📚 Core Topics — Conceptual Anchors 60-second scan
The chapter builds from the microscopic picture (force on a single moving charge) → circuit-level machinery (galvanometer, cyclotron). Twelve conceptual anchors — a scan of them gives you the whole chapter shape before diving into formulas.
1 Lorentz force
Total EM force on a moving charge is the sum of electric and magnetic parts. Only the electric part does work.
\[\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B)\]
2 Circular motion in B
When \(\mathbf v \perp \mathbf B\), qvB supplies centripetal force. Speed and KE stay constant — direction changes only.
\[r = \dfrac{mv}{qB},\quad T = \dfrac{2\pi m}{qB}\]
3 Helical path & velocity selector
Angled \(\mathbf v\) gives a helix (r from \(v_\perp\), pitch from \(v_\parallel\)). Crossed E and B pass one v undeviated.
\[\text{pitch} = \dfrac{2\pi m v\cos\theta}{qB},\quad v_{\text{sel}} = \dfrac{E}{B}\]
4 Cyclotron
Period is independent of v (T = 2πm/qB) — a fixed-frequency AC voltage between the dees keeps ions in resonance every half-cycle.
\[f_c = \dfrac{qB}{2\pi m},\quad K_{\max} = \dfrac{q^2 B^2 R^2}{2m}\]
5 Biot–Savart law
Every current element contributes a small \(d\mathbf B\). Direction by \(d\mathbf l \times \hat r\); magnitude falls as 1/r².
\[d\mathbf B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf l \times \hat r}{r^2}\]
6 Field of a straight wire
Field lines are concentric circles by right-hand thumb rule. \(\mu_0/(4\pi) = 10^{-7}\) T·m/A.
\[B = \dfrac{\mu_0 I}{2\pi a}\]
7 Field of a circular loop
At centre: 2R, not 2πR. Loop acts as a magnetic dipole far along its axis.
\[B_{\text{c}} = \dfrac{\mu_0 N I}{2R},\; B_{\text{axis}} = \dfrac{\mu_0 N I R^2}{2(R^2+x^2)^{3/2}}\]
8 Ampère's circuital law
Line integral of B around any closed loop = μ₀ × enclosed current. Always true; only useful when symmetry lets you pull B out of the integral.
\[\oint \mathbf B\cdot d\mathbf l = \mu_0 I_{\text{enc}}\]
9 Solenoid & toroid
Long solenoid: uniform inside, zero outside; μ₀nI/2 at each end. Toroid: field only inside the ring, zero elsewhere.
\[B_{\text{sol}} = \mu_0 n I,\quad B_{\text{tor}} = \dfrac{\mu_0 N I}{2\pi r}\]
10 Force on current-carrying wire
Perpendicular to both I and B. Net force on any closed loop in uniform B = 0.
\[\mathbf F = I\mathbf L \times \mathbf B,\quad F = BIL\sin\theta\]
11 Force between parallel wires
Same-direction currents attract; opposite currents repel. Definition of ampere: 2×10⁻⁷ N/m between two 1-A wires 1 m apart.
\[\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}\]
12 Torque, magnetic moment & MCG
Loop in B feels torque (max when plane ∥ B). Moving-coil galvanometer converts to ammeter (parallel shunt) or voltmeter (series R).
\[\boldsymbol\tau = \mathbf m \times \mathbf B,\; m = NIA,\; U = -\mathbf m\cdot\mathbf B\]
🧮 NEET Formula Bank — Every Formula, Grouped
Every formula NEET could test from this chapter, organised into seven alphabetical sub-groups. Each formula card carries symbols · SI unit · condition. Use as reference / final-week sheet.
A · Force on Charged Particles
1 Lorentz force
Total EM force on a moving charge
\[\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B)\]
q in C · v in m/s · B in T · magnetic part is ⟂ v, so it does no work.
