Electromagnetism: Fields, Forces & Induction

Core Concepts

Magnetic Fields and Forces

Magnetic Field (B): An invisible field around a magnet or moving charge that exerts forces on other magnets or charges. It is a vector field, meaning it has both magnitude and direction.

Key Definitions:

• Magnetic Field Lines: Imaginary lines that represent the direction and strength of a magnetic field. They emerge from the North pole and enter the South pole.
• Magnetic Force on a Current-Carrying Conductor: A conductor carrying current in a magnetic field experiences a force given by F = I (L × B), where:
  • F = Force (N)
  • I = Current (A)
  • L = Length of the conductor (m)
  • B = Magnetic field (T)
  • × denotes the cross product (direction given by the right-hand rule).

Shows field lines emerging from the North pole and entering the South pole. (Image: Original version:Yzarc-pan This version:Chetvorno, CC BY-SA 4.0, via Wikimedia Commons)

Illustrates how to determine the direction of the force using the right-hand rule. (Image: OpenStax, CC BY 4.0, via Wikimedia Commons)

Worked Example: Force on a Wire in a Magnetic Field

A wire of length 0.5 m carries a current of 3 A and is placed perpendicular to a magnetic field of 0.4 T. Calculate the force acting on the wire.

Solution: Using F = I (L × B), since the wire is perpendicular to the field, the angle θ = 90° and sin(90°) = 1. $$F=3\text{A}\times 0.5\text{m}\times 0.4\text{T}\times \sin (90°)=0.6\text{N}$$ Direction: Use the right-hand rule to determine the direction of the force.

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Biot-Savart Law and Ampère’s Law

Biot-Savart Law

Describes the magnetic field generated by a steady current. For a small current element Idl, the magnetic field dB at a point is given by: $$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Id\mathbf{l}\times \hat{\mathbf{r}}}{r^2}$$

• μ₀ = Permeability of free space (4π × 10⁻⁷ T·m/A)
• I = Current (A)
• dl = Infinitesimal length element (m)
• r = Distance from the current element to the point (m)
• r̂ = Unit vector pointing from the current element to the point

Ampère’s Law

Relates the magnetic field around a closed loop to the current passing through the loop: $$\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\text{enc}}$$

• Useful for calculating magnetic fields in symmetric situations (e.g., solenoids, toroids).

Shows circular field lines around the wire. (Image: OpenStax, CC BY 4.0, via Wikimedia Commons)

Illustrates how a solenoid creates a uniform magnetic field inside. (Image: MikeRun, CC BY-SA 4.0, via Wikimedia Commons)

Worked Example: Magnetic Field Inside a Solenoid

A solenoid has 500 turns/m and carries a current of 2 A. Calculate the magnetic field inside the solenoid.

Solution: Using Ampère’s law for a solenoid: $$B={\mu }_{0}nI$$ where:

• n = Number of turns per unit length (500 turns/m)
• I = Current (2 A) $$B=(4\pi \times 10^{-7})\times 500\times 2=1.2566\times 10^{-3}\text{T}=1.2566\text{mT}$$

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Faraday’s Law of Electromagnetic Induction

States that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop: $$\mathcal{E}=-\frac{d{\Phi }_B}{dt}$$ where:

• Φ_B = Magnetic flux (T·m²)
• ε = Induced emf (V)

Magnetic Flux (Φ_B): $${\Phi }_B=\int \mathbf{B}\cdot d\mathbf{A}$$

Lenz’s Law

The direction of the induced current is such that it opposes the change in magnetic flux that produced it.

Shows a coil with changing magnetic flux inducing a current. (Image: Zahoornoman0, CC BY-SA 4.0, via Wikimedia Commons)

Worked Example: Induced EMF in a Loop

A rectangular loop of area 0.1 m² is placed in a magnetic field of 0.5 T. If the magnetic field is reduced to 0 T in 0.2 s, calculate the induced emf.

