Magnetic Effects of Current – Detailed Examination Notes

Page 1 – Historical Background & Fundamental Definitions

• 1820: Hans Christian Øersted discovers that electric current in a straight conductor deflects a compass needle.
  • Deflection ∝ magnitude of current.
  • Reversing current reverses deflection.
• Ampère’s “Swimming Rule” (Mnemonic – SNOW)
  1. Imagine a man swimming along the wire in the direction of current; current enters his feet, leaves his head.
  2. With his face toward the needle, the north pole of the compass is deflected toward his left hand.
  3. SNOW: Current from South to North over a magnetic needle deflects its North pole toward West.
• Magnetic field (magnetic induction, magnetic flux density) B
  • Region around a current-carrying conductor or magnet where magnetic effects are felt.
  • Disappears when current stops (moving charge needed).
  • Moving charge acts as a source of both electric and magnetic fields.
  • $\vec B$ is a vector.
• Biot–Savart Law – qualitative statement
  • Deals with magnetic induction at a point due to elemental current segment $I\,d\vec l$.
  • Geometry: point P at distance $r$ from element; angle between $d\vec l$ and $\vec r$ is $\theta$.

Page 2 – Biot–Savart Law: Mathematical Form & Consequences

• Proportionalities for elemental field $d\vec B$:
  1. $dB \propto I$
  2. $dB \propto dl$
  3. $dB \propto \sin \theta$
  4. $dB \propto \frac{1}{r^{2}}$
• Combining and inserting the constant $\mu<em>0/4\pi$ (SI) or $k</em>m$ (CGS):
  $d\vec B = \frac{\mu_0}{4\pi}\, \frac{I\, d\vec l \times \hat r}{r^{2}}$
  where $\hat r$ is the unit vector from element to P.
• $\mu_0$ (absolute permeability of free space) = $4\pi \times 10^{-7}\;\mathrm{Wb\,A^{-1}m^{-1}}$.
• Direction: right-hand screw rule ($d\vec l$ → tip of screw, rotation gives $\vec B$).
• For entire current path: integrate
  $\vec B = \frac{\mu_0}{4\pi} \int \frac{I\, d\vec l \times \hat r}{r^{2}}$
• Current density form (use $\vec J\, dV = I d\vec l$):
  $\vec B = \frac{\mu_0}{4\pi} \int \frac{\vec J \times \hat r}{r^{2}}\, dV$
• Comparisons with Coulomb’s law:
  1. Both long-range ( $\propto 1/r^{2}$ ).
  2. Superposition principle holds.
  3. Electrostatic source is scalar charge $q$; magnetostatic source is vector $I d\vec l$.
  4. Electrostatic field along $\vec r$; magnetic field perpendicular to plane of $I d\vec l$ and $\vec r$.
  5. Angle factor $\sin\theta$ present only in Biot–Savart.

Page 3 – Relation among $\mu<em>0$, $\varepsilon</em>0$, and $c$

• $\frac{1}{4\pi\varepsilon_0}=9\times10^{9}\;\mathrm{N\,m^{2}C^{-2}}$
• $\mu_0 = 4\pi \times 10^{-7}\;\mathrm{Wb\,A^{-1}m^{-1}}$
• Using Maxwell’s theory: $\mu<em>0\varepsilon</em>0 = \frac{1}{c^{2}}$
  → $c = \frac{1}{\sqrt{\mu<em>0\varepsilon</em>0}} \approx 3\times10^{8}\;\mathrm{m\,s^{-1}}$

Page 3 (cont.) – Magnetic Field on Axis of a Circular Coil

• Coil radius $a$, current $I$, point P on axis at distance $x$ from center.
• From two diametrically opposite elements $d\vec l$, transverse components cancel; axial components add.
• Result for single turn:
  $B = \frac{\mu_0 I a^{2}}{2\,(a^{2}+x^{2})^{3/2}}$
• For N turns:
  $B = \frac{\mu_0 N I a^{2}}{2\,(a^{2}+x^{2})^{3/2}}$
• Special cases
  1. At centre ($x=0$): $B = \frac{\mu_0 N I}{2a}$
  2. Far-field ($x \gg a$): $B \approx \frac{\mu_0 N I a^{2}}{2x^{3}}$ (dipole behaviour).

