⚡ Magnetic Effects & Magnetism
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PART A moving charge · field
1Oersted's Experiment
No formula — current-carrying conductor produces magnetic field (compass needle deflects).
2Biot–Savart Law
$$d\vec{B} = \frac{\mu_0}{4\pi}\,\frac{I\,d\vec{l} \times \hat{r}}{r^2}$$
\( \vec{B} \) = Magnetic field |
\( \mu_0 \) = Permeability of free space (\(4\pi\times10^{-7}\)) |
\( I \) = Current |
\( d\vec{l} \) = Length element |
\( \hat{r} \) = Unit vector from wire to point |
\( r \) = Distance
$$dB = \frac{\mu_0}{4\pi}\,\frac{I\,dl\sin\theta}{r^2}$$
\( \theta \) = Angle between \(d\vec{l}\) and \(\hat{r}\)
Field at centre of circular loop:
$$B = \frac{\mu_0 I}{2R}$$
\( R \) = Radius of the loop
Field on the axis of circular loop:
$$B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$$
\( x \) = Distance from centre of loop on the axis
TRICK Set \(x=0\) → axial formula collapses to \(\mu_0 I/2R\). Algebra check.
3Ampere's Circuital Law
$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$$
\( \oint \) = Closed line integral |
\( I_{\text{enclosed}} \) = Net current enclosed by the loop
Infinite straight wire:
$$B = \frac{\mu_0 I}{2\pi r}$$
\( r \) = Perpendicular distance from the wire
Ideal solenoid:
$$B = \mu_0 n I$$
\( n \) = Number of turns per unit length (\(N/L\))
TRICK Only current enclosed by Amperian loop matters.
4Lorentz Force
$$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$$
\( q \) = Charge |
\( \vec{E} \) = Electric field |
\( \vec{v} \) = Velocity of charge |
\( \vec{B} \) = Magnetic field
$$F_{\text{mag}} = qvB\sin\theta$$
\( \theta \) = Angle between \(\vec{v}\) and \(\vec{B}\)
5Charged Particle in Perpendicular B — Cyclotron
$$r = \frac{mv}{qB}$$
\( r \) = Radius of circular path |
\( m \) = Mass of particle |
\( v \) = Speed |
\( q \) = Charge
$$T = \frac{2\pi m}{qB} \qquad f = \frac{qB}{2\pi m}$$
\( T \) = Time period |
\( f \) = Frequency (cyclotron frequency)
TRICK \(T, f\) independent of \(v, r\) — cyclotron frequency constant.
6Force on Current-Carrying Conductor
$$F = BIL\sin\theta$$
\( L \) = Length of conductor in magnetic field |
\( \theta \) = Angle between conductor and field
TRICK "FBI" — Fleming's Left Hand: Thumb=Force, First=Field, Middle=Current.
7Force Between Two Parallel Conductors
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
\( F/L \) = Force per unit length |
\( I_1, I_2 \) = Currents in the two wires |
\( d \) = Distance between wires
Same direction → attract; opposite → repel. Defines ampere.
8Torque on a Current Loop
$$\vec{\tau} = NIA\,\vec{B}\sin\theta = \vec{m}\times\vec{B}$$
\( N \) = Number of turns |
\( A \) = Area of the loop |
\( \vec{m} \) = Magnetic moment |
\( \theta \) = Angle between normal and field
$$\vec{m} = NIA$$
Galvanometer: \(NIAB = k\theta\)
$$I_s = \frac{NAB}{k} \qquad V_s = \frac{NAB}{kR}$$
\( I_s \) = Current sensitivity |
\( V_s \) = Voltage sensitivity |
\( k \) = Torsional constant |
\( R \) = Resistance
TRICK Raising \(A\) also raises \(R\), so \(V_s\) gain is not automatic.
9Converting a Galvanometer
$$\text{Shunt (Ammeter): } S = \frac{I_g G}{I-I_g}$$
\( S \) = Shunt resistance |
\( I_g \) = Galvanometer full-scale current |
\( G \) = Galvanometer resistance |
\( I \) = Desired full-scale current
$$\text{Series R (Voltmeter): } R = \frac{V}{I_g} - G$$
\( R \) = Series resistance |
\( V \) = Desired full-scale voltage
TRICK "A → parallel shunt (low R)"; "V → Very high series R."
PART B magnetism · matter
10Current Loop as Magnetic Dipole
$$m = NIA$$
\( m \) = Magnetic dipole moment |
\( N \) = Number of turns |
\( I \) = Current |
\( A \) = Area of loop
Direction by right-hand rule, normal to the loop.
11Magnetic Moment of Revolving Electron
$$m = \frac{evr}{2} = \left(\frac{e}{2m}\right)L$$
\( e \) = Charge of electron |
\( v \) = Speed |
\( r \) = Radius of orbit |
\( L \) = Angular momentum |
\( e/2m \) = Gyromagnetic ratio
12Bar Magnet Field
$$B_{\text{axial}} = \frac{\mu_0}{4\pi}\frac{2m}{x^3}$$
$$B_{\text{equatorial}} = \frac{\mu_0}{4\pi}\frac{m}{x^3}$$
\( m \) = Magnetic moment of bar magnet |
\( x \) = Distance from centre of magnet
TRICK \(B_{\text{axial}} = 2\,B_{\text{equatorial}}\) at same \(x\).
