Magnetic Effects of Current & Magnetism

1. Biot-Savart Law

Magnetic field dB due to current element Idl at position r.

dB = (μ₀ / 4π) (I dl × r) / r³
dB = (μ₀ / 4π) (I dl sinθ) / r²
μ₀ = 4π × 10^-7 T m/A.

Applications

Straight Wire: B = (μ₀I / 2πr) (Infinite length).
Circular Loop (Center): B = μ₀I / 2R.
Circular Loop (Axis): B = μ₀I R² / 2(R² + x²)^3/2.

(Asked in NEET 2018, 2022)

2. Ampere's Circuital Law

Line integral of magnetic field around a closed loop equals μ₀ times net current enclosed.

∮ B ⋅ dl = μ₀ I_enclosed

Solenoid & Toroid

Long Solenoid: B = μ₀ n I (Inside), B = 0 (Outside).
Toroid: B = μ₀ N I / 2πr = μ₀ n I.

3. Motion in Magnetic Field

Lorentz Force

F = q (v × B)
F = qvB sinθ
Force is perpendicular to both v and B. Work done by magnetic force is Zero.

Cyclotron Motion

Charged particle moving perpendicular to B-field describes circular path.

Radius r = mv / qB
Time Period T = 2πm / qB (Independent of speed/radius)
Kinetic Energy K = q²B²r² / 2m

(Asked in NEET 2016, 2019)

4. Force on Current & Galvanometer

Force on Conductor

F = I (l × B) = BIl sinθ
Parallel Wires:
F/l = μ₀ I₁ I₂ / 2πd
(Same direction → Attract, Opposite → Repel).

Moving Coil Galvanometer

Current carrying coil in B-field experiences torque.

Torque τ = NIAB sinθ (Radial field θ=90, so τ = NIAB)
Restoring Torque τ = kα
Current Sensitivity: φ/I = NAB / k

(Asked in NEET 2018, 2020, 2022)

5. Magnetism and Matter

Earth's Magnetism

Declination (α): Angle between geographic and magnetic meridian.
Dip (δ): Angle with horizontal.
B_H = B cosδ, B_V = B sinδ
tanδ = B_V / B_H.

Magnetic Properties

Diamagnetic: Repelled by magnet (Bi, Cu, H₂O). χ is small -ve.
Paramagnetic: Weakly attracted (Al, Na, O₂). χ is small +ve.
Ferromagnetic: Strongly attracted (Fe, Co, Ni). χ is large +ve.
Curie Law: χ ∝ 1/T (Paramagnetic).

Numericals: Magnetic Effects

Q1. A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?

Solution:
B = μ₀I / 2πr
μ₀/2π = 2 × 10^-7 Tm/A.
I = 35 A, r = 0.2 m.
B = (2 × 10^-7 × 35) / 0.2
B = 70 × 10^-7 / 0.2 = 350 × 10^-7 = 3.5 × 10^-5 T.

Q2. A circular coil of wire consisting of 100 turns, each of radius 8 cm, carries a current of 0.40 A. What is B at center?

Solution:
B = μ₀ N I / 2R
B = (4π × 10^-7 × 100 × 0.4) / (2 × 0.08)
B = (16π × 10^-6) / 0.16
B = 100π × 10^-6 = 3.14 × 10^-4 T.

Q3. An alpha particle (q = +2e) travels at 5 × 10⁵ m/s perpendicular to a magnetic field of 0.2 T. Calculate force.

Solution:
F = qvB (sin 90° = 1)
q = 2 × 1.6 × 10^-19 = 3.2 × 10^-19 C.
F = 3.2 × 10^-19 × 5 × 10⁵ × 0.2
F = 3.2 × 1 × 10^-14 = 3.2 × 10^-14 N.

Q4. A 40 cm long solenoid has 500 turns and carries 2 A current. Find B inside.

Solution:
n = N/l = 500 / 0.4 = 1250 turns/m.
B = μ₀ n I
B = 4π × 10^-7 × 1250 × 2
B = 10000π × 10^-7 = π × 10^-3 ≈ 3.14 mT.

Q5. A proton moves in a circular orbit of radius 20 cm in B-field 0.5 T. Find its velocity. (m_p = 1.67 × 10^-27 kg)

Solution:
r = mv / qB → v = qBr / m
v = (1.6 × 10^-19 × 0.5 × 0.2) / (1.67 × 10^-27)
v = 0.16 × 10^-19 / 1.67 × 10^-27
v ≈ 0.096 × 10⁸ = 9.6 × 10⁶ m/s.

Q6. Two parallel wires 1 m apart carry 1 A current in same direction. Find force per unit length.

