Magnetism

Study of magnetic fields, forces, and their relationship to electric currents.

Definition

Magnetism is a physical phenomenon produced by the motion of electric charge, resulting in attractive and repulsive forces between objects. Magnetic fields are vector fields that exert forces on moving charges and magnetic materials.

Direction Rules

Right Hand Grip Rule (RHR)

For a current-carrying straight wire or solenoid:

  1. Thumb points in the direction of conventional current $I$ ($+$ to $-$).
  2. Curled fingers show the direction of magnetic field $\vec{B}$.
  3. The direction of $\vec{B}$ at any point is tangent to the field line.

For a solenoid or coil, the thumb points toward the North pole (direction of $\vec{B}$ inside).

2D Vector Notation

Symbol Meaning
$\odot$ Out of the page (arrow head coming toward you)
$\otimes$ Into the page (arrow tail moving away)

These apply to current $I$, magnetic field $\vec{B}$, force $\vec{F}$, and velocity $\vec{v}$.

Direction relationships:

  • $I$ out of page → $\vec{B}$ is counter-clockwise
  • $I$ into page → $\vec{B}$ is clockwise

Key Concepts

  • Magnetic Field (B) — field that exerts force on moving charges
  • Magnetic Field Lines — form closed loops, from N to S pole
  • Lorentz Force — force on moving charge: $$\vec{F} = q\vec{v} \times \vec{B}$$ Magnitude: $|F_B| = |q|vB\sin\theta$
  • Direction Rules — RHR, LHR (Fleming's); for $-q$, reverse the force direction or swap $v$ and $B$
  • Stationary Charge — no magnetic force when $v = 0$
  • Cyclotron Motion — circular motion when $v \perp B$
  • Helical Motion — spiral path when $v$ is at an angle to $B$
  • Velocity Selector — crossed $E$ and $B$ fields; particles with $v = E/B$ pass undeflected
  • Mass Spectrometer — uses velocity selector + mass selector to separate ions by mass
  • Force on Current-Carrying Wire — $\vec{F} = I\vec{L} \times \vec{B}$
  • Force Between Parallel Wires — same-direction currents attract, opposite-direction currents repel; magnitude $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$; per-unit-length force $f = \frac{\mu_0 I_1 I_2}{2\pi d}$
  • Ampere's Law — relates current to magnetic field: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
  • Magnetic Field of Solenoid — $B = \mu_0 nI$ (nearly uniform inside; external field $\approx 0$ near centre when $L \gg r$)
  • Magnetic Field of Circular Loop (centre) — $B = \frac{\mu_0 NI}{2r}$ ($N=1$ by default; note there is no $\pi$ in this formula)
  • Magnetic Field Inside Wire — for $r < R$, $B = \frac{\mu_0 I r}{2\pi R^2}$; enclosed current scales with area ratio $I_{enc} = \frac{\pi r^2}{\pi R^2}I$
  • Superposition Principle — net $\vec{B}$ is the vector sum of fields from all current elements
  • Magnetic Dipole Moment — $\vec{\mu} = N I\vec{A}$ for an $N$-turn coil
  • Torque on Current Loop — $\vec{\tau} = \vec{\mu} \times \vec{B}$; magnitude $\tau = N I A B \sin\theta$ (valid for any planar shape)
  • Torque Direction — Fleming's rule or RHR; torque acts to align $\vec{A}$ with $\vec{B}$
  • DC Motor Commutator — split-ring reverses current every half-rotation, enabling continuous rotation

Force Comparison: Moving Charge vs Current-Carrying Wire

flowchart LR
    subgraph MC["Force on Moving Charge"]
        direction TB
        MC1["F = q(v × B)"]
        MC2["|F| = |q|vB sin θ"]
        MC3["Direction: RHR (reverse for -q)"]
    end

    subgraph CW["Force on Current-Carrying Wire"]
        direction TB
        CW1["F = I(L × B)"]
        CW2["|F| = ILB sin θ"]
        CW3["Direction: RHR"]
    end

    MC1 -.->|"Analogous"| CW1

Key Formulas

Formula Description
$ F_B
$r = \frac{mv}{qB}$ Cyclotron radius
$T = \frac{2\pi m}{qB}$ Period of circular motion (independent of $v$)
$\omega = \frac{qB}{m}$ Angular frequency (cyclotron)
$v = \frac{E}{B}$ Velocity selector condition
$m = \frac{qrB^2}{E}$ Mass selector (mass spectrometer)
$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$ Force between parallel wires
$f = \frac{\mu_0 I_1 I_2}{2\pi d}$ Force per unit length between parallel wires
$B = \frac{\mu_0 I}{2\pi r}$ Field around long straight wire ($r \ge R$)
$B = \frac{\mu_0 I r}{2\pi R^2}$ Field inside long straight wire ($r < R$)
$B = \frac{\mu_0 NI}{2r}$ Field at centre of circular loop ($N=1$ default)
$B = \mu_0 nI$ Solenoid interior field
$B = \frac{\mu_0 IR^2}{2(R^2 + x^2)^{3/2}}$ Field on axis of loop
$\tau = N I A B \sin\theta$ Torque on $N$-turn current loop
$\tau = \mu B\sin\theta$ Torque via dipole moment
$\mu = NIA$ Magnetic dipole moment ($N$-turn coil)

