Magnetic Effects of Current and Magnetism Formula Sheet
All key magnetism formulas for JEE - Biot-Savart, Ampere's law, Lorentz force, torque, dipoles, galvanometers, and magnetic materials. Quick revision for JEE Main & Advanced.
Every must-know formula from this chapter in one scannable place. Use it for last-minute revision: each result below is drawn directly from the chapter topics on Biot-Savart law, Ampere’s law, Lorentz force, force on conductors, galvanometers, magnetic dipoles, and magnetic materials.
Memorize $\mu_0 = 4\pi \times 10^{-7}$ T·m/A and the shortcut $\dfrac{\mu_0}{4\pi} = 10^{-7}$ T·m/A. The second form saves real time in almost every numerical.
Biot-Savart Law and Magnetic Field of Currents
Field due to a current element:
$$\boxed{d\vec{B} = \frac{\mu_0}{4\pi}\cdot\frac{I\,d\vec{l}\times\vec{r}}{r^3}} \qquad dB = \frac{\mu_0}{4\pi}\cdot\frac{I\,dl\sin\theta}{r^2}$$| Configuration | Magnetic field | Key point |
|---|---|---|
| Infinite straight wire | $B = \dfrac{\mu_0 I}{2\pi r}$ | Most common building block |
| Finite wire (angles $\alpha,\beta$) | $B = \dfrac{\mu_0 I}{4\pi r}(\sin\alpha + \sin\beta)$ | Measured from the ends |
| Semi-infinite wire | $B = \dfrac{\mu_0 I}{4\pi r}$ | Half of the infinite wire |
| Circular loop (center) | $B = \dfrac{\mu_0 N I}{2R}$ | $\perp$ to plane of loop |
| Circular loop (on axis) | $B = \dfrac{\mu_0 N I R^2}{2(R^2+x^2)^{3/2}}$ | The $3/2$ power |
| Loop, far on axis ($x \gg R$) | $B \approx \dfrac{\mu_0 I R^2}{2x^3} = \dfrac{\mu_0 M}{2\pi x^3}$ | $M = I\pi R^2$ |
| Arc subtending $\theta$ at center | $B = \dfrac{\mu_0 I \theta}{4\pi R}$ | $\theta$ in radians |
| Square loop, side $a$ (center) | $B = \dfrac{2\sqrt{2}\,\mu_0 I}{\pi a}$ | Sum of 4 sides |
Arc special cases (from $B = \dfrac{\mu_0 I\theta}{4\pi R}$):
$$\text{Semicircle: } B = \frac{\mu_0 I}{4R} \qquad \text{Quarter: } B = \frac{\mu_0 I}{8R} \qquad \text{Full circle: } B = \frac{\mu_0 I}{2R}$$Right-hand thumb rule: point the thumb along the current, the curled fingers give the circular field direction. The field around a straight wire forms loops, it never points radially.
Ampere’s Circuital Law
$$\boxed{\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}}$$Maxwell-corrected form (with displacement current):
$$\oint \vec{B}\cdot d\vec{l} = \mu_0\left(I_{\text{enc}} + \epsilon_0\frac{d\Phi_E}{dt}\right)$$| Configuration | Region | Magnetic field |
|---|---|---|
| Infinite wire | Outside | $B = \dfrac{\mu_0 I}{2\pi r}$ |
| Solid cylinder (radius $R$) | Inside, $r| $B = \dfrac{\mu_0 I r}{2\pi R^2}$ | |
| Solid cylinder | Outside, $r>R$ | $B = \dfrac{\mu_0 I}{2\pi r}$ |
| Solenoid | Inside | $B = \mu_0 n I$ |
| Solenoid | Outside (ideal) | $B = 0$ |
| Toroid | Inside | $B = \dfrac{\mu_0 N I}{2\pi r}$ |
| Toroid | Outside | $B = 0$ |
| Infinite current sheet | Either side | $B = \dfrac{\mu_0 K}{2}$ |
Hollow cylinder (inner $a$, outer $b$), current within the conductor ($a Use Ampere’s law only when there is cylindrical, planar, or translational symmetry (infinite wire, solenoid, toroid, cylinder, current sheet). For finite or bent wires, loops, and arcs, fall back to Biot-Savart. Complete Lorentz force (electric + magnetic): The magnetic force is always perpendicular to $\vec{v}$, so $W = \vec{F}\cdot\vec{v} = 0$. The magnetic field changes a particle’s direction, never its speed or kinetic energy. Motion summary by angle between $\vec{v}$ and $\vec{B}$: $\theta = 0^\circ$ gives a straight line ($F=0$); $\theta = 90^\circ$ gives a circle; any other angle gives a helix. Particles accelerated through the