N2 Radius (v ⟂ B)
Four equivalent forms
\[r = \dfrac{mv}{qB} = \dfrac{p}{qB} = \dfrac{\sqrt{2mK}}{qB} = \dfrac{1}{B}\sqrt{\dfrac{2mV}{q}}\]
Use the form matching the given quantity: v · p · K · V.
m3 Period · frequency · angular frequency
All independent of v and r
\[T = \dfrac{2\pi m}{qB},\; f = \dfrac{qB}{2\pi m},\; \omega = \dfrac{qB}{m}\]
Fast particles trace bigger circles in the same time — cyclotron principle.
s · Hz · rad/s4 Helical pitch
v at angle θ to B
\[\text{pitch} = v\cos\theta \cdot T = \dfrac{2\pi m v\cos\theta}{qB}\]
Radius uses only v sin θ; pitch uses only v cos θ.
m5 Velocity selector
Crossed E and B
\[v = \dfrac{E}{B}\]
Undeviated regardless of q and m — pure velocity filter.
m/s6 Cyclotron — frequency & max KE
Fixed AC drives resonance
\[f_c = \dfrac{qB}{2\pi m},\quad K_{\max} = \dfrac{q^2 B^2 R^2}{2m}\]
Cannot accelerate: neutrons (no q) or electrons (relativistic mass rise breaks resonance).
Hz · JB · Field due to Currents — Biot–Savart & Simple Geometries
7 Biot–Savart law
Element-level field
\[d\mathbf B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf l \times \hat r}{r^2}\]
μ₀/(4π) = 10⁻⁷ T·m/A · dB = 0 when dl ∥ r̂ (θ = 0 or π).
T8 Infinite straight wire
⟂ distance a
\[B = \dfrac{\mu_0 I}{2\pi a}\]
Field lines are concentric circles by right-hand thumb rule.
T9 Finite straight wire
θ₁, θ₂ from ⟂ at ends
\[B = \dfrac{\mu_0 I}{4\pi a}(\sin\theta_1 + \sin\theta_2)\]
Reduces to μ₀I/(2πa) when θ₁ = θ₂ = 90° (infinite wire limit).
T10 Centre of circular coil (N turns)
Radius R
\[B = \dfrac{\mu_0 N I}{2R}\]
2R, NOT 2πR — classic trap. Direction ⟂ loop plane, right-hand curl.
T11 Arc subtending φ (radians)
Fraction of a full loop
\[B = \dfrac{\mu_0 I \varphi}{4\pi R}\]
Semicircle (φ = π): μ₀I/(4R). Quarter: μ₀I/(8R). Full: μ₀I/(2R).
T12 Axis of loop at distance x
Along the perpendicular axis
\[B = \dfrac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}\]
Recovers μ₀NI/(2R) at x = 0 · for x ≫ R: dipole approx \(B \approx \mu_0(2m)/(4\pi x^3)\).
T13 Square loop of side a — centre
Sum of 4 finite-wire contributions
\[B = \dfrac{2\sqrt2\,\mu_0 I}{\pi a}\]
Derived from the finite-wire formula with each half-side subtending 45° at centre.
TC · Ampère's Law & Its Applications
14 Ampère's circuital law
Around any closed loop
\[\oint \mathbf B\cdot d\mathbf l = \mu_0 I_{\text{enc}}\]
Line integral depends only on enclosed current — not on shape/size of loop or currents outside.
T·m15 Long solenoid (inside)
n = turns per unit length
\[B = \mu_0 n I\]
Uniform inside, ~zero outside. At either end: μ₀nI/2. n = N/L.
T16 Toroid
Total turns N, mean radius r
\[B = \dfrac{\mu_0 N I}{2\pi r}\]
B = 0 outside AND in the empty region enclosed by the ring. Only inside the ring is B ≠ 0.
T17 Thick current cylinder (inside)
r < R
\[B = \dfrac{\mu_0 I r}{2\pi R^2}\]
Linear rise inside (0 at axis → μ₀I/(2πR) at surface), then 1/r decay outside.
TD · Forces on Current-Carrying Conductors
18 Force on a straight wire
Vector form + magnitude
\[\mathbf F = I\mathbf L \times \mathbf B \;\Rightarrow\; F = BIL\sin\theta\]
Direction by right-hand rule (Fleming's left-hand rule for force). Net force on any closed loop in uniform B = 0.
N19 Force between two parallel wires
Separation d
\[\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}\]
Same direction ⇒ attract · opposite ⇒ repel. Opposite of electrostatic same-charge rule.
N/m20 Definition of ampere
SI base-unit definition
\[F/L = 2 \times 10^{-7}\;\text{N/m}\]
Force per metre between two infinitely long ∥ conductors 1 m apart, each carrying 1 A.
