Magnetic moment: M = IA = I x pi r^2
Same current I in both coils.
M proportional to r^2
Radii ratio r1:r2 = 1:2
M1/M2 = r1^2/r2^2 = (1)^2/(2)^2 = 1/4
Ratio M1:M2 = 1:4
A current-carrying loop acts as a magnetic dipole. Magnetic moment M = NIA. N = number of turns, I = current (A), A = area (m^2). For circular coil: A = pi r^2. Direction: perpendicular to plane, right-hand rule (curl fingers in current direction, thumb points along M). M is a vector quantity. In external magnetic field B: torque tau = M x B. Potential energy U = -M.B.
B at centre = mu0 I / (2R). For N turns: B = mu0 N I / (2R). Inversely proportional to radius. At axis at distance x from centre: B = mu0 I R^2 / (2(R^2 + x^2)^(3/2)). At large distance (x >> R): B = mu0 M / (2pi x^3) - looks like a dipole field.
Integral of B.dl around any closed loop = mu0 I_enclosed. Useful for symmetric situations. For a long straight wire: B = mu0 I / (2pi r). For solenoid: B = mu0 n I (n = turns per unit length). For toroid: B = mu0 N I / (2pi r) inside, B = 0 outside. Ampere law is the magnetostatic equivalent of Gauss law.
dB = (mu0/4pi)(I dl x r_hat)/r^2. Used when Ampere law cannot be applied (non-symmetric configurations). For circular loop at centre: B = mu0 I/(2R). For infinite straight wire: B = mu0 I/(2pi r) (same as Ampere law, consistent). For finite wire: B = (mu0 I/4pi r)(sin theta1 + sin theta2).
Torque: tau = MB sin theta = M x B. For theta = 90 deg: maximum torque tau = MB. For theta = 0 or 180 deg: zero torque (stable/unstable equilibrium). Work done rotating from theta1 to theta2: W = MB(cos theta1 - cos theta2). Potential energy: U = -MB cos theta = -M.B. Analogy: exactly like electric dipole p in electric field E, replacing p with M and E with B.
Moving coil galvanometer: current-carrying coil in radial magnetic field. Torque due to B: tau = NIAB (radial field ensures constant torque). Restoring torque from spiral spring: tau = C theta. Equilibrium: NIAB = C theta. Deflection theta = NIAB/C. Sensitivity: theta/I = NAB/C. To convert to ammeter: add low resistance shunt in parallel. To convert to voltmeter: add high resistance in series. Ballistic galvanometer: measures charge (first-throw deflection proportional to charge).
Smallest unit of magnetic moment for electron. mu_B = eh/(4pi m_e) = 9.27 x 10^-24 J/T. Orbital magnetic moment of electron in nth Bohr orbit: M = n x mu_B. Spin magnetic moment of electron: M = sqrt(s(s+1)) x g_s x mu_B ≈ mu_B (g_s = 2 for electron). Total magnetic moment determines paramagnetism, ferromagnetism of materials.
Diamagnetic: weak repulsion from B field. All materials have diamagnetism. Induced moments oppose field. Examples: Bi, Cu, water. Paramagnetic: weak attraction. Random magnetic dipoles partially align with B field. Examples: Al, Mn, O2. Ferromagnetic: strong attraction. Magnetic domains align spontaneously. Hysteresis loop. Examples: Fe, Ni, Co. Retentivity and coercivity important for permanent magnets. Curie temperature: above this ferromagnets become paramagnetic.