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Magnetic Effects of Current and Magnetism | PHYSICS | CUET | JEE | NEET

     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:

dB=μ04πIdl×r^r2d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{r}}{r^2}

or in magnitude:

dB=μ04πIdlsinθr2dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2}
  • 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):
    B=μ0NI2rB = \frac{\mu_0 N I}{2r}
  • On the axis of the loop (distance x from centre):
    B=μ0NIr22(r2+x2)3/2B = \frac{\mu_0 N I r^2}{2(r^2 + x^2)^{3/2}}
  • Straight infinite conductor at perpendicular distance d:
    B=μ0I2πdB = \frac{\mu_0 I}{2\pi d}
    Direction: Concentric circles around the wire (Right-Hand Thumb Rule: Thumb along current, fingers curl in direction of B).
  • Finite straight conductor:
    B=μ0I4πd(sinϕ1+sinϕ2)B = \frac{\mu_0 I}{4\pi d} (\sin\phi_1 + \sin\phi_2)
    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:

Bdl=μ0Ienclosed\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}
  • Useful for symmetric current distributions (infinite straight wire, solenoid, toroid).
  • Inside a long solenoid (n turns per unit length):
    B=μ0nIB = \mu_0 n I
    (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:

FB=q(v×B)\mathbf{F}_B = q (\mathbf{v} \times \mathbf{B})

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):
    F=q(E+v×B)\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})

Motion in Uniform Magnetic Field:

  • If v ⊥ B: Circular path with radius
    r=mvqB(cyclotron radius)r = \frac{m v}{q B} \quad (\text{cyclotron radius})
    Cyclotron frequency:
    ν=qB2πm\nu = \frac{q B}{2\pi m}
    (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:

F=I(l×B)\mathbf{F} = I (\mathbf{l} \times \mathbf{B})

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:
    Fl=μ0I1I22πd\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d}
    (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:

τ=NIA×B=m×B\boldsymbol{\tau} = N I \mathbf{A} \times \mathbf{B} = \mathbf{m} \times \mathbf{B}

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):
    B=μ0(H+M)\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})
  • 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

  1. 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.
  2. 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.
  3. 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:

BdA=0\oint \mathbf{B} \cdot d\mathbf{A} = 0

(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: dB=μ04πIdlsinθr2 dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}
  • Ampere’s Law: Bdl=μ0I \oint B \cdot dl = \mu_0 I
  • Force on charge: F=qvBsinθ F = q v B \sin\theta
  • Radius in B-field: r=mvqB r = \frac{mv}{qB}
  • Force on wire: F=IlBsinθ F = I l B \sin\theta
  • Torque on dipole: τ=mBsinθ \tau = m B \sin\theta
  • B due to solenoid: B=μ0nI B = \mu_0 n I
  • Relation: B=μ0(H+M) B = \mu_0 (H + M) , μr=1+χ \mu_r = 1 + \chi
  • 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).

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