N/m at 1 A, 1 m apartE · Magnetic Moment · Torque · Dipole
21 Magnetic moment of a loop
N turns · area A
\[m = NIA,\;\; \mathbf m = NI\mathbf A\]
Direction ⟂ loop plane by right-hand curl rule (curl fingers in direction of I, thumb = m).
A·m²22 Torque on loop in B
Uniform field
\[\boldsymbol\tau = \mathbf m \times \mathbf B,\;\; \tau = NIAB\sin\theta\]
θ = angle between m and B. τ max when loop-plane ∥ B (m ⟂ B). τ = 0 when plane ⟂ B (m ∥ B).
N·m23 PE of magnetic dipole
Signed by orientation
\[U = -\mathbf m\cdot\mathbf B = -mB\cos\theta\]
Stable eqm at θ = 0 (U min). Unstable at θ = π (U max).
J24 Revolving electron — magnetic moment
Orbital contribution
\[\mu_l = \dfrac{evr}{2} = \dfrac{e}{2m_e}\,L\]
Gyromagnetic ratio e/2m_e ≈ 8.8 × 10¹⁰ C/kg. L = orbital angular momentum.
A·m²25 Bohr magneton
Natural atomic magnetic-moment unit · smallest orbital moment (n = 1)
\[\mu_B = \dfrac{eh}{4\pi\, m_e} = \dfrac{e\hbar}{2 m_e} \approx 9.27 \times 10^{-24}\ \text{J T}^{-1}\]
Derivation: for smallest Bohr orbit L = ℏ = h/(2π); substitute into μl = (e/2me) L. All atomic magnetic moments are quoted as multiples of μB. Unit: 1 J T−1 = 1 A m2.
J T⁻¹ = A m²F · Moving Coil Galvanometer & Conversions
26 Restoring balance
k = torsional constant
\[k\theta = NIAB \;\Rightarrow\; \theta = \dfrac{NAB}{k}\,I\]
Deflection θ is linearly proportional to I.
rad27 Current sensitivity
Deflection per unit current
\[\dfrac{\theta}{I} = \dfrac{NAB}{k}\]
Raise N ⇒ raises CS. But also raises coil R.
rad/A28 Voltage sensitivity
R = coil resistance
\[\dfrac{\theta}{V} = \dfrac{NAB}{k R}\]
Raising N raises CS but also R ⇒ VS may not rise. Classic NEET trap.
rad/V29 Galvanometer → Ammeter
Shunt in parallel
\[S = \dfrac{I_g\,G}{I - I_g}\]
S is small (parallel) · connect ammeter in series with the load · ideal ammeter: R → 0.
Ω30 Galvanometer → Voltmeter
High-value R in series
\[R_s = \dfrac{V}{I_g} - G\]
R_s is large (series) · connect voltmeter in parallel across element · ideal voltmeter: R → ∞.
ΩG · Quick Reference — Constants & Standard Values
μ₀ = 4π × 10⁻⁷ T·m·A⁻¹ · μ₀/(4π) = 10⁻⁷ T·m·A⁻¹ · e = 1.6 × 10⁻¹⁹ C · m_e = 9.1 × 10⁻³¹ kg · m_p = 1.67 × 10⁻²⁷ kg (≈ 1836 m_e)
Bohr magneton μ_B = 9.27 × 10⁻²⁴ A·m² · Gyromagnetic ratio e/2m_e ≈ 8.8 × 10¹⁰ C·kg⁻¹
Earth's magnetic field ~ 30–60 μT · Force per m between 1-A parallel wires 1 m apart = 2 × 10⁻⁷ N/m · Cyclotron freq of proton in 1 T ≈ 15.2 MHz
Bohr magneton μ_B = 9.27 × 10⁻²⁴ A·m² · Gyromagnetic ratio e/2m_e ≈ 8.8 × 10¹⁰ C·kg⁻¹
Earth's magnetic field ~ 30–60 μT · Force per m between 1-A parallel wires 1 m apart = 2 × 10⁻⁷ N/m · Cyclotron freq of proton in 1 T ≈ 15.2 MHz
📊 PYQ Weightage Map — Last 10–15 Years ≈ 3 Q/yr
Ranked by how often NEET has pulled from each sub-topic (2010–2024). MCM sits in the top-4 highest-yield Physics chapters — Lorentz force + circular motion combinations dominate.
