Magnetic Field and Electromagnetism

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Magnetic Field and Electromagnetism — Key Facts

• Force on a Moving Charge in a Magnetic Field: $F = Bqv\sin\theta$, where $B$ = magnetic flux density (tesla, T), $q$ = charge (coulomb, C), $v$ = velocity (m/s), $\theta$ = angle between $B$ and $v$.
• Force on a Current-Carrying Conductor: $F = BIL\sin\theta$ in a magnetic field.
• Fleming’s Left-Hand Rule: Thumb = force (motion), First finger = magnetic field (north to south), Second finger = current (positive to negative).
• Magnetic Flux: $\Phi = BA\cos\theta$ in webers (Wb); flux linkage $N\Phi = NBA\cos\theta$.
• Faraday’s Law of Electromagnetic Induction: $E = -\frac{Nd\Phi}{dt}$ volts.
• Lenz’s Law: The induced e.m.f. opposes the change in flux that produces it (conservation of energy).
• Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ (for an ideal step-up/step-down transformer).

⚡ WAEC Exam Tip: Remember that magnetic force $F = Bqv\sin\theta$ acts perpendicular to both $B$ and $v$. This means it changes the direction of motion but NOT the speed — the path becomes circular. If $\theta = 0°$ or $180°$ (charge moving parallel to field), $F = 0$. If $\theta = 90°$, $F = Bqv$.

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Magnetic Field and Electromagnetism — WAEC WASSCE Study Guide

Magnetic Fields Around Currents

A current-carrying conductor produces a magnetic field. The direction is given by the right-hand grip rule (or Maxwell’s corkscrew rule): if you grip the wire with your right hand with the thumb pointing in the direction of conventional current, your fingers curl in the direction of the magnetic field.

• Straight wire: Circular field lines centred on the wire. Magnetic field strength $B = \frac{\mu_0 I}{2\pi r}$ at distance $r$ from the wire, where $\mu_0 = 4\pi \times 10^{-7}$ H m⁻¹ (permeability of free space).
• Solenoid: A coil of many turns. Inside a long solenoid, $B = \mu_0 nI$ where $n = \frac{N}{L}$ = number of turns per unit length. The field inside is uniform; outside it resembles that of a bar magnet.

Force on a Moving Charge

When a charged particle moves perpendicularly to a uniform magnetic field, it moves in a circle. The centripetal force is provided by the magnetic force:

$$Bqv = \frac{mv^2}{r} \quad \Rightarrow \quad r = \frac{mv}{Bq}$$

The time period of one revolution: $T = \frac{2\pi m}{Bq}$. Note this is independent of velocity $v$ — faster particles travel further in the same time, so the radius increases but the period is constant. This principle is used in a cyclotron (particle accelerator).

Electromagnetic Induction

When a conductor moves through a magnetic field, or when the magnetic flux through a circuit changes, an e.m.f. is induced. The magnitude is given by Faraday’s law:

$$E = N\left|\frac{d\Phi}{dt}\right|$$

The direction is given by Lenz’s law and Fleming’s right-hand rule (generator rule): Thumb = motion, First finger = field (N to S), Second finger = induced current.

Alternating Current (A.C.) Generator

A rotating coil in a magnetic field generates an alternating e.m.f. The e.m.f. varies as $E = E_0\sin\omega t$ where $E_0 = BAN\omega$ (maximum e.m.f.), $N$ = number of turns, $A$ = coil area, $\omega$ = angular velocity in rad/s, and $T = \frac{2\pi}{\omega}$.

Self-Induction

When the current in a coil changes, the changing magnetic flux links with the same coil and induces an e.m.f. This is self-induction. The self-inductance $L$ of a coil is defined as:

$$L = \frac{N\Phi}{I}$$

The induced e.m.f. is $E = -L\frac{dI}{dt}$. The unit is the henry (H).

Energy Stored in an Inductor

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

This energy is magnetic energy stored in the magnetic field of the inductor.

Comparison Table — Magnetic Force Equations

Situation	Formula	Notes
Force on moving charge	$F = Bqv\sin\theta$	Perpendicular to both $B$ and $v$
Force on current-carrying wire	$F = BIL\sin\theta$	Perpendicular to $B$ and wire
Force between two parallel wires	$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$	Attraction if currents in same direction
Magnetic field near straight wire	$B = \frac{\mu_0 I}{2\pi r}$	Circular field lines
Magnetic field inside solenoid	$B = \mu_0 nI$	Uniform field inside

Earth’s Magnetic Field

The Earth acts as a giant bar magnet. The Earth’s magnetic field has a horizontal component $B_H = B\cos\theta$, a vertical component $B_V = B\sin\theta$, where $\theta$ is the angle of dip. The angle of declination is the angle between magnetic and geographic north. At the magnetic equator, $\theta = 0°$ and $B_V = 0$. At the magnetic poles, $\theta = 90°$ and $B_H = 0$.

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🔴 Extended — Deep Study (3mo+)

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Magnetic Field and Electromagnetism — Comprehensive WAEC Physics Notes

Derivation: E.M.F. Induced in a Rotating Coil (A.C. Generator)

Consider a coil of area $A$ and $N$ turns rotating with angular velocity $\omega$ in a uniform magnetic field $B$. At time $t$, the angle between the normal to the coil and the magnetic field is $\theta = \omega t$.

