Electromagnetism & Magnetic Forces: Fields, Forces & Applications

TAKEAWAYS:

• Magnetic fields arise from moving charges (currents) and exert forces on other moving charges via the Lorentz force $\mathbf{F}=q(\mathbf{v}\times \mathbf{B})$.
• Biot-Savart law and Ampère’s law mathematically describe magnetic field generation by currents, with the latter simplifying symmetric cases.
• Magnetic dipole moments (e.g., current loops, atoms) create fields and experience torque $\mathbf{\tau }=\mathbf{m}\times \mathbf{B}$ in external fields.
• Applications include electric motors (torque conversion), mass spectrometers (charge/mass analysis), and MRI machines (nuclear spin alignment).
• Key equations: Lorentz force, magnetic field from a wire ($B=\frac{{\mu }_{0}I}{2\pi r}$), torque on a loop ($\tau =NIAB\sin \theta$).
• Real-world tie: Kathmandu’s traffic signals use electromagnetic relays (current-induced magnetic fields to switch circuits), while Ncell’s base stations employ magnetic shielding to protect electronics.

Core Concepts & Visuals

1. Magnetic Fields: Sources and Properties

Magnetic fields ($\mathbf{B}$) are generated by:

• Moving charges (currents) via the Biot-Savart law: $$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Id\mathbf{l}\times \hat{\mathbf{r}}}{r^2}$$
• Permanent magnets (atomic dipole moments aligned).
• Changing electric fields (Maxwell’s extension, not in syllabus but noted for context).

Key Properties:

• Field lines emerge from north poles, loop into south poles (never intersect).
• Strength ($B$) measured in tesla (T) or gauss (1 T = 10⁴ G).
• Direction: Right-hand rule for currents (thumb = current, fingers = $\mathbf{B}$).

graph LR
    A["Moving Charge (Current)"] --> B["Magnetic Field (Biot-Savart)"]
    C["Permanent Magnet"] --> B
    D["Changing E-field"] --> B
    B --> E["Force on Moving Charges (Lorentz)"]
    B --> F["Torque on Current Loops"]

Shows circular field lines around a straight wire, with right-hand rule arrows. (Image: Chetvorno, CC0, via Wikimedia Commons)

Worked Example: Field from a Long Wire A copper wire carries 10 A. Find $B$ at 5 cm from it. Solution: Use Ampère’s law for a straight wire: $$B=\frac{{\mu }_{0}I}{2\pi r}=\frac{(4\pi \times 10^{-7})(10)}{2\pi (0.05)}=4\times 10^{-5}\text{T}=0.4\text{G}$$ Real-world tie: NTC’s power lines generate magnetic fields; birds avoid flying too close due to disorientation (studies show fields > 10 G affect navigation).

2. Lorentz Force: Force on Charged Particles

A charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ experiences: $$\mathbf{F}=q(\mathbf{v}\times \mathbf{B})$$ Key Points:

• Direction: Perpendicular to both $\mathbf{v}$ and $\mathbf{B}$ (right-hand rule).
• Magnitude: $F=qvB\sin \theta$.
• No work done: Force is always perpendicular to displacement ($\mathbf{F}\cdot \mathbf{v}=0$).

Worked Example: Proton in a Mass Spectrometer A proton ($q=+e$) moves at $\mathbf{v}=(3\times 10^5\mathbf{i}+7\times 10^5\mathbf{k})\text{m/s}$ in $\mathbf{B}=0.4\mathbf{j}\text{T}$. Find $\mathbf{F}$. Solution: $$\mathbf{v}\times \mathbf{B}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 3\times 10^5& 0& 7\times 10^5\\ 0& 0.4& 0\end{array}\right|=-2.8\times 10^5\mathbf{i}+1.2\times 10^5\mathbf{k}\text{N/C}$$ $$\mathbf{F}=e(-2.8\times 10^5\mathbf{i}+1.2\times 10^5\mathbf{k})=-4.48\times 10^{-14}\mathbf{i}+1.92\times 10^{-14}\mathbf{k}\text{N}$$ Real-world tie: Hospitals use cyclotrons (like Nepal’s B.P. Koirala Memorial Hospital’s research labs) to accelerate protons/charged particles via magnetic fields for cancer therapy.

3. Magnetic Dipole Moment and Torque

A current loop of area $A$ carrying current $I$ has a magnetic dipole moment: $$\mathbf{m}=I\mathbf{A}$$ Torque on a Dipole: In an external field $\mathbf{B}$, the loop experiences: $$\mathbf{\tau }=\mathbf{m}\times \mathbf{B}$$ Key Points:

• Stable equilibrium: $\mathbf{m}\parallel \mathbf{B}$ (minimum potential energy).
• Unstable equilibrium: $\mathbf{m}$ antiparallel to $\mathbf{B}$.
• Applications: Electric motors, galvanometers, MRI machines.

Worked Example: Electric Motor Coil A rectangular coil (2 cm × 3 cm, 50 turns) carries 2 A in a 0.5 T field. Find the maximum torque. Solution: $$A=0.02\times 0.03=6\times 10^{-4}{\text{m}}^2$$ $${\tau }_{\text{max}}=NIAB=50\times 2\times 6\times 10^{-4}\times 0.5=3\times 10^{-2}\text{Nm}$$ Real-world tie: Pathao’s electric scooters use DC motors where torque from current loops propels the vehicle. The battery supplies current to coils in a magnetic field, converting electrical energy to mechanical rotation.