1. Force on a charged particle · circular motion in B (\(r = mv/qB\))
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≈ 1 Q/yr · protons vs α-particle traps
2. Biot-Savart applications — centre of loop / arc / straight wire
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≈ 1 Q/yr · 2R vs 2πR trap
3. Ampere's law — solenoid & toroid (\(B = \mu_0 n I\))
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Inside vs outside toroid; end of solenoid
4. Torque on current loop & magnetic moment
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τ = NIAB sinθ; stable/unstable eqm
5. Force on current-carrying wire · parallel-wire force
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Attract vs repel; define ampere
6. Cyclotron — frequency, K_max, limitations
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Can't accelerate neutron/electron
7. Moving-coil galvanometer → ammeter/voltmeter conversion
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Shunt in parallel; series R for V
8. Current sensitivity vs voltage sensitivity
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Adding turns raises CS but maybe not VS
9. Helical path · velocity selector (\(v = E/B\))
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Pitch = v cosθ · T
10. Magnetic dipole moment of revolving electron · Bohr magneton
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μ_B = 9.27 × 10⁻²⁴ A m²
11. Field at axis of a coil (\(x \gg R\) dipole approx)
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Rare direct Q
12. Square/polygonal loop field derivations
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2√2 μ₀I/πa for side a
🔥 Priority order: Lorentz + circular motion → Biot-Savart (loop/arc/wire) → Ampere (solenoid/toroid) → Torque on loop → Parallel wires → Cyclotron → Galvanometer. Master the first four blocks and you lock ~ 2.5 of 3 chapter Qs almost every NEET.
🧮 Master Formula Sheet Every symbol · unit · condition
A · Magnetic field due to currents 🔥🔥🔥🔥🔥
| Situation | Formula | Symbols · Units · Condition |
|---|---|---|
| Biot-Savart | \[d\mathbf B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf l \times \hat r}{r^2}\] | μ₀ = 4π×10⁻⁷ T·m·A⁻¹ · I in A · dl in m · r in m · dB in T · zero when dl ∥ r̂ (θ = 0 or π) |
| Infinite straight wire | \[B = \dfrac{\mu_0 I}{2\pi a}\] | a = ⟂ distance from wire · field lines concentric circles · right-hand thumb rule |
| Finite straight wire | \[B = \dfrac{\mu_0 I}{4\pi a}(\sin\theta_1 + \sin\theta_2)\] | θ₁, θ₂ measured from ⟂ to wire at the two ends |
| Centre of circular coil | \[B = \dfrac{\mu_0 N I}{2R}\] | N turns · R = radius · 2R not 2πR (classic trap) |
| Arc subtending φ (rad) | \[B = \dfrac{\mu_0 I \varphi}{4\pi R}\] | Semicircle (φ = π) → \(B = \mu_0 I/4R\); quarter circle → \(\mu_0 I/8R\) |
| Axis of loop (distance x) | \[B = \dfrac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}\] | x = distance along axis · at x=0 recovers centre formula · x ≫ R: \(B \approx \mu_0 (2m)/(4\pi x^3)\) (dipole) |
| Square loop of side a (at centre) | \[B = \dfrac{2\sqrt2\,\mu_0 I}{\pi a}\] | Sum of 4 finite-wire contributions |
| Ampere's law | \[\oint \mathbf B\cdot d\mathbf l = \mu_0 I_{\text{enc}}\] | Line integral around any closed loop = μ₀ × enclosed current · shape-independent |
| Long solenoid (inside) | \[B = \mu_0 n I\] | n = turns per unit length · uniform inside, ~zero outside. At either end: μ₀nI/2 |
| Toroid | \[B = \dfrac{\mu_0 N I}{2\pi r}\] | N = total turns · r = mean radius · B = 0 outside AND in the empty region inside the ring |