Page 5 – Direction Rules & Variation Along Axis

• Right-hand thumb rule: curl fingers along current, thumb gives $\vec B$.
• “Clock rule” for faces of a coil
  • Anticlockwise current → north pole (outward $\vec B$).
  • Clockwise current → south pole.
• Variation of $B$ along axis (graphically decreases from centre to zero at infinity as above formula).

Page 5 (end) & Page 6 – Centre of Loop & Ampère’s Circuital Law

• For loop of radius $R$, centre field:
  $B = \frac{\mu_0 N I}{2R}$
• Ampère’s Circuital Law: for closed path $\oint \vec B \cdot d\vec l = \mu<em>0 I</em>{\text{enc}}$.
  • Proof sketched with infinitely long straight conductor: choose circular Amperian loop radius $r$.
  • Since $B$ constant on loop and tangential: $B(2\pi r)=\mu<em>0 I \Rightarrow B=\frac{\mu</em>0 I}{2\pi r}$.

Page 7 – Infinite Straight Wire & Solenoid

• Infinite straight wire: $B=\mu_0 I / (2\pi r)$ (direction via right-hand rule around wire).
• Solenoid (length $l\gg$ diameter, N total turns): turn density $n=N/l$.
  • Inside (near axis): uniform $B=\mu<em>0 n I = \mu</em>0 N I / l$.
  • At ends ≈ $\tfrac{1}{2} \mu_0 n I$.
• Finite solenoid: outside field weak because contributions cancel; inside strong & uniform.

Page 8 – Toroid

• Solenoid bent into closed ring (toroid), N turns, current I.
• Inside core (radius r between inner & outer boundaries): $B=\mu_0 N I /(2\pi r)$ (non-uniform, inversely with r).
• Outside toroid: $B=0$ using Ampère’s law (net current enclosed =0).

Page 9 – Magnetic Force on a Moving Charge

• Lorentz magnetic force: $\vec F = q\, \vec v \times \vec B$
  • Magnitude $F = q v B \sin\theta$.
  • $\theta$ = angle between $\vec v$ and $\vec B$.
  • Direction from right-hand-screw or Fleming’s left-hand rule (for conventional current).
• Special cases
  1. $\theta = 0,\,\pi$ → $F=0$ (motion parallel/antiparallel to $\vec B$).
  2. $\theta = 90^{\circ}$ → $F_{\max}=qvB$.
• SI unit of $B$ (Tesla): $1\,\mathrm{T}=1\,\mathrm{N\,A^{-1}m^{-1}}=1\,\mathrm{Wb\,m^{-2}}$. Defined such that $q=1\,\mathrm{C}, v=1\,\mathrm{m/s}, F=1\,\mathrm{N}, \theta=90^{\circ}$.

Page 10 – Motion in a Uniform Electric Field

• Particle (+q, mass m) enters region length x with initial horizontal speed v, field $\vec E$ vertical.
• Force $qE$ upward → acceleration $a=qE/m$.
• Time inside field $t=x/v$.
• Vertical displacement $y = \tfrac{1}{2} a t^{2} = \frac{qE x^{2}}{2 m v^{2}}$.
• Trajectory is a parabola: $x^{2} = \frac{2m v^{2}}{qE}\, y$.

Page 11 – Motion in a Uniform Magnetic Field

• Velocity components: $v<em>{\parallel}=v \cos\theta$ (along $\vec B$), $v</em>{\perp}=v \sin\theta$ (perpendicular).
• Perpendicular component causes circular motion (centripetal force from magnetic Lorentz force):
  $\frac{m v<em>{\perp}^{2}}{r} = q v</em>{\perp} B \Rightarrow r = \frac{m v \sin\theta}{q B}$.
• Angular (cyclotron) frequency and time period:
  $\omega = \frac{qB}{m},\qquad T = \frac{2\pi m}{qB}$ (independent of speed and radius).
• Parallel component → uniform motion → helix; pitch (advance per turn):
  $p = v \cos\theta \; T = \frac{2\pi m v \cos\theta}{qB}$.

Page 12 – Lorentz Force in Combined $\vec E$ and $\vec B$ Fields

• General force: $\vec F = q(\vec E + \vec v \times \vec B)$.
• Special situations
  1. $\vec E\, ,\vec B, \vec v$ collinear → magnetic part zero, particle accelerates linearly by $\vec E$.
  2. $\vec E\perp\vec B$ and $\vec v$ adjusted so that $qE = qvB$ (opposite directions) → net force zero → particle passes undeflected. Velocity selector condition: $v = E/B$.

