Chapter 4 - Moving Charges and Magnetism

In 1820, Oersted observed that when current passes through wire, nearby compass needle deflects. This proved that moving charges or currents produce magnetic field in space around them. Magnetic field B is vector field, similar to electric field E but different properties. SI unit Tesla T: 1 T = 1 N/(A m). Another unit Gauss G: 1 G=10⁻⁴ T. Earth's magnetic field ~10⁻⁴ T = 1 G small, strong lab magnet ~1-2 T.

Electric field produced by charges, magnetic field produced by moving charges or currents or magnets. Stationary charge produces only electric field, moving charge produces both electric and magnetic fields. Magnetic field can be detected by its force on moving charge or current carrying conductor or magnetic needle.

Lorentz force: If charge q moves with velocity v in region having electric field E and magnetic field B, total force F = qE + q(v×B). First term electric force along E, second magnetic force perpendicular to both v and B given by right hand rule. Magnitude magnetic part Fm = q v B sinθ where θ angle between v and B. Direction by Fleming's left hand rule or right hand rule: Stretch fingers v to B, thumb gives force for positive charge, opposite for negative.

Magnetic field lines form closed loops unlike electric field lines which start on positive and end on negative. Magnetic monopoles do not exist isolated north or south always exists as dipole.

Consider charge q moving with velocity v entering uniform magnetic field B. Magnetic force Fm=q v B sinθ perpendicular to both v and B, so it does no work because force ⊥ displacement instantaneous, speed magnitude remains constant, kinetic energy unchanged only direction changes.

Three cases:

Case 1: θ=0° or 180° velocity parallel to B sinθ=0 Fm=0 particle moves straight line along B unaffected.

Case 2: θ=90° velocity perpendicular to B. Force provides centripetal force q v B=m v²/r so radius r=mv/qB. Particle moves in circular path. Period T=2πr/v=2πm/qB independent of velocity v, frequency f=qB/2πm cyclotron frequency. Angular frequency ω=qB/m cyclotron frequency. This is used in cyclotron to accelerate particles.

Case 3: θ between 0° and 90° velocity has components v_parallel along B and v_perp perpendicular. v_parallel moves straight along B, v_perp moves circular perpendicular B, combined helical path. Radius of helix r=m v_perp/qB=m v sinθ/qB, pitch p=v_parallel T=v cosθ × 2πm/qB distance moved along B in one revolution.

Cyclotron is device to accelerate charged particles like protons to high energy using perpendicular magnetic field and alternating electric field in two Dees. Frequency of AC = cyclotron frequency qB/2πm resonance.

Biot-Savart law gives magnetic field due to small current element Idl similar to Coulomb law gives electric field due to charge element dq.

Statement: Magnetic field dB due to current element Idl at point distance r making angle θ between dl and r vector is dB = (μ0/4π) I dl × r̂ / r² magnitude dB=μ0/4π I dl sinθ / r² direction perpendicular to plane containing dl and r given by right hand screw rule. μ0=4π×10⁻⁷ Tm/A permeability of free space similar to ϵ0 for electric.

Applications:

1. Straight infinite wire: At perpendicular distance a, field B=μ0 I/(2π a) circular around wire direction by right hand thumb rule thumb along current fingers curl give B circular loops.

2. Circular loop radius R center: Field at center B=μ0 I/(2R) direction along axis by right hand curl fingers along current thumb gives B direction.

3. On axis of circular loop at distance x from center: B=μ0 I R²/[2(R²+x²)^(3/2)] along axis. At center x=0 reduces to μ0 I/2R. Far away x>>R B≈μ0 I R²/(2 x³)=μ0/(4π) 2m/x³ where m=I A=I πR² magnetic moment similar to dipole field.

Biot-Savart is basic law valid for any shape current.

Ampere's circuital law is equivalent of Gauss law for magnetism, useful for highly symmetric currents.

Statement: Line integral of magnetic field B around closed loop (Amperian loop) equals μ0 times total current enclosed: ∮ B·dl = μ0 I_enclosed. I_enclosed is algebraic sum of currents crossing surface bounded by loop, sign by right hand rule thumb along positive current fingers curl give positive loop direction.