| Thick cylindrical conductor (inside) | \[B = \dfrac{\mu_0 I r}{2\pi R^2}\;(r < R)\] | Linear rise inside · 1/r outside |
B · Motion of a charged particle 🔥🔥🔥🔥🔥
| Quantity | Formula | Symbols · Notes |
|---|---|---|
| Lorentz force | \[\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B)\] | Magnetic part ⊥ v ⇒ no work; speed & KE constant · Newton on unit charge |
| Radius (v ⊥ B) | \[r = \dfrac{mv}{qB} = \dfrac{p}{qB} = \dfrac{\sqrt{2mK}}{qB}\] | Also \(r = \dfrac{1}{B}\sqrt{\dfrac{2mV}{q}}\) if accelerated through V |
| Period · frequency · ω | \[T = \dfrac{2\pi m}{qB},\; f = \dfrac{qB}{2\pi m},\; \omega = \dfrac{qB}{m}\] | All independent of v and r — cyclotron principle |
| Helical path (angle θ to B) | \[\text{pitch} = v\cos\theta \cdot T = \dfrac{2\pi m v\cos\theta}{qB}\] | Radius uses only v⊥ = v sinθ |
| Velocity selector (crossed E, B) | \[v = \dfrac{E}{B}\] | Undeviated · regardless of q and m |
| Cyclotron | \[f = \dfrac{qB}{2\pi m},\; K_{\max} = \dfrac{q^2 B^2 R^2}{2m}\] | R = dee radius · cannot accelerate n (no q) or e⁻ (relativistic) |
C · Force · torque · magnetic dipole 🔥🔥🔥🔥
| Quantity | Formula | Symbols · Notes |
|---|---|---|
| Force on current-carrying wire | \[\mathbf F = I\mathbf L \times \mathbf B \Rightarrow F = BIL\sin\theta\] | L = length vector along current · net force on closed loop in uniform B = 0 |
| Force between parallel wires | \[\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}\] | Same direction ⇒ attract · opposite ⇒ repel · d = separation |
| Definition of ampere | \[F/L = 2\times 10^{-7}\text{ N/m}\] | Between two ∥ wires 1 m apart, each carrying 1 A |
| Magnetic moment (loop) | \[m = NIA,\; \mathbf m = NI\mathbf A\] | Direction ⟂ loop plane by right-hand curl rule · SI unit: A·m² |
| Torque on loop in B | \[\boldsymbol\tau = \mathbf m \times \mathbf B \Rightarrow \tau = NIAB\sin\theta\] | θ = angle between m and B · τ max when loop-plane ∥ B; τ = 0 when plane ⟂ B |
| PE of dipole | \[U = -\mathbf m\cdot\mathbf B = -mB\cos\theta\] | Stable at θ = 0 (U min); unstable at θ = π |
| Revolving electron (μl) | \[\mu_l = \dfrac{evr}{2} = \dfrac{e}{2 m_e}\,L\] | Gyromagnetic ratio e/(2me) ≈ 8.8 × 10¹⁰ C·kg⁻¹ |
| Bohr magneton | \[\mu_B = \dfrac{eh}{4\pi\, m_e} = \dfrac{e\hbar}{2 m_e} \approx 9.27 \times 10^{-24}\ \text{J T}^{-1}\] | Smallest orbital moment (n = 1) · 1 J T⁻¹ = 1 A m² |
D · Moving coil galvanometer 🔥🔥🔥
| Quantity | Formula | Notes |
|---|---|---|
| Restoring torque | \[k\theta = NIAB \Rightarrow \theta = \dfrac{NAB}{k}\cdot I\] | k = torsional const. · θ = deflection · A = coil area |
| Current sensitivity | \[\dfrac{\theta}{I} = \dfrac{NAB}{k}\] | Deflection per unit current |
| Voltage sensitivity | \[\dfrac{\theta}{V} = \dfrac{NAB}{kR}\] | R = coil resistance · raising N raises CS but also R ⇒ VS may not rise |
| Galvanometer → Ammeter | \[S = \dfrac{I_g\, G}{I - I_g}\] | Shunt S in parallel; low S · connect in series with load |
| Galvanometer → Voltmeter | \[R_s = \dfrac{V}{I_g} - G\] | High-value R in series; connect in parallel with element |
📐 Key Derivations (pivot points)
Field at centre of circular loop (from Biot-Savart)
Start: \(d\mathbf B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf l\times\hat r}{r^2}\). For every element, \(d\mathbf l \perp \hat r\) so \(|d\mathbf l\times\hat r| = dl\), and every dB is along the axis. Integrate around circle: \(\oint dl = 2\pi R\). Result: \(B = \dfrac{\mu_0 I(2\pi R)}{4\pi R^2} = \dfrac{\mu_0 I}{2R}\).