The magnetic flux through one turn: $\Phi = BA\cos\theta = BA\cos\omega t$

By Faraday’s law, the induced e.m.f.: $$E = -N\frac{d\Phi}{dt} = -N\frac{d}{dt}(BA\cos\omega t) = NBA\omega\sin\omega t$$

Thus $E = E_0\sin\omega t$ where $E_0 = NBA\omega$ is the peak e.m.f.

The r.m.s. value: $E_{\text{rms}} = \frac{E_0}{\sqrt{2}} = \frac{NBA\omega}{\sqrt{2}}$

Derivation: Force on a Current-Carrying Conductor in a Magnetic Field

A conductor of length $L$ carrying current $I$ in a magnetic field $B$: the drift velocity of electrons is $v_d$. Each electron experiences force $F_e = Bev\sin\theta$. With $n$ electrons per unit volume in a conductor of cross-section $A$, the total force on the conductor is:

$$F = (nAL) \times Bev\sin\theta = (nAe v_d)B L\sin\theta$$

But current $I = nAe v_d$, therefore:

$$F = BIL\sin\theta$$

Derivation: Radius of Circular Path of a Charged Particle

For a charge $q$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$, the magnetic force $Bqv$ acts as a centripetal force:

$$Bqv = \frac{mv^2}{r} \quad \Rightarrow \quad r = \frac{mv}{Bq}$$

The angular frequency $\omega = \frac{v}{r} = \frac{Bq}{m}$, and the period $T = \frac{2\pi}{\omega} = \frac{2\pi m}{Bq}$.

Induced Electric Fields (Faraday’s Law in Differential Form)

For a changing magnetic flux, the induced e.m.f. around a closed path is:

$$\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$$

This is Faraday’s law in integral form and shows that a changing magnetic field produces a circulating electric field. This is fundamentally different from electrostatic fields (where $\oint \mathbf{E} \cdot d\mathbf{l} = 0$).

Mutual Inductance

When the current in one coil changes, an e.m.f. is induced in a nearby coil. The mutual inductance $M$ is defined by:

$$E_2 = -M\frac{dI_1}{dt}$$

For two coils wound on a common core: $M = \frac{\mu_0 N_1 N_2 A}{l}$ where $l$ = length of magnetic circuit, $A$ = cross-sectional area. Also $M = k\sqrt{L_1 L_2}$ where $k$ is the coupling coefficient (0 ≤ $k$ ≤ 1).

Eddy Currents

When a bulk conductor moves through a magnetic field (or when magnetic flux changes in a conductor), circulating currents called eddy currents are induced. These dissipate energy as heat (Joule heating). Eddy currents are minimised by using laminated cores (thin sheets of metal insulated from each other), as used in transformers and alternating current machines.

Magnetic Materials — Detailed Classification

Type	Behaviour	Atomic Explanation	Examples
Diamagnetic	Weakly repelled by magnetic field	No permanent dipole; induced dipole opposes applied field	Bismuth, copper, water
Paramagnetic	Weakly attracted	Unpaired electrons give permanent moments; randomly oriented	Aluminium, oxygen, manganese
Ferromagnetic	Strongly attracted, can be magnetised	Domains aligned parallel below Curie temperature	Iron, nickel, cobalt

Above the Curie temperature $T_C$, all ferromagnetic materials become paramagnetic because thermal energy destroys the domain alignment.

Hysteresis Loop

When a ferromagnetic material is magnetised and demagnetised in a cyclic process, the magnetisation $M$ (or magnetic flux density $B$) lags behind the magnetising field $H$. The area of the hysteresis loop represents energy loss per unit volume per cycle. Soft iron has a narrow loop (low energy loss, easily magnetised and demagnetised) — ideal for transformer cores and electromagnets. Steel has a wide loop (high energy loss, high coercivity) — used for permanent magnets because it retains its magnetism.

Transformer Power and Efficiency

For an ideal transformer: $P_{\text{input}} = P_{\text{output}}$ → $V_p I_p = V_s I_s$

In practice, transformers have losses: copper losses ($I^2R$ in windings), core losses (eddy currents + hysteresis). The efficiency $\eta = \frac{P_{\text{output}}}{P_{\text{input}}} \times 100%$.

WAEC Past Question Patterns

Typical WAEC questions include:

1. Applying $F = BIL$ to calculate force on conductors in magnetic fields
2. Using Fleming’s left-hand and right-hand rules for direction determination
3. Calculating induced e.m.f. using $E = BAN\omega\sin\omega t$ or $E = \frac{Nd\Phi}{dt}$
4. Transformer ratio calculations (step-up/step-down)
5. Explaining electromagnetic induction phenomena (moving coil microphone, moving iron loudspeaker, eddy current brake)
6. Calculations involving the radius and period of charged particles in magnetic fields (cyclotron)

⚡ WAEC Exam Tip: In transformer questions, always check whether the question asks for the ideal case or accounts for efficiency. In real transformers, $V_s I_s < V_p I_p$ due to losses. Also remember that for a step-up transformer, $N_s > N_p$ and $V_s > V_p$, but $I_s < I_p$. Power is conserved (ignoring losses), so $V_p I_p \approx V_s I_s$.

Numerical Worked Example (WAEC-style)

A straight wire of length 0.5 m carrying a current of 10 A is placed at right angles to a magnetic field of flux density 0.2 T. Calculate the force on the wire.

Solution: $F = BIL\sin\theta = 0.2 \times 10 \times 0.5 \times \sin 90°$ $F = 0.2 \times 10 \times 0.5 \times 1 = 1.0$ N

Answer: Force = 1.0 N

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