Field Formula Decision Flowchart

flowchart TD
    A["Calculate B"] --> B{"Geometry?"}
    B -->|"Long straight wire"| C{"Location?"}
    B -->|"Circular loop"| D{"Location?"}
    B -->|"Solenoid"| E["B = μ₀nI (inside, uniform)"]
    C -->|"Outside (r ≥ R)"| F["B = μ₀I / 2πr"]
    C -->|"Inside (r < R)"| G["B = μ₀Ir / 2πR²"]
    D -->|"At centre"| H["B = μ₀NI / 2r"]
    D -->|"On axis (distance x)"| I["B = μ₀IR² / 2(R²+x²)^(3/2)"]

Physical Constants

  • Permeability of free space: $\mu_0 = 4\pi \times 10^{-7}; \text{T}\cdot\text{m}\cdot\text{A}^{-1}$
  • Unit conversion: $1,\text{G} = 10^{-4},\text{T}$

Force Between Parallel Wires

When two long straight parallel wires carry currents, each wire produces a magnetic field that exerts a force on the other.

  • Same-direction currents → attract
  • Opposite-direction currents → repel

The magnitude of the force on a length $L$ of either wire is:

$$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$

where $d$ is the perpendicular separation between the wires. The force per unit length is:

$$f = \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

The two forces form a Newton's third-law pair: $F_{21} = F_{12}$.

Definition of the Ampere

The ampere (A) is the SI base unit of electric current. One ampere is defined as the constant current that, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed $1,\text{m}$ apart in vacuum, would produce between these conductors a force of $2 \times 10^{-7},\text{N}$ per metre of length. This definition fixes the value of the permeability of free space at exactly:

$$\mu_0 = 4\pi \times 10^{-7};\text{T}\cdot\text{m}\cdot\text{A}^{-1}$$

Permeability vs Permittivity

Aspect Permittivity ($\varepsilon$) Permeability ($\mu$)
Definition Ability of a material to polarize in response to an external electric field Ability of a material to magnetize in response to an external magnetic field
Symbol $\varepsilon$ $\mu$
SI unit Fm$^{-1}$ Hm$^{-1}$ (kg$\cdot$m$\cdot$s$^{-2}\cdot$A$^{-2}$)
Free-space value $8.85 \times 10^{-12}$ Fm$^{-1}$ $1.26 \times 10^{-6}$ Hm$^{-1}$
Related to Electric fields Magnetic fields
Application Dielectric materials in capacitors Transformer cores and inductors

Note: The free-space permeability value $1.26 \times 10^{-6}$ Hm$^{-1}$ is the approximate value of $\mu_0 = 4\pi \times 10^{-7}$ T$\cdot$m$\cdot$A$^{-1}$ used in calculations.

Torque on Current Loops

When a current-carrying coil is placed in a uniform magnetic field, the field exerts forces on opposite sides of the loop, creating a torque.

Maximum and Minimum Torque

Condition Geometry Torque
Maximum Plane of coil parallel to $\vec{B}$ (or $\vec{B} \perp \vec{A}$) $\tau_{max} = N I A B$ ($\sin 90^\circ = 1$)
Minimum Plane of coil perpendicular to $\vec{B}$ (or $\vec{B} \parallel \vec{A}$) $\tau_{min} = 0$ ($\sin 0^\circ = 0$)

DC Motors

A loop carrying DC in a uniform $\vec{B}$ oscillates back and forth (~$90^\circ$) rather than rotating continuously. A split-ring commutator reverses the current direction every half-rotation, allowing the loop to spin continuously.

DC Motor Operation State Diagram

stateDiagram-v2
    [*] --> MaxTorque : Start (θ ≈ 0°)
    MaxTorque --> ZeroTorque_90 : Rotate 0° to 90°
    ZeroTorque_90 --> ReverseRisk : Rotate 90° to 180°
    note right of ReverseRisk
        Without commutator, torque would reverse here
    end note
    ReverseRisk --> MaxTorque_Reversed : Commutator reverses current
    note right of MaxTorque_Reversed
        Split-ring swaps contacts to keep rotation clockwise
    end note
    MaxTorque_Reversed --> ZeroTorque_270 : Rotate 180° to 270°
    ZeroTorque_270 --> MaxTorque : Rotate 270° to 360°<br/>Commutator reverses again

Related Concepts

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