Magnetic Effects of Current & Magnetism

Topics covered: Magnetic field due to currents, Biot–Savart law, Lorentz force, motion of charged particles, Ampere’s law, force on current-carrying conductors.

1. Magnetic Effect of Current

• Oersted experimentally discovered that a current-carrying conductor produces a magnetic field.
• A stationary charge produces only an electric field.
• A moving charge produces both electric and magnetic fields.

2. Biot–Savart Law

Magnetic field due to a small current element:

dB = (μ₀ / 4π) · (I dl × r̂) / r²
  

• Magnetic field is zero along the line of the current (θ = 0 or π).

Magnetic Field Due to Straight Current-Carrying Wire

Finite length wire:

B = (μ₀ I / 4πd) (sinφ₁ + sinφ₂)
  

Infinite long wire:

B = μ₀I / (2πd)
  

Magnetic Field Due to Circular Current Loop

At the centre of a loop:

B = μ₀I / (2R)
  

For a circular arc subtending angle φ (in radians):

B = (μ₀Iφ) / (4πR)
  

For a semicircular loop:

B = μ₀I / (4R)
  

Magnetic Field Due to Solenoid

At centre of a long solenoid:

B = μ₀ n I
  

At the ends:

B = (μ₀ n I) / 2
  

3. Lorentz Force

Force on a charged particle moving in magnetic field:

F = q (v × B)
  

• Force is perpendicular to both velocity and magnetic field.
• No work is done by magnetic force.
• If charge is at rest, no magnetic force acts.

4. Motion of Charged Particle in Magnetic Field

Case A: v ⟂ B (Circular Motion)

qvB = mv² / r
r = mv / (qB)
  

Time period:

T = 2πm / (qB)
  

Frequency:

f = qB / (2πm)
  

Angular frequency (cyclotron frequency):

ω = qB / m
  

Case B: v at an angle to B (Helical Path)

r = m v sinθ / (qB)
  

Pitch of helix:

p = v cosθ · T = (2πm v cosθ) / (qB)
  

5. Ampere’s Circuital Law

∮ B · dl = μ₀ I
  

Applications

Long solid conductor (radius R):

• For r < R:

B = (μ₀ I r) / (2πR²)
  

• For r ≥ R:

B = μ₀ I / (2πr)
  

Hollow conductor:

• Magnetic field inside the hollow region is zero.

6. Force on Current-Carrying Conductor

Force on a current element:

dF = I (dl × B)
  

For a straight conductor of length l:

F = IlB sinθ
  

7. Force Between Two Parallel Current-Carrying Wires

• Same direction → attraction
• Opposite direction → repulsion

Force per unit length:

F / L = (μ₀ I₁ I₂) / (2πd)
  

8. Magnetic Field Due to Moving Charge

B = (μ₀ / 4π) · (q v × r̂) / r²