Comprehensive Physics Study Notes: Magnetic Effects of Current

Biot-Savart Law and Ampere's Circuital Law\n\n* The Biot-Savart Law: The magnetic field contribution $dB$ produced by a small current element $dl$ carrying current $I$ at a point located at distance $r$ is given by:\n $dB = \frac{\mu_0}{4 \pi} \frac{I \, dl \sin(\theta)}{r^2}$\n\n* Vector Form of Biot-Savart Law:\n $d\mathbf{B} = \frac{\mu_0}{4 \pi} \frac{I (d\mathbf{l} \times \mathbf{r})}{r^3}$\n\n* Ampere's Circuital Law: The line integral of the magnetic field $\mathbf{B}$ around any closed loop is equal to $\mu_0$ times the total current $I$ passing through the surface enclosed by the loop:\n $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$\n\n# Magnetic Fields Produced by Current-Carrying Conductors\n\n* Straight Current-Carrying Conductor: For a wire of finite length with angles $\phi_1$ and $\phi_2$ subtended at the point of observation located at a perpendicular distance $a$ from the wire:\n $B = \frac{\mu_0 I}{4 \pi a} (\sin(\phi_1) + \sin(\phi_2))$\n\n* Infinite Long Current-Carrying Conductor: For an infinitely long straight wire, the magnetic field is:\n $B = \frac{\mu_0 I}{2 \pi a}$\n\n* Semi-Infinite Wire: The magnetic field at one end of a semi-infinite wire is:\n $B = \frac{\mu_0 I}{4 \pi a}$\n\n* Circular Current-Carrying Loop:\n * At the center of the loop (radius $R$):\n $B = \frac{\mu_0 I}{2 R}$\n * At a point on the axis of the loop at a distance $x$ from the center:\n $B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}$\n\n* Current-Carrying Arc: For an arc of a circle with radius $R$ subtending an angle $\theta$ at the center:\n $B = \frac{\mu_0 I}{2 R} \times \frac{\theta}{360}$\n\n* Solenoid:\n * Magnetic field inside a solenoid: $B = \mu_0 n I$, where $n = \frac{N}{l}$ (number of turns per unit length).\n * Magnetic field at the end of a solenoid: $B = \frac{\mu_0 n I}{2}$\n\n* Toroidal Solenoid: The magnetic field inside the core of a toroid is:\n $B = \mu_0 n I$, where $n = \frac{N}{2 \pi R}$\n\n# Force Experienced by Charges and Conductors\n\n* Force on a Moving Charge: The magnetic force $\mathbf{F}$ on a charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ is:\n $\mathbf{F} = q (\mathbf{v} \times \mathbf{B})$\n $F = q v B \sin(\theta)$\n\n* Force on a Current-Carrying Conductor: A conductor of length $l$ carrying current $I$ in a magnetic field $\mathbf{B}$ experiences a force:\n $\mathbf{F} = I (\mathbf{l} \times \mathbf{B})$\n $F = I l B \sin(\theta)$\n\n* Force Between Two Parallel Current-Carrying Conductors:\n * Total force on length $l$ with separation distance $d$:\n $F = \frac{\mu_0 I_1 I_2 l}{2 \pi d}$\n * Force per unit length:\n $\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2 \pi d}$\n * Direction of Force: If the currents flow in the same direction, the conductors experience an attractive force. If the currents flow in opposite directions, the conductors experience a repulsive force.\n\n# Motion of a Charged Particle in a Magnetic Field\n\n* Case 1: $\theta = 0$: The particle moves in a straight line.\n\n* Case 2: $\theta = 90^\circ$: The particle follows a circular path.\n * Radius of the circle: $r = \frac{m v}{q B}$\n * Time Period: $T = \frac{2 \pi m}{q B}$\n\n* Case 3: $0 < \theta < 90^\circ$: The particle follows a helical path ($\theta$ is the angle between velocity $\mathbf{v}$ and magnetic field $\mathbf{B}$).\n * Radius of the helix: $r = \frac{m v \sin(\theta)}{q B}$\n * Pitch of the helix (linear distance covered per rotation): $p = v \cos(\theta) \times T$\n\n# Magnetic Dipole Moment and Torque\n\n* Magnetic Dipole Moment ($M$ or $m$):\n * For a single current loop: $M = I A$\n * For a coil with $N$ turns: $M = N I A$\n\n* Torque on a Current-Carrying Coil: A coil in a magnetic field experiences a torque $\tau$ defined as:\n $\tau = N I B A \sin(\theta)$\n $\tau = M B \sin(\theta)$\n $\mathbf{\tau} = \mathbf{M} \times \mathbf{B}$