7.4 Magnetism, Electromagnetic Induction & AC Circuits

7.4 Magnetism, Electromagnetic Induction & AC Circuits

Electromagnetism connects electric fields and magnetic fields, forming the basis for electrical power generation, transformers, and alternating current (AC) circuit dynamics. This section delivers essential concepts, laws, and equations for the AMC Physics test.

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1. Magnetic Force & Magnetic Field ((\vec{B}))

Lorentz Magnetic Force on a Charge

A charge $q$ moving with velocity $\vec{v}$ through a uniform magnetic field $\vec{B}$ experiences a magnetic Lorentz force $\vec{F}_B$:

$\vec{F}_B = q (\vec{v} \times \vec{B}) \implies F_B = q v B \sin\theta$

where $\theta$ is the angle between velocity vector $\vec{v}$ and magnetic field vector $\vec{B}$.

Key Trajectory Cases for AMC:

1. $\theta = 0^\circ$ or $180^\circ$ (Parallel/Antiparallel): $F_B = 0$. The particle continues in a straight line un-deflected.
2. $\theta = 90^\circ$ (Perpendicular): $F_B = q v B$ acts as a centripetal force ($q v B = \frac{m v^2}{r}$), forcing the particle into a circular orbit of radius: $r = \frac{m v}{q B}$ Cyclotron frequency of revolution is $f = \frac{q B}{2\pi m}$.
3. Arbitrary angle $\theta$: The particle executes a helical path.

Magnetic Force on a Current-Carrying Conductor

A straight conductor of length vector $\vec{L}$ carrying current $I$ in a magnetic field $\vec{B}$ experiences force:

$\vec{F} = I (\vec{L} \times \vec{B}) \implies F = I L B \sin\theta$

Ampere's Law & Solenoids

Ampere's Circuital Law relates integrated magnetic field around a closed loop to enclosed current:

$\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enclosed}$

• Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$.
• Inside Solenoid: $B = \mu_0 n I$ (where $n = \frac{N}{L}$ is turns per unit length).

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2. Electromagnetic Induction & Faraday's Law

Magnetic Flux ((\Phi_B))

Magnetic flux measures net magnetic field lines penetrating a surface area $A$:

$\Phi_B = \vec{B} \cdot \vec{A} = B A \cos\theta \quad [\text{SI Unit: Weber (Wb) = T}\cdot\text{m}^2]$

Faraday's Law of Induction

Whenever magnetic flux linked with a closed coil changes over time, an induced electromotive force (EMF) $\mathcal{E}$ is generated across the coil:

$\mathcal{E} = -N \frac{\Delta \Phi_B}{\Delta t}$

where $N$ is number of coil turns.

Motional EMF

When a conductor of length $L$ moves at velocity $v$ perpendicular to magnetic field $B$:

$\mathcal{E} = v B L$

Lenz's Law & Conservation of Energy

The negative sign in Faraday's Law reflects Lenz's Law: The direction of induced current always flows such that its own magnetic field opposes the original change in magnetic flux that generated it.

Physical Principle: Lenz's Law is a direct consequence of the Law of Conservation of Energy.

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3. Inductance & Transformers

Self-Inductance ($L$) & Mutual Inductance ($M$)

• Self-Inductance: Induced EMF in a coil opposing current change in itself: $\mathcal{E} = -L \frac{\Delta I}{\Delta t}$ (Unit: Henry, H).
• Energy stored in magnetic field of an inductor: $U_B = \frac{1}{2} L I^2$.
• Mutual Inductance: Induced EMF in secondary coil due to primary current change: $\mathcal{E}_s = -M \frac{\Delta I_p}{\Delta t}$.

AC Transformer Principles

A transformer alters alternating voltages via magnetic induction across iron cores:

$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$

• Step-Up Transformer: $N_s > N_p \implies V_s > V_p$ and $I_s < I_p$.
• Step-Down Transformer: $N_s < N_p \implies V_s < V_p$ and $I_s > I_p$.
• Ideal Transformer Efficiency: $P_{in} = P_{out} \implies V_p I_p = V_s I_s$.

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4. Alternating Current (AC) Circuits & Resonance

RMS Values

For sinusoidal AC voltage $V(t) = V_0 \sin(\omega t)$:

$V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0, \quad I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$

In Pakistan, standard AC domestic mains supply is $220 \text{ V (rms)}$ at $50 \text{ Hz}$. Peak voltage is $V_0 = 220 \times \sqrt{2} \approx 311 \text{ V}$.

Reactance & Phase Relationships

1. Pure Resistor: Voltage and current are in phase ($\phi = 0^\circ$). Resistance $R$.
2. Pure Inductor: Voltage leads current by $90^\circ$ ($\pi/2$). Inductive Reactance $X_L = \omega L = 2\pi f L$.
3. Pure Capacitor: Current leads voltage by $90^\circ$ ($\pi/2$). Capacitive Reactance $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$.

Series RLC Circuit & Resonance

Total opposition to current flow in a series RLC circuit is Impedance $Z$:

$Z = \sqrt{R^2 + (X_L - X_C)^2} \quad [\text{Unit: Ohm (}\Omega\text{)}]$

Phase angle $\phi$ between voltage and current: $\tan\phi = \frac{X_L - X_C}{R}$. Average power dissipated: $P_{avg} = V_{rms} I_{rms} \cos\phi$, where $\cos\phi$ is the power factor.

Electrical Resonance Condition:

Resonance occurs when inductive reactance equals capacitive reactance ($X_L = X_C$):

$2\pi f_r L = \frac{1}{2\pi f_r C} \implies f_r = \frac{1}{2\pi \sqrt{L C}}$

At Resonance:

• Impedance is minimum: $Z = R$.
• Current is maximum: $I_{max} = \frac{V_{rms}}{R}$.
• Power factor is unity: $\cos\phi = 1$.
• Circuit is purely resistive.

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5. Worked Numerical Examples for AMC Candidates

Example 1: Cyclotron Radius of Moving Electron

Problem: An electron ($m = 9.1 \times 10^{-31} \text{ kg}, q = 1.6 \times 10^{-19} \text{ C}$) enters a magnetic field $B = 0.01 \text{ T}$ perpendicularly at velocity $v = 2 \times 10^6 \text{ m/s}$. Find its orbit radius.

Solution: $r = \frac{m v}{q B} = \frac{(9.1 \times 10^{-31} \text{ kg}) (2 \times 10^6 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C}) (0.01 \text{ T})} = \frac{1.82 \times 10^{-24}}{1.6 \times 10^{-21}} = 1.1375 \times 10^{-3} \text{ m} \approx 1.14 \text{ mm}$

Example 2: Transformer Output Calculation

Problem: A step-down transformer has primary turns $N_p = 1000$ and secondary turns $N_s = 100$. If primary voltage is $220 \text{ V}$, find secondary voltage.

Solution: $\frac{V_s}{V_p} = \frac{N_s}{N_p} \implies V_s = 220 \times \left(\frac{100}{1000}\right) = 220 \times 0.1 = 22 \text{ V}$