Magnetic Effects of Current and Magnetism
1. Introduction
The chapters explore the intimate relationship between electricity and magnetism. A moving electric charge (or current) produces a magnetic field, and a magnetic field exerts force on moving charges or currents. This interconnection, first demonstrated by Oersted’s experiment in 1820, laid the foundation for electromagnetism.
- Oersted’s Experiment: A current-carrying wire deflects a nearby magnetic compass needle. The deflection reverses when current direction is reversed, proving that electric current produces a magnetic field.
- Magnetic field (B) is a vector quantity. SI unit: Tesla (T) or Weber/m² (Wb/m²). 1 T = 1 N/(A·m). CGS unit: Gauss (1 T = 10⁴ Gauss).
Magnetic effects are central to devices like motors, generators, transformers, cyclotrons, and particle accelerators.
2. Magnetic Field Due to Moving Charges and Currents
Biot-Savart’s Law
This law gives the magnetic field (dB) at a point due to a small current element Idl:
or in magnitude:
- μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space).
- θ is the angle between dl and the position vector r from the element to the point.
- Direction of dB is perpendicular to the plane of dl and r (right-hand rule).
Applications of Biot-Savart’s Law:
- Magnetic field at the centre of a current-carrying circular loop (radius r, N turns):
- On the axis of the loop (distance x from centre):
- Straight infinite conductor at perpendicular distance d:Direction: Concentric circles around the wire (Right-Hand Thumb Rule: Thumb along current, fingers curl in direction of B).
- Finite straight conductor:where ϕ₁ and ϕ₂ are angles subtended by the ends at the point.
Ampere’s Circuital Law
This is analogous to Gauss’s law in electrostatics:
- Useful for symmetric current distributions (infinite straight wire, solenoid, toroid).
- Inside a long solenoid (n turns per unit length):(Uniform inside, nearly zero outside).
- Toroid: B = μ₀ N I / (2πr) inside the core.
Note: Ampere’s law holds for steady currents and is combined with Biot-Savart for complex cases.
3. Force on Moving Charges in Magnetic Field (Lorentz Force)
The force on a charge q moving with velocity v in magnetic field B is:
Magnitude: F = q v B sinθ (θ between v and B).
- Magnetic force is always perpendicular to both v and B; it does no work (speed unchanged, only direction changes).
- When v is parallel or anti-parallel to B, F = 0.
- Combined with electric field (Lorentz force):
Motion in Uniform Magnetic Field:
- If v ⊥ B: Circular path with radiusCyclotron frequency:(independent of v; basis of cyclotron accelerator).
- If v has component parallel to B: Helical path.
Force on a Current-Carrying Conductor: A wire of length l carrying current I in field B experiences:
Magnitude: F = I l B sinθ.
- Direction: Fleming’s Left-Hand Rule (Forefinger: B, Middle finger: I, Thumb: Force).
Force Between Two Parallel Current-Carrying Wires:
- Wires attract if currents are in same direction; repel if opposite.
- Force per unit length:(Definition of ampere is based on this).
4. Torque on Current Loop and Magnetic Dipole
A rectangular loop (area A, current I, N turns) in uniform B experiences torque:
where magnetic moment m = N I A (direction by right-hand rule: fingers along current, thumb along m; or from S to N pole).
- Magnitude: τ = m B sinθ.
- The loop tends to align with m parallel to B (stable equilibrium when θ = 0).
Moving Coil Galvanometer:
- Torque due to current balanced by restoring torque of suspension: θ ∝ I.
- Current sensitivity and voltage sensitivity depend on design.
Conversion:
- To ammeter: Low shunt resistance in parallel.
- To voltmeter: High series resistance.
5. Magnetism and Matter (Magnetic Properties of Materials)
Materials respond differently to external magnetic fields due to atomic currents (orbital and spin motions of electrons).
Key Terms
- Magnetisation (M): Magnetic moment per unit volume. Unit: A/m.
- Magnetic Intensity (H):
- Magnetic Susceptibility (χ): M = χ H (dimensionless).
- Relative Permeability (μ_r): μ = μ_r μ₀ = μ₀ (1 + χ).
- Magnetic Permeability (μ).
Types of Magnetic Materials
- Diamagnetic (χ negative, small; μ_r < 1):
- Weakly repelled by magnetic field (e.g., water, copper, bismuth).
- Induced moment opposes applied field.
- No permanent magnetism; atoms have no net moment.
- Paramagnetic (χ positive, small; μ_r > 1):
- Weakly attracted (e.g., aluminium, oxygen).
- Atoms have permanent moments randomly oriented; align weakly with field.
- Follow Curie’s law: χ ∝ 1/T.
- Ferromagnetic (χ large positive; μ_r >> 1):
- Strongly attracted; retain magnetism (e.g., iron, nickel, cobalt).
- Domains (regions of aligned atomic moments) align with field.
- Hysteresis: B-H curve shows retentivity and coercivity.
- Curie temperature: Above which ferromagnetic becomes paramagnetic.
Hysteresis Loop: Important for permanent magnets (high retentivity, high coercivity) vs. transformer cores (low hysteresis loss).
Magnetic Dipole and Bar Magnet
- Magnetic dipole moment m = m × 2l (pole strength × length).
- Field due to bar magnet (short dipole approximation):
- Axial line: B = (μ₀ / 4π) (2m / r³)
- Equatorial line: B = (μ₀ / 4π) (m / r³)
Gauss’s Law for Magnetism:
(No magnetic monopoles; magnetic field lines are closed loops).
Earth’s Magnetism
- Earth behaves like a giant bar magnet (south magnetic pole near geographic north).
- Elements:
- Declination (θ): Angle between geographic and magnetic meridian.
- Dip (Inclination, δ): Angle that total field makes with horizontal. tan δ = V/H (V vertical, H horizontal component).
- Horizontal component H = B cos δ.
- Total field B ≈ 0.3 to 0.6 Gauss at surface.
6. Important Formula Summary
- Biot-Savart:
- Ampere’s Law:
- Force on charge:
- Radius in B-field:
- Force on wire:
- Torque on dipole:
- B due to solenoid:
- Relation: ,
- Bar magnet axial/equatorial fields as above.
7. Applications and Advanced Insights
- Cyclotron: Accelerates charged particles using perpendicular B and alternating E.
- Velocity Selector: Crossed E and B fields select particles with v = E/B.
- Electromagnets and Transformers: Use soft iron cores (high permeability, low hysteresis).
- Magnetic Levitation and MRI: Exploit strong fields and diamagnetic/ferromagnetic properties.
- Domain Theory: Explains ferromagnetism; domains grow or rotate in applied field.
Common Misconceptions:
- Magnetic force does no work (changes direction, not speed).
- Magnetic monopoles do not exist (Gauss’s law).
- All materials show some magnetic response; classification is by strength.
Examination Tips:
- Master direction rules (Right-hand thumb, Fleming’s left/right, Maxwell’s cork-screw).
- Practice derivations of B for wire, loop, solenoid.
- Solve numericals on force, torque, radius of path, and conversion of galvanometer.
- Understand hysteresis for qualitative questions.
- Link with previous chapters (Current Electricity for I, Electrostatics for analogies).