Applications:

1. Infinite straight wire: Choose Amperian loop circle radius r around wire, B constant magnitude along loop parallel to dl, ∮ B·dl = B×2πr = μ0 I so B=μ0 I/2πr same as Biot-Savart easier.

2. Solenoid: Long solenoid N turns length L turns per length n=N/L. Field inside uniform parallel to axis B=μ0 n I direction by right hand, outside ~0. Choose rectangular Amperian loop partly inside partly outside, ∮ B·dl = B L = μ0 n L I so B=μ0 n I inside uniform.

3. Toroid: Doughnut shaped solenoid N turns total, radius r inside toroid mean radius. Choose circular Amperian loop radius r inside toroid, B×2πr=μ0 N I so B=μ0 N I/(2πr) inside toroid, outside zero. Field non-uniform inside 1/r dependence.

Ampere law useful only when high symmetry allows B constant along part of loop and perpendicular elsewhere.

Consider two long parallel wires I1 I2 distance r apart. Wire 1 produces magnetic field at location of wire 2: B1=μ0 I1/(2π r) direction perpendicular to plane by right hand.

Wire 2 carrying current I2 length L in field B1 experiences force F=I2 L×B1 magnitude F= I2 L B1 sin90=I2 L μ0 I1/(2π r)=μ0 I1 I2 L/(2π r). Direction by Fleming left hand rule or right hand: If currents parallel same direction wires attract, antiparallel opposite direction repel. Because field due to one interacts with current in other.

Force per unit length f=F/L=μ0 I1 I2/(2π r) N/m.

Definition of ampere: 1 ampere is that current which when flowing in each of two infinitely long parallel conductors 1 meter apart in vacuum produces force of 2×10⁻⁷ N per meter length on each conductor. Derivation: Put I1=I2=1A r=1m in f=μ0 I1 I2/2πr μ0=4π×10⁻⁷ so f=4π×10⁻⁷×1×1/(2π×1)=2×10⁻⁷ N/m. This is used to define ampere.

Consider rectangular loop N turns area A current I in uniform magnetic field B at angle θ between normal to loop and B. Sides of loop experience forces F=I L×B equal opposite forming couple torque.

Torque magnitude τ=N I A B sinθ direction perpendicular to plane containing normal and B. Vector form τ=N I A × B where A vector area magnitude A direction normal to loop by right hand rule curl fingers along current thumb gives A direction.

Define magnetic dipole moment m=N I A vector same direction as A. So m magnitude N I A direction normal by right hand. Then torque τ=m×B magnitude m B sinθ. Loop behaves like magnetic dipole like bar magnet.

Potential energy of dipole in magnetic field U=-m·B=-m B cosθ minimum at θ=0° m parallel B stable equilibrium τ=0, maximum at θ=180° antiparallel unstable τ=0 but unstable. Work done rotating dipole from θ1 to θ2 W=m B(cosθ1-cosθ2).

Examples: Compass needle is magnetic dipole aligns with Earth's magnetic field to minimize energy.

Moving coil galvanometer MCG is sensitive device to detect small current. Construction: Rectangular coil N turns area A wound on soft iron core free to rotate between pole pieces of permanent magnet providing radial magnetic field always parallel to plane of coil so sinθ=1 always torque maximum τ=N I A B.

Spring attached provides counter torque k φ where φ deflection angle and k torsional constant spring. At equilibrium N I A B = k φ so φ=(N A B/k) I deflection proportional to current linear scale. Pointer attached shows deflection on scale.

Current sensitivity Is=φ/I=N A B/k rad/A deflection per unit current. Voltage sensitivity Vs=φ/V=Is/R= N A B/(k R) rad/V deflection per unit voltage where R resistance galvanometer. Increasing current sensitivity by increasing N A B decreasing k but voltage sensitivity may decrease if R increases.

Conversion to ammeter: Galvanometer can tolerate small current Ig full scale. To measure large current I, small resistance shunt S connected in parallel with galvanometer. Ig G=(I-Ig)S where G resistance galvanometer. So S=Ig G/(I-Ig) small resistance. Ammeter low resistance connected in series with circuit ideal ammeter zero resistance.

Conversion to voltmeter: To measure large voltage V, high resistance R connected in series with galvanometer. V=Ig(G+R) so R=V/Ig - G high resistance. Voltmeter high resistance connected in parallel with component ideal voltmeter infinite resistance.