Explore the basic tenets of electromagnetism (electromagnetic force, induction and electromagnetic waves)

A current-carrying conductor in a magnetic field experiences a force because it consists of charges in motion. For a straight wire of length l carrying current I in a uniform magnetic field B:

$F = IBl\sin\theta$

where θ is the angle between the conductor and the magnetic field. This principle is used in devices like moving-coil galvanometers and electric motors.

For a rectangular coil of area A with N turns carrying current I in a magnetic field B, the torque is:

$\tau = NIAB\sin\theta$

The product IA is called the magnetic dipole moment, μ.

Electromagnetic induction is the process whereby a changing magnetic flux through a circuit induces an electromotive force (e.m.f.) in that circuit.

Magnetic Flux

Magnetic flux Φ_B through an area A in a uniform magnetic field B is:

$\Phi_B = BA\cos\theta$

where θ is the angle between the magnetic field and the normal to the surface. The SI unit is weber (Wb).

Faraday's Law of Induction

Faraday's law states that the induced e.m.f. in a circuit is directly proportional to the rate of change of magnetic flux through the circuit:

$\varepsilon = -\frac{d\Phi_B}{dt}$

The negative sign expresses Lenz's law: the induced current always opposes the change that produces it.

Methods of Inducing EMF

An e.m.f. can be induced by changing:

1. The magnetic field strength (B)
2. The area enclosed by the loop (A)
3. The angle between the field and the normal to the area (θ)

Induced EMF in a Moving Conductor

For a straight conductor of length l moving with velocity v perpendicular to a uniform magnetic field B:

$\varepsilon = Blv$

Induced EMF in a Rotating Coil

For a coil rotating in a uniform magnetic field with angular velocity ω:

$\varepsilon = \omega NAB\sin\omega t$

The maximum (peak) e.m.f. is ε₀ = ωNAB.

Self-induction occurs when a changing current in a coil induces an e.m.f. in the same coil. The self-induced e.m.f. is:

$\varepsilon = -L\frac{dI}{dt}$

where L is the self-inductance. The energy stored in an inductor is:

$U = \frac{1}{2}LI^2$

Mutual induction occurs when a changing current in one coil induces an e.m.f. in a nearby coil. The mutual inductance M between two coils is:

$M = \frac{\mu_0 N_1 N_2 A}{l}$

A proton moves at 8.5 × 10⁷ m/s perpendicular to a magnetic field and travels in a circular path of radius 0.68 m. Determine the magnetic field strength. (Mass of proton = 1.67 × 10⁻²⁷ kg, charge = 1.6 × 10⁻¹⁹ C)

Solution

For circular motion, magnetic force provides the centripetal force:

$qvB = \frac{mv^2}{r}$

Solving for B:

$B = \frac{mv}{qr}$

$B = \frac{1.67 \times 10^{-27} \text{ kg} \times 8.5 \times 10^7 \text{ m/s}}{1.6 \times 10^{-19} \text{ C} \times 0.68 \text{ m}}$

$B = \frac{1.42 \times 10^{-19}}{1.09 \times 10^{-19}}$

$B = 1.3 \text{ T}$

Electromagnetic principles are applied in numerous devices:

• Generators: Convert mechanical energy to electrical energy
• Transformers: Change voltage levels in AC circuits
• Cyclotrons: Accelerate charged particles for research
• Mass spectrometers: Separate ions by charge-to-mass ratio
• Moving-coil galvanometers: Measure current and voltage

In Tanzania, electromagnetic induction is used in the electricity generators at the Julius Nyerere Hydropower Station at Stiegler's Gorge. When water flows through the turbines, it rotates coils of wire within strong magnetic fields, inducing an e.m.f. that produces electricity distributed across the national grid. This same principle applies to small-scale hydro projects in rural areas like those in Mbeya and Kilimanjaro regions, providing power for local communities and small businesses.