same potential $V$ then bent in field $B$: For any shape of wire in a uniform field, the force equals that on a straight wire joining the endpoints: $\vec{F} = I\vec{L}_{\text{eff}}\times\vec{B}$. So a semicircle uses $L_{\text{eff}} = 2R$ (diameter), and a closed loop has $L_{\text{eff}} = 0$, hence zero net force (but possibly nonzero torque). Same-direction currents attract, opposite-direction currents repel. Torque is zero at equilibrium ($\theta=0$), maximum at $\theta=90^\circ$. The potential energy keeps its negative sign: $U = -MB\cos\theta$. Definition of the ampere: two infinite parallel wires 1 m apart each carrying 1 A experience a force of $2\times 10^{-7}$ N/m. Conversion to ammeter (shunt $S$ in parallel, $n = I/I_g$): Conversion to voltmeter (resistance $R$ in series, $m = V/(I_g G)$): To find the galvanometer resistance, halve the deflection by adding a shunt; at half-deflection $G = S$, because equal currents flow through equal parallel resistances. General point at angle $\theta$ to the axis: Force between two coaxial dipoles falls as $r^{-4}$: $F = \dfrac{6\mu_0 M_1 M_2}{4\pi r^4}$ (coaxial), $F = \dfrac{3\mu_0 M_1 M_2}{4\pi r^4}$ (side by side). Bohr magneton: $\mu_B = \dfrac{e\hbar}{2m_e} = 9.27\times 10^{-24}$ A·m². Axial field is twice the equatorial field (not the other way around). The energy sign is negative: $U = -MB\cos\theta$. Cutting a magnet always yields two magnets - no isolated monopoles - and cutting in half either way gives moment $M/2$. Earth’s moment $M_E \approx 8\times 10^{22}$ A·m²; field near equator $B_E \approx 3\times 10^{-5}$ T (30 μT). Curie temperatures: Iron 1043 K, Cobalt 1394 K, Nickel 631 K. Above $T_C$ a ferromagnet becomes paramagnetic. Diamagnetic $\chi$ is negative. The permeability relation is $\mu_r = 1 + \chi$ (plus sign). Paramagnetic $\chi$ decreases with temperature. Domains exist always in a ferromagnet; the external field merely aligns them.Lorentz Force and Motion of Charged Particles
$$\boxed{\vec{F} = q(\vec{v}\times\vec{B})} \qquad F = qvB\sin\theta$$Quantity Formula Note Radius (circular, $v\perp B$) $r = \dfrac{mv}{qB}$ Larger mass and speed widen the circle Time period $T = \dfrac{2\pi m}{qB}$ Independent of $v$ and $r$ Cyclotron frequency $f = \dfrac{qB}{2\pi m}$ Independent of $v$ Angular frequency $\omega = \dfrac{qB}{m}$ Simplest form Radius (helical) $r = \dfrac{mv\sin\theta}{qB}$ Uses $v_\perp$ only Pitch (helical) $p = \dfrac{2\pi m v\cos\theta}{qB}$ Uses $v_\parallel$ Applications (Crossed and Cyclotron Fields)
Device Key formula Note Velocity selector $v = \dfrac{E}{B}$ Only this speed passes undeflected Cyclotron AC frequency $f = \dfrac{qB}{2\pi m}$ Independent of radius Cyclotron max KE $KE_{\max} = \dfrac{q^2 B^2 R^2}{2m}$ At dee radius $R$ Mass spectrometer $m = \dfrac{qB^2 r}{E}$ After velocity selection $v=E/B$ Magnetic focusing $d = \dfrac{\pi m v}{qB}$ Distance between focuses $E \parallel B$ accelerating helix Helix with increasing pitch $E \perp B$ (from rest) drift $v_{\text{drift}} = \dfrac{E}{B}$ Cycloidal motion Force and Torque on Current-Carrying Conductors
$$\boxed{d\vec{F} = I\,d\vec{l}\times\vec{B}} \qquad \boxed{\vec{F} = I\vec{L}\times\vec{B}} \qquad F = BIL\sin\theta$$Quantity Formula Note Force on straight wire $F = BIL\sin\theta$ Max ($BIL$) when $I\perp B$ Force between parallel wires $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ Same direction $\to$ attract Torque on coil $\tau = NIAB\sin\theta$ Max when plane $\parallel B$ Magnetic moment $M = NIA$ $\vec{\tau} = \vec{M}\times\vec{B}$ Potential energy $U = -\vec{M}\cdot\vec{B} = -MB\cos\theta$ Min at $\theta = 0^\circ$ Work to rotate $W = MB(\cos\theta_1 - \cos\theta_2)$ $\theta_1\to\theta_2$ Moving Coil Galvanometer
