Board exam focus — Moving Charges and Magnetism (CBSE & HBSE)

CBSE emphasizes Biot-Savart law derivations, Ampere's law applications, motion of charged particles, and galvanometer conversion. HBSE focuses on force on current-carrying conductor, magnetic field formulas, and cyclotron working.

Biot-Savart Law and Ampere's Law

Magnetic Field and Oersted's Discovery

In 1820, Oersted discovered that a current-carrying conductor produces a magnetic field around it. This established the link between electricity and magnetism.

Biot-Savart Law

The magnetic field dB⃗ due to a small current element Idl⃗ at a field point P at distance r:

dB⃗ = (μ₀/4π) · (Idl⃗ × r̂)/r²

Magnitude: dB = (μ₀/4π) · Idl sinθ/r²

where μ₀ = 4π×10⁻⁷ T·m/A (permeability of free space) θ = angle between dl⃗ and r̂

Magnetic Field due to Long Straight Wire

B = μ₀I/(2πr)

Directed in circles around the wire (Right-hand screw rule or Thumb rule).

Magnetic Field on Axis of Circular Loop

For a circular loop of radius R carrying current I, at axial distance x:

B = μ₀IR²/[2(R²+x²)^(3/2)]

At center (x=0): B = μ₀I/(2R)

Ampere's Circuital Law

The line integral of B⃗ around any closed path (Amperian loop) equals μ₀ times the total current enclosed:

∮B⃗·dl⃗ = μ₀I_enc

Field of a Solenoid (using Ampere's law)

For a solenoid with n turns per unit length, current I:

B = μ₀nI (inside, uniform, parallel to axis)

B ≈ 0 (outside, for ideal solenoid)

Field of a Toroid

B = μ₀NI/(2πr) (inside toroid, r = mean radius)

B = 0 (in the hole and outside toroid)

Comparison: Biot-Savart vs Ampere's Law

Diagram Indicator: [Solenoid cross-section showing field lines parallel inside and nearly zero outside; circular field lines around a long straight wire with right-hand thumb rule.]

Motion of Charged Particle in Magnetic Field

Force on a Moving Charge (Lorentz Force)

F⃗ = qv⃗ × B⃗

Magnitude: F = qvB sinθ

• θ = angle between v⃗ and B⃗
• θ = 90°: F = qvB (maximum)
• θ = 0° or 180°: F = 0

The magnetic force is always perpendicular to v⃗ → no work done → no change in KE → only changes direction.

Circular Motion in Magnetic Field

When v⃗ ⊥ B⃗, the particle moves in a circle. Magnetic force provides centripetal force:

qvB = mv²/r

Radius: r = mv/(qB)

Angular frequency (cyclotron frequency): ω = qB/m

Time period: T = 2πm/(qB) (independent of v and r!)

Helical Motion

When v⃗ makes angle θ with B⃗:

• Component v cosθ (along B) → uniform motion (no force)
• Component v sinθ (⊥ B) → circular motion
• Combined: helical motion
• Pitch = v cosθ × T = 2πmv cosθ/(qB)

Force on Current-Carrying Conductor in Magnetic Field

F⃗ = IL⃗ × B⃗

Magnitude: F = BIL sinθ

• I = current, L = length of conductor in field
• Fleming's Left-Hand Rule: thumb=force, index=field, middle=current

Force between Parallel Current-Carrying Conductors

Two parallel wires carrying I₁ and I₂, separated by distance d:

F/L = μ₀I₁I₂/(2πd)

• Parallel currents: attract
• Anti-parallel currents: repel
• Definition of Ampere based on this force!

Diagram Indicator: [Charged particle spiraling in a magnetic field (helical motion); also two parallel wires with force arrows showing attraction for same-direction currents.]

Cyclotron, Galvanometer, Ammeter, Voltmeter

Cyclotron

A cyclotron is a device that accelerates charged particles (protons, deuterons, α-particles) to high energies using crossed electric and magnetic fields.

Principle: Charged particles spiral outward in D-shaped electrodes (Dees) under alternating electric field, while magnetic field keeps them in circular paths.

Cyclotron Frequency: f_c = qB/(2πm) — independent of particle speed!

Maximum Energy: KE_max = q²B²R²/(2m), where R = radius of Dee

Resonance condition: AC frequency must equal cyclotron frequency.

Limitation of Cyclotron:

• Cannot accelerate electrons (relativistic mass increase)
• Cannot accelerate neutral particles
• At very high energies, relativistic mass increase desynchronizes frequency

Moving Coil Galvanometer (MCG)

Detects small currents using the force on a current-carrying coil in a magnetic field.

Principle: Torque τ = NBIA (deflecting) = kθ (restoring)

At equilibrium: NBIA = kθ → I = kθ/(NBA)

Current sensitivity: θ/I = NBA/k Voltage sensitivity: θ/V = NBA/(kR_G)

Converting Galvanometer to Ammeter

Connect a low resistance shunt S in parallel with the galvanometer:

S = I_g × G / (I − I_g)

where I_g = full-scale deflection current, G = galvanometer resistance, I = desired range.

An ammeter has very low resistance and is connected in series.

Converting Galvanometer to Voltmeter

Connect a high resistance R in series with the galvanometer:

R = V/I_g − G

where V = desired voltage range.

A voltmeter has very high resistance and is connected in parallel.

Comparison: Ammeter vs Voltmeter

Diagram Indicator: [Cyclotron cross-section showing Dees D₁ and D₂, particle spiral paths, magnetic field B perpendicular into page, alternating electric field E between Dees; also galvanometer conversion circuits.]

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