, and r is distance in meters (convert cm to m by dividing by 100).
• How do we apply the permeability of free space constant? Apply the constant for the permeability of free space (μ₀ = 4π × 10⁻⁷ Tm/A) to solve electromagnetic field equations, using the π cancellation shortcut (B = 2 × 10⁻⁷ × I / r) to simplify expressions.
• How do we determine the magnetic field inside a solenoid? Determine the strength of the magnetic field inside a solenoid using B = μ₀nI, where n = N/L is the number of turns per unit length (total turns divided by length in meters), and understand that the field is uniform and parallel to the solenoid axis.
• How do we relate Tesla to other units? Relate the SI unit Tesla (T) to other fundamental units (1 T = 1 N/A·m = 1 kg/A·s²), and perform multi-step unit conversions in magnetism problems including converting cm to m (divide by 100) and ensuring current is in amperes.

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