Quantity Formula Note Deflection $\theta = \dfrac{NBA}{k}\,I$ Linear; $\propto I$ (radial field) Current sensitivity $S_I = \dfrac{\theta}{I} = \dfrac{NBA}{k}$ Higher is more sensitive Voltage sensitivity $S_V = \dfrac{\theta}{V} = \dfrac{NBA}{kG}$ Depends on $G$ too Figure of merit $k_g = \dfrac{1}{S_I}$ Lower is better Property Ammeter Voltmeter Add Shunt in parallel (low R) Resistance in series (high R) Ideal resistance $R_A \to 0$ $R_V \to \infty$ Quality — $\dfrac{R_V}{V} = \dfrac{1}{I_g}$ ($\Omega$/V) Magnetic Dipole and Bar Magnet
$$\boxed{\vec{M} = NI\vec{A}} \qquad M = m\times 2l \ (\text{bar magnet})$$Quantity Formula Note Torque $\tau = MB\sin\theta$ Max at $\theta = 90^\circ$ Potential energy $U = -MB\cos\theta$ Min (stable) at $\theta = 0^\circ$ Work to rotate $W = MB(\cos\theta_1 - \cos\theta_2)$ $\theta_1\to\theta_2$ Energy gap (stable$\to$unstable) $\Delta U = 2MB$ $0^\circ \to 180^\circ$ Oscillation period $T = 2\pi\sqrt{\dfrac{I}{MB}}$ Like a torsion pendulum Angular frequency $\omega = \sqrt{\dfrac{MB}{I}}$ $I$ = moment of inertia Axial field ($r\gg$ length) $B = \dfrac{\mu_0}{4\pi}\dfrac{2M}{r^3}$ Along axis Equatorial field $B = \dfrac{\mu_0}{4\pi}\dfrac{M}{r^3}$ Opposite to $\vec{M}$ Field ratio $\dfrac{B_{\text{axial}}}{B_{\text{equatorial}}} = 2$ At same distance Earth’s Magnetic Field
Quantity Relation Note Angle of dip $\tan\theta = \dfrac{B_V}{B_H}$ $0^\circ$ at equator, $90^\circ$ at poles Horizontal component $B_H = B\cos\theta$ Vertical component $B_V = B\sin\theta$ Magnetic Materials
Quantity Formula Note Magnetic intensity $\vec{H} = \dfrac{\vec{B}}{\mu_0} - \vec{M}$ Unit A/m Solenoid intensity $H = nI$ Depends only on current Magnetization $M = \chi H$ Moment per unit volume Susceptibility $\chi = \mu_r - 1$ Dimensionless Relative permeability $\mu_r = 1 + \chi$ $\mu_r = \mu/\mu_0$ B-H relation $B = \mu_0\mu_r H$ In a medium B-M relation $B = \mu_0(H + M) = \mu_0(1+\chi)H$ Alternative form Curie law (para) $\chi = \dfrac{C}{T}$ $\chi \propto 1/T$ Curie-Weiss (ferro) $\chi = \dfrac{C}{T - T_C}$ Valid above $T_C$ Hysteresis loss $E = \oint H\,dB$ Loop area = energy/cycle/volume Classification of Materials
Property Diamagnetic Paramagnetic Ferromagnetic Susceptibility $\chi$ Negative ($\sim -10^{-5}$) Small positive ($10^{-5}$ to $10^{-3}$) Large positive ($10^2$ to $10^5$) Permeability $\mu_r$ $<1$ $>1$ (slightly) $\gg 1$ (1000 to 100000) Behavior in field Weakly repelled Weakly attracted Strongly attracted Temperature effect Independent $\chi \propto 1/T$ $\chi \propto 1/(T-T_C)$ Examples Bi, Cu, Ag, Au, H₂O Al, Na, O₂, Pt Fe, Co, Ni Hysteresis Terms
Property Soft magnetic Hard magnetic Hysteresis loop Narrow (small area) Wide (large area) Retentivity / Coercivity Low High Energy loss Low High Use Transformer cores, electromagnets Permanent magnets, data storage Examples Soft iron, permalloy Steel, alnico, NdFeB Chapter Map
graph TD
A[Magnetism] --> B[Field of currents]
A --> C[Forces and torque]
A --> D[Dipoles]
A --> E[Materials]
B --> B1["Biot-Savart: dB = (mu0/4pi) I dl x r / r^3"]
B --> B2["Ampere: closed integral B.dl = mu0 I_enc"]
C --> C1["Lorentz: F = q(v x B)"]
C --> C2["Conductor: F = I L x B"]
C --> C3["Galvanometer: theta = NBA I / k"]
D --> D1["M = NIA, tau = M x B, U = -M.B"]
D --> D2["Axial = 2 x Equatorial field"]
E --> E1["chi = mu_r - 1, B = mu0(H+M)"]
E --> E2["Curie & hysteresis"]Last-Minute Reminders
Further Reading