4. Ampère’s Law and Symmetry

Ampère’s law relates magnetic fields to current: $$\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\text{enc}}$$ Applications:

• Solenoids: $B={\mu }_{0}nI$ (ideal, infinite length).
• Toroids: Confined field inside the ring.
• Straight wires: $B=\frac{{\mu }_{0}I}{2\pi r}$.

Comparison Table:

Shows uniform field inside, zero outside (ideal case). (Image: OpenStax, CC BY 4.0, via Wikimedia Commons)

Worked Example: Magnetic Shielding in Electronics A toroidal coil (100 turns, radius 5 cm) carries 0.1 A. Find $B$ inside. Solution: $$B=\frac{{\mu }_{0}NI}{2\pi r}=\frac{(4\pi \times 10^{-7})(100)(0.1)}{2\pi (0.05)}=4\times 10^{-5}\text{T}$$ Real-world tie: Ncell’s base stations use mu-metal (high-permeability) shields to block external magnetic interference, ensuring stable signal transmission.

5. Magnetic Forces on Current-Carrying Wires

A wire of length $L$ carrying current $I$ in a magnetic field $\mathbf{B}$ experiences: $$\mathbf{F}=I(\mathbf{L}\times \mathbf{B})$$ Key Points:

• Direction: Perpendicular to both $\mathbf{L}$ and $\mathbf{B}$.
• Applications: Electric meters, loudspeakers, homopolar motors.

Shows wire perpendicular to $\mathbf{B}$, with $\mathbf{F}$ direction marked. (Image: MikeRun, CC BY-SA 4.0, via Wikimedia Commons)

Worked Example: Force on a Power Line A 100 m aluminum wire (mass = 0.8 kg) carries 50 A horizontally in Earth’s $B=5\times 10^{-5}\text{T}$ field (north). Find the vertical force. Solution: $$F=ILB\sin 90^{\circ }=50\times 100\times 5\times 10^{-5}=0.25\text{N}$$ Real-world tie: NTC’s overhead power lines sag slightly due to this force; engineers account for it in design to prevent short circuits.

## In the Real World

1. eSewa & Khalti Payments:

   • Magnetic Stripe Cards: Older versions use magnetic strips where data is stored as tiny magnetized regions. A magnetic head (electromagnet) reads the pattern as the card swipes past.
   • How it uses this unit: The Lorentz force induces a voltage in the read head’s coil when the magnetized strip passes by, converting magnetic patterns into electrical signals for processing.

2. Daraz’s Logistics & Inventory Systems:

   • Barcode Scanners: Many scanners use magneto-optical sensors where a magnetic field modulates reflected light. The interaction between current (in the sensor) and magnetic fields (from barcodes) changes light polarization, enabling reading.
   • Worked example tie: If a Daraz warehouse uses a conveyor belt with metal tags (current loops), an external $B$-field could induce eddy currents, creating a torque to sort items automatically.

3. Nepal’s Railway Electrification (Future Project):

   • Electric Trains: Use DC motors where torque is generated by current loops in a magnetic field. For example, a 100-turn coil (area = 0.01 m²) in a 0.2 T field with 5 A current produces: $$\tau =NIAB=100\times 5\times 0.01\times 0.2=0.1\text{Nm}$$ This torque accelerates the train’s wheels via gears.

## Exam Tip

1. Memorize Key Equations:

   • Lorentz force: $\mathbf{F}=q(\mathbf{v}\times \mathbf{B})$.
   • Magnetic field from a wire: $B=\frac{{\mu }_{0}I}{2\pi r}$.
   • Torque on a loop: $\tau =NIAB\sin \theta$.
   • Always draw diagrams for cross products (right-hand rule).

2. Unit Consistency:

   • Convert all units to SI (A, m, T, s). For example, $1\text{G}=10^{-4}\text{T}$.

3. Common Pitfalls:

   • Direction mistakes: Use the right-hand rule for $\mathbf{v}\times \mathbf{B}$ and $\mathbf{m}\times \mathbf{B}$.
   • Sign errors: Negative charges reverse the force direction.
   • Symmetry in Ampère’s law: Only use it for symmetric cases (straight wires, solenoids).

4. Problem-Solving Strategy:

   • Step 1: Identify moving charges/current loops.
   • Step 2: Sketch $\mathbf{v}$, $\mathbf{B}$, and $\mathbf{F}$ or $\mathbf{\tau }$.
   • Step 3: Apply the relevant equation (Lorentz, Biot-Savart, or torque).
   • Step 4: Check units and reasonableness (e.g., force should be in Newtons).

5. Past Exam Patterns:

   • Numerical problems: 60% focus on calculating forces/torques (e.g., proton in a field, wire in a motor).
   • Conceptual questions: 30% on dipole moments, field directions, or applications (e.g., "Why does a compass needle align with Earth’s field?").
   • Derivations: 10% (e.g., derive $B$ for a solenoid using Ampère’s law).

Final Note: Magnetic forces are non-conservative (work done depends on path), unlike electric forces. Always verify whether the question asks for magnitude or vector answers—partial credit is lost for missing directions!