MHT-CET Magnetism Cheatsheet

Cheatsheet Content

1. Magnetic Fields and Forces Magnetic Force on a Moving Charge (Lorentz Force): $ \vec{F} = q(\vec{v} \times \vec{B}) $ Magnitude: $ F = qvB \sin\theta $ Direction: Given by the right-hand rule. If $ \vec{v} \parallel \vec{B} $, then $ F=0 $. If $ \vec{v} \perp \vec{B} $, then $ F = qvB $. Motion of Charge in Uniform Magnetic Field: Circular path if $ \vec{v} \perp \vec{B} $: Radius: $ r = \frac{mv}{qB} $ Angular frequency (cyclotron frequency): $ \omega = \frac{qB}{m} $ Time period: $ T = \frac{2\pi m}{qB} $ Frequency: $ f = \frac{qB}{2\pi m} $ Helical path if $ \vec{v} $ has components parallel and perpendicular to $ \vec{B} $. Magnetic Force on a Current-Carrying Conductor: $ \vec{F} = I(\vec{L} \times \vec{B}) $ Magnitude: $ F = ILB \sin\theta $ For a straight wire in uniform $ \vec{B} $. Magnetic Force Between Two Parallel Current-Carrying Conductors: $ F = \frac{\mu_0 I_1 I_2 L}{2\pi d} $ Attractive if currents are in the same direction. Repulsive if currents are in opposite directions. $ \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} $ (Permeability of free space). 2. Sources of Magnetic Field Biot-Savart Law: $ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2} $ $ \hat{r} $ is a unit vector from $ d\vec{l} $ to the point where $ \vec{B} $ is calculated. Magnetic Field due to a Straight Current-Carrying Wire: Infinite wire: $ B = \frac{\mu_0 I}{2\pi r} $ Finite wire: $ B = \frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2) $ Magnetic Field at the Centre of a Circular Loop: $ B = \frac{\mu_0 I}{2R} $ For N turns: $ B = \frac{\mu_0 N I}{2R} $ Magnetic Field on the Axis of a Circular Loop: $ B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} $ At $ x=0 $, reduces to center formula. Ampere's Circuital Law: $ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} $ $ I_{enc} $ is the net current enclosed by the Amperian loop. Magnetic Field of a Solenoid: Inside (long solenoid): $ B = \mu_0 n I $ ($n$ is turns per unit length) At ends: $ B = \frac{1}{2}\mu_0 n I $ Magnetic Field of a Toroid: $ B = \frac{\mu_0 N I}{2\pi r} $ ($N$ is total turns, $r$ is average radius) 3. Torque on a Current Loop, Magnetic Dipole Moment Torque on a Current Loop in a Magnetic Field: $ \vec{\tau} = \vec{M} \times \vec{B} $ Magnitude: $ \tau = M B \sin\theta $ $ \theta $ is the angle between the magnetic dipole moment $ \vec{M} $ and $ \vec{B} $. Magnetic Dipole Moment of a Current Loop: $ \vec{M} = I \vec{A} $ For N turns: $ M = NIA $ Direction: Perpendicular to the plane of the loop, given by right-hand rule. Potential Energy of a Magnetic Dipole: $ U = -\vec{M} \cdot \vec{B} = -MB \cos\theta $ Minimum energy when $ \vec{M} \parallel \vec{B} $ (stable equilibrium). Maximum energy when $ \vec{M} \parallel -\vec{B} $ (unstable equilibrium). 4. Galvanometers, Ammeters, Voltmeters Moving Coil Galvanometer: $ I \propto \phi $ (deflection angle) Current sensitivity: $ S_I = \frac{\phi}{I} = \frac{NBA}{k} $ ($k$ is torsional constant) Voltage sensitivity: $ S_V = \frac{\phi}{V} = \frac{NBA}{kR} $ Conversion of Galvanometer to Ammeter: A low resistance shunt ($R_s$) is connected in parallel. $ R_s = \frac{I_g R_g}{I - I_g} $ Ideal ammeter has zero resistance. Conversion of Galvanometer to Voltmeter: A high resistance ($R_H$) is connected in series. $ R_H = \frac{V}{I_g} - R_g $ Ideal voltmeter has infinite resistance. 5. Earth's Magnetism Magnetic Elements of Earth: Magnetic Declination ($ \alpha $): Angle between geographic meridian and magnetic meridian. Magnetic Dip or Inclination ($ \delta $): Angle made by the resultant magnetic field of Earth with the horizontal direction. $ B_H = B \cos\delta $, $ B_V = B \sin\delta $ $ \tan\delta = \frac{B_V}{B_H} $ Horizontal Component of Earth's Magnetic Field ($ B_H $). Neutral Points: Points where the net magnetic field is zero. 6. Magnetic Properties of Materials Magnetic Intensity ($ H $): $ H = \frac{B}{\mu} = \frac{B}{\mu_0 \mu_r} $ (Ampere-turns/meter) Magnetization ($ M $): Net magnetic dipole moment per unit volume. Magnetic Susceptibility ($ \chi $): $ M = \chi H $ (dimensionless) Relative Permeability ($ \mu_r $): $ \mu_r = 1 + \chi $ Permeability ($ \mu $): $ \mu = \mu_0 \mu_r = \mu_0 (1 + \chi) $ Relation between $ B, H, M $: $ B = \mu_0(H + M) = \mu_0(1 + \chi)H = \mu_0 \mu_r H $ Classification of Magnetic Materials: Diamagnetic: $ \chi $ is small, negative ($ -1 Paramagnetic: $ \chi $ is small, positive ($ 0 1 $. Weakly attracted by magnets. Examples: Al, Na, O$_2$. Ferromagnetic: $ \chi $ is large, positive ($ \chi \gg 1 $). $ \mu_r \gg 1 $. Strongly attracted by magnets. Exhibit hysteresis. Examples: Fe, Ni, Co. Curie's Law for Paramagnetism: $ \chi = \frac{C}{T} $ (C is Curie constant, T is absolute temperature). Curie Temperature ($ T_C $): Temperature above which ferromagnetic substances become paramagnetic.