Solution: Change in magnetic flux: $$\Delta {\Phi }_B={\Phi }_{B,f}-{\Phi }_{B,i}=(0\times 0.1)-(0.5\times 0.1)=-0.05\text{Wb}$$ Rate of change of flux: $$\frac{d{\Phi }_B}{dt}=\frac{-0.05\text{Wb}}{0.2\text{s}}=-0.25\text{Wb/s}$$ Induced emf (magnitude): $$\mathcal{E}=\left|-\frac{d{\Phi }_B}{dt}\right|=0.25\text{V}$$ Direction: Use Lenz’s law to determine the direction of the induced current.

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Applications of Electromagnetic Induction

Generators

Convert mechanical energy into electrical energy using Faraday’s law. A coil rotates in a magnetic field, inducing an alternating current (AC).

Shows how mechanical rotation induces AC in a coil. (Image: MikeRun, CC BY-SA 4.0, via Wikimedia Commons)

Transformers

Transfer electrical energy between circuits through electromagnetic induction. Consists of two coils (primary and secondary) wound around a common core.

Transformer Equation: $$\frac{V_p}{V_s}=\frac{N_p}{N_s}$$ where:

• V_p, V_s = Primary and secondary voltages
• N_p, N_s = Number of turns in primary and secondary coils

Worked Example: Transformer Voltage Calculation

A transformer has 200 turns in the primary coil and 1000 turns in the secondary coil. If the primary voltage is 220 V, calculate the secondary voltage.

Solution: Using the transformer equation: $$\frac{V_p}{V_s}=\frac{N_p}{N_s}⟹\frac{220}{V_s}=\frac{200}{1000}$$ $$V_s=\frac{220\times 1000}{200}=1100\text{V}$$

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In the Real World

1. Ncell and NTC (Nepal Telecommunications Company):

   • Electromagnetic Induction in Wireless Charging: Modern smartphones (e.g., those using Ncell or NTC SIMs) support wireless charging via electromagnetic induction. A charging pad creates an alternating magnetic field, which induces a current in the phone’s receiver coil, converting magnetic energy into electrical energy to charge the battery.

2. eSewa and Khalti (Digital Payments):

   • Magnetic Stripe Cards: Older versions of eSewa or Khalti payment cards used magnetic stripes that store data in tiny magnetic particles. When swiped, the magnetic field induces a current in the card reader, allowing data to be read and transactions to be processed.

3. Pathao and Daraz (Logistics and Delivery):

   • Electric Motors in Delivery Vehicles: Pathao’s electric scooters and Daraz’s delivery vehicles use electric motors powered by batteries. These motors operate on the principle of electromagnetic induction, where an electric current in a coil interacts with a magnetic field to produce rotational motion, enabling efficient and eco-friendly deliveries.

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Exam Tip

1. Memorize Key Equations:

   • Magnetic force on a current-carrying conductor: F = I (L × B)
   • Biot-Savart law and Ampère’s law for magnetic fields.
   • Faraday’s law: ε = -dΦ_B/dt
   • Transformer equation: V_p/V_s = N_p/N_s

2. Direction Matters:

   • Always use the right-hand rule for magnetic forces and Lenz’s law for induced currents. Examiners often test your ability to determine directions correctly.

3. Real-World Applications:

   • Relate theoretical concepts to real devices like generators, transformers, and electric motors. For example:
     • In a Nepal Electricity Authority (NEA) power plant, turbines rotate coils in a magnetic field to generate electricity (Faraday’s law).
     • Ncell’s base stations use transformers to step up or step down voltages for efficient power transmission.

4. Graphical Questions:

   • Be prepared to sketch magnetic field lines around current-carrying conductors, solenoids, and bar magnets. Label directions clearly.

5. Numerical Problems:

   • Practice calculating magnetic forces, induced emfs, and transformer voltages. Always check units and significant figures.

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