13Torque and Potential Energy of Dipole
$$\tau = mB\sin\theta$$
\( \theta \) = Angle between \(\vec{m}\) and \(\vec{B}\)
$$U = -\vec{m}\cdot\vec{B} = -mB\cos\theta$$
\( U \) = Potential energy
Stable equilibrium at \(\theta = 0°\), unstable at \(180°\).
14Magnetic Properties of Materials
$$\mu = \frac{B}{H} \qquad \chi = \frac{M}{H} \qquad \mu_r = 1+\chi \qquad M = \frac{m}{V} \qquad B = \mu_0(H+M)$$
\( \mu \) = Permeability |
\( \mu_r \) = Relative permeability |
\( \chi \) = Magnetic susceptibility |
\( H \) = Magnetizing field |
\( M \) = Magnetization |
\( V \) = Volume
15Hysteresis (B–H Loop)
Loop area = energy dissipated per cycle. Retentivity = \(B\) remaining when \(H\to0\). Coercivity = reverse \(H\) needed to bring \(B\to0\).
TRICK Hard materials (steel) → fat loop → permanent magnets. Soft iron → thin loop → transformers.
16Earth's Magnetic Elements
$$B_H = B\cos\delta \qquad B_V = B\sin\delta \qquad \tan\delta = \frac{B_V}{B_H}$$
\( B_H \) = Horizontal component |
\( B_V \) = Vertical component |
\( \delta \) = Angle of dip |
\( B \) = Total magnetic field
Declination = angle between geographic north and magnetic north.
17Dia / Para / Ferro-Magnetic Substances
| Type | Susceptibility \(\chi\) | Examples | Behaviour |
|---|---|---|---|
| Diamagnetic | small, negative | Bismuth, Copper, Water | weakly repelled |
| Paramagnetic | small, positive | Aluminium, Sodium, Platinum | weakly attracted |
| Ferromagnetic | large, positive | Iron, Cobalt, Nickel | strongly attracted, retains magnetism |
18Electromagnets
Strength increases with number of turns \(N\), current \(I\), and a soft-iron core (high \(\mu\), low retentivity).
📐 Units & Dimensions
| Quantity | SI Unit | Alternative / Equivalent | Dimension |
|---|---|---|---|
| Magnetic field \(B\) | T (tesla) | Wb/m², N/(A·m) | \(M\,T^{-2}\,A^{-1}\) |
| Magnetic flux \(\Phi\) | Wb (weber) | T·m², V·s | \(M\,L^2\,T^{-2}\,A^{-1}\) |
| Magnetic moment \(m\) | A·m² | J/T | \(L^2\,A\) |
| Magnetization \(M\) | A/m | — | \(L^{-1}\,A\) |
| Magnetic field intensity \(H\) | A/m | — | \(L^{-1}\,A\) |
| Permeability \(\mu\) | H/m (henry/m) | N/A² | \(M\,L\,T^{-2}\,A^{-2}\) |
| Relative permeability \(\mu_r\) | dimensionless | — | 1 |
| Susceptibility \(\chi\) | dimensionless | — | 1 |
| Current \(I\) | A (ampere) | C/s | \(A\) |
| Charge \(q\) | C (coulomb) | A·s | \(T\,A\) |
| Force \(F\) | N (newton) | kg·m/s² | \(M\,L\,T^{-2}\) |
| Torque \(\tau\) | N·m | J | \(M\,L^2\,T^{-2}\) |
| Energy \(U\) | J (joule) | N·m, V·C | \(M\,L^2\,T^{-2}\) |
| Cyclotron frequency \(f\) | Hz (s⁻¹) | — | \(T^{-1}\) |
| Resistance \(R\) | Ω (ohm) | V/A | \(M\,L^2\,T^{-3}\,A^{-2}\) |
⚡ Quick Recall
| Biot–Savart | \(dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}\) |
| Circular loop, centre | \(B = \dfrac{\mu_0 I}{2R}\) |
| Straight wire | \(B = \dfrac{\mu_0 I}{2\pi r}\) |
| Solenoid | \(B = \mu_0 nI\) |
| Lorentz force | \(F = q(\vec{E}+\vec{v}\times\vec{B})\) |
| Cyclotron radius | \(r = \dfrac{mv}{qB}\) |
| Cyclotron frequency | \(f = \dfrac{qB}{2\pi m}\) |
| Force on conductor | \(F = BIL\sin\theta\) |
| Force/length, parallel wires | \(\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}\) |
| Torque on loop | \(\tau = NIAB\sin\theta\) |
| Current sensitivity | \(I_s = \dfrac{NAB}{k}\) |
| Shunt (ammeter) | \(S = \dfrac{I_gG}{I-I_g}\) |
| Series R (voltmeter) | \(R = \dfrac{V}{I_g}-G\) |
| Axial field, bar magnet | \(B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}\) |
| Equatorial field, bar magnet | \(B = \dfrac{\mu_0}{4\pi}\dfrac{m}{x^3}\) |
| Torque on dipole | \(\tau = mB\sin\theta\) |
| Potential energy | \(U = -mB\cos\theta\) |
| Susceptibility | \(\chi = \dfrac{M}{H}\) |
| Relative permeability | \(\mu_r = 1+\chi\) |
📘 Magnetic Effects · full formula sheet with symbol meanings · updated 10 Aug 2026
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