Solution:
F/l = μ₀ I₁ I₂ / 2πd
F/l = (4π × 10^-7 × 1 × 1) / (2π × 1)
F/l = 2 × 10^-7 N/m. (Attractive).

Q7. A coil of 50 turns and area 20 cm² carries 2 A in B = 0.5 T. Find max torque.

Solution:
Max torque when sinθ = 1.
τ = NIAB
τ = 50 × 2 × (20×10^-4) × 0.5
τ = 100 × 10 × 10^-4 × 0.5 = 1000 × 10^-4 × 0.5
τ = 0.1 × 0.5 = 0.05 Nm.

Q8. Galvanometer with resistance G=50 Ω gives full deflection for 2 mA. Convert to Ammeter of range 0-5 A.

Solution:
Shunt S = I_gG / (I - I_g)
S = (2×10^-3 × 50) / (5 - 0.002)
S ≈ 0.1 / 5 = 0.02 Ω.

Q9. At a place, the horizontal component of Earth's field is 0.3 G and dip angle is 30°. Find total field.

Solution:
B_H = B cosδ → B = B_H / cosδ
B = 0.3 / cos(30°) = 0.3 / (0.866)
B ≈ 0.346 Gauss.

Q10. Calculate magnetic moment of a hydrogen electron orbiting with speed 2.2 × 10⁶ m/s in radius 0.53 Å.

Solution:
M = IA = (ev/2πr) × πr² = evr/2
M = (1.6×10^-19 × 2.2×10⁶ × 0.53×10^-10) / 2
M = (1.865 × 10^-23) / 2
M = 9.3 × 10^-24 Am² (Bohr Magneton).

Important Formulae

1. Magnetic Fields

Long Straight Wire:

B = μ₀ I / 2πr

Circular Coil Center:

B = μ₀ N I / 2R

Solenoid (Inside):

B = μ₀ n I

2. Magnetic Force

Lorentz Force (Charge):

F = q v B sinθ

Ampere Force (Wire):

F = B I l sinθ

Force per unit length (Parallel):

F/l = μ₀ I₁ I₂ / 2πd

3. Circular Motion

r = mv / qB = √(2mK) / qB

T = 2πm / qB
f = qB / 2πm (Cyclotron Frequency)

4. Dipole & Earth

Magnetic Moment:

M = N I A

Torque:

τ = M B sinθ

Earth's Field:

B_H = B cosδ, B_V = B sinδ

20 NEET Golden Facts

1. No Magnetic Monopoles: Isolated magnetic poles do not exist. Gauss's law for magnetism: ∮ B.dS = 0.
2. Work Done Zero: Magnetic force is always perpendicular to velocity. Hence no work is done and Speed/KE remains constant.
3. Cyclotron Limit: Cannot accelerate neutral particles (neutrons) or electrons (relativistic mass increase issue).
4. Parallel Wires: Currents in same direction Attract. Opposite direction Repel (Unlike charges which repel for like types).
5. Solenoid Ends: B-field at ends of a long solenoid is half of that at the center ( μ₀nI / 2 ).
6. Ammeter Conversion: Connect low resistance (Shunt) in Parallel to Galvanometer.
7. Voltmeter Conversion: Connect high resistance in Series to Galvanometer.
8. Radial Field: In Moving Coil Galvanometer, soft iron core and concave pole pieces creates radial field so θ=90 always.
9. Sensitivity: Increasing turns N increases Voltage Sensitivity but NOT necessarily Current Sensitivity (Resistance also increases).
10. Earth's B-Field: Horizontal component is zero at Poles. Vertical component is zero at Equator.
11. Curie Temperature: Temperature above which Ferromagnetic material becomes Paramagnetic.
12. Superconductors: Perfect Diamagnets (χ = -1). They expel magnetic field lines (Meissner Effect).
13. Hysteresis Loop: Area under loop represents energy loss per cycle (Heat). Soft iron has narrow loop.
14. Angle of Dip: 90° at Poles, 0° at Equator. In northern hemisphere, north pole dips down.
15. Magnetic Shielding: Soft iron ring can shield inner region from external magnetic fields.
16. Unit of B: 1 Tesla = 10⁴ Gauss.
17. Frequency Indep.: B-field of infinite wire is independent of diameter of wire (outside).
18. Neutral Points: Points where Earth's B-field is exactly canceled by magnet's B-field.
19. Electromagnets: Soft iron is used because high permeability, low retentivity. Permanent magnets use Steel (high retentivity).
20. Torque & PE: Torque τ = M×B. Potential Energy U = -M⋅B.

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