Page 13 – Force on a Current-Carrying Conductor

• Consider conductor length $l$ carrying current $I$ at angle $\theta$ to field $\vec B$.
• Force: $\vec F = I \,(\vec l \times \vec B)$; magnitude $F = B I l \sin\theta$.
• Microscopic view: current $I = n e A v_d$ (drift). Using $N$ electrons segment, force derives to same macroscopic form.
• Direction by Fleming’s left-hand rule or right-hand palm rule.

Page 15 – Force Between Two Parallel Currents & Definition of Ampere

• Currents $I<em>1, I</em>2$ separated by distance $d$.
• Field from wire 1 at wire 2: $B<em>1 = \mu</em>0 I_1 / (2\pi d)$.
• Force per unit length on wire 2: $\frac{F}{l}= I<em>2 B</em>1 = \frac{\mu<em>0 I</em>1 I_2}{2\pi d}$.
  • Same magnitude (Newton’s 3rd), opposite direction.
• Currents same direction → attractive; opposite → repulsive.
• Ampere (definition): currents of $1\,\mathrm{A}$ in two infinite parallel conductors 1 m apart produce force $2 \times 10^{-7}\;\mathrm{N\,m^{-1}}$.

Page 17 – Torque on a Current Loop; Magnetic Dipole Moment

• Rectangular loop (area $A=lb$) in uniform field $\vec B$; angle between area vector $\vec A$ and $\vec B$ is $\theta$.
• Forces on opposite sides produce couple; net torque:
  $\tau = B I A \sin\theta$.
• For N turns: $\tau = B I A N \sin\theta$.
• Define magnetic dipole moment:
  $\vec m = I N \vec A$.
  Then $\vec \tau = \vec m \times \vec B$, $|\tau_{\max}| = B I A N$ (when loop plane perpendicular to field).

Page 18 – Moving-Coil Galvanometer (MCG)

• Principle: current-carrying coil in radial magnetic field experiences torque $\tau = B I N A$ (since $\sin 90^{\circ}=1$).
• Construction Elements
  1. Rectangular multi-turn coil, suspended by phosphor-bronze strip.
  2. Cylindrical pole pieces + soft-iron core → strong radial field (lines radial, plane of coil always parallel to field).
  3. Mirror or pointer attached; torsion spring provides restoring torque $\tau_r = k \phi$.
• Working equilibrium: $B N A I = k \phi \Rightarrow I = (k/ B N A) \phi = G \phi$.
  • Galvanometer constant (figure of merit) $G = k / (B N A) \;\mathrm{A/div}$.
• Sensitivities
  1. Current sensitivity $\phi/I = B N A / k$ (↑ by ↑N, B, A or ↓k).
  2. Voltage sensitivity $\phi/V = (\phi/I)/R_g$ (total series resistance).

Page 20 – Converting Galvanometer to Ammeter

• Need low overall resistance; connect shunt $S$ parallel to galvanometer (resistance $G$).
• Let $I_g$ be full-scale galvanometer current, desired full-scale $I$.
• Same potential across parallel paths: $(I-I<em>g) S = I</em>g G$ →
  $S = \frac{I<em>g G}{I - I</em>g}$ (≪ G).
• Resulting ammeter resistance $R_A = \frac{G S}{G + S} \approx S$ (low).

Page 21 – Converting Galvanometer to Voltmeter

• Need high resistance; connect series resistor $R_s$ with galvanometer.
• For full-scale voltage $V$:
  $V = I<em>g (G + R</em>s) \Rightarrow R<em>s = \frac{V}{I</em>g} - G$.
• Voltmeter resistance $R<em>V = G + R</em>s$ (≫ circuit resistance to minimize loading).
• Ideal voltmeter: $R<em>V \to \infty$; ideal ammeter: $R</em>A \to 0$.

Page 21 (addendum) – Magnetic Dipole Moment of Bar Magnet

• For magnet poles $+p<em>m, -p</em>m$ separated by length $2l$:
  $M = p_m (2l)$ directed from south to north pole.
• For current loop, same $M = I A N$ (equivalence principle).