Ampere's law → solenoid field
Amperian loop: rectangle with one long side of length L inside solenoid (parallel to axis), the parallel side outside (B≈0), short sides ⟂ to B. Only the inside side contributes: \(\oint \mathbf B\cdot d\mathbf l = BL\). Enclosed current: \(nL\cdot I\). Result: \(BL = \mu_0 nLI \Rightarrow B = \mu_0 nI\).
Force between two parallel wires
Wire 1 produces \(B_1 = \dfrac{\mu_0 I_1}{2\pi d}\) at wire 2. Force on length L of wire 2: \(F = I_2 L B_1 = \dfrac{\mu_0 I_1 I_2 L}{2\pi d}\). Same direction ⇒ B field lines encircle both wires so the wires "pull" toward each other ⇒ attract. Opposite ⇒ repel.
Torque on current loop
Rectangular loop, sides a·b, in uniform B. Forces on the two sides ∥ B are zero. Forces on the two sides ⟂ B are equal & opposite (F = BIa) but at distance b apart. Torque = F · b sinθ = BI(ab)sinθ = BIA sinθ. For N turns: τ = NIAB sinθ. Vector form: \(\boldsymbol\tau = \mathbf m \times \mathbf B\) with m = NIA.
Cyclotron frequency
Radius equation: \(r = mv/qB\). Period T: \(T = 2\pi r/v = 2\pi m/qB\). Notice: T does not contain v or r ⇒ the particle takes the same time for every half-circle regardless of speed. That's why a fixed-frequency AC voltage between the dees keeps it in resonance. f_c = qB/2πm.
💡 Concept Essentials
- Magnetic force does no work. Speed & KE of a charged particle in pure B are constant — direction changes only.
- Field lines from currents are closed loops (no magnetic monopoles). Curl direction by right-hand thumb rule.
- Ampere's law works only when there's symmetry to make the line integral trivial (long straight wire, solenoid, toroid). It's always true, just not always useful.
- Right-hand rule twice: once for field from current (thumb = I, fingers = B). Once for force (fingers point along v, curl toward B, thumb = F, for +ve charge). Reverse F for electron.
- Torque on loop is max when the plane of the coil is parallel to B (m ⟂ B) and zero when the plane is perpendicular to B (m ∥ B).
- In a uniform B, net force on a closed loop = 0 · in a non-uniform B, net force = ∇(m·B) ≠ 0.
- Cyclotron works because T is independent of v; a fixed-frequency AC voltage keeps the ion in phase every half cycle.
- Galvanometer converts to ammeter with a low-R shunt in parallel; to voltmeter with a high-R in series.
- Cyclotron can't accelerate: neutrons (no q), electrons (relativistic mass rise breaks resonance).
🎯 Problem Types & Methods
(a) 🔥 Radius/period of a charged particle in B
Given: q, m, v (or K or V), B. Ask: r or T. Method: Pick the right radius form — if v given: r = mv/qB; if K given: r = √(2mK)/qB; if accelerated through V: r = (1/B)√(2mV/q). T = 2πm/qB (no v!). For α vs proton: same K ⇒ r_α/r_p = √(m_α/m_p) × (q_p/q_α) = √4 × (1/2) = 1 (surprising equality).
(b) 🔥 Field due to Biot-Savart configurations
Step 1: Break the wire path into standard pieces (finite wires + arcs). Step 2: Apply the piece-specific formula (μ₀I/2πa for straight, μ₀Iφ/4πR for arc, 0 for radial pieces where dl ∥ r̂). Step 3: Add vectorially — usually all pieces contribute along the same axis so it's just algebra.
(c) 🔥 Ampere's law — solenoid / toroid / cylinder
Solenoid inside: B = μ₀nI. Solenoid at end: B = μ₀nI/2. Toroid inside: B = μ₀NI/(2πr). Toroid outside / inside empty region: B = 0. Thick cylinder (r < R): B = μ₀Ir/(2πR²) (linear in r); r > R: μ₀I/(2πr).
(d) 🔥 Torque on rectangular / circular loop
Compute m = NIA (unit-vector normal to plane by right-hand curl). Then τ = mB sinθ where θ is angle between m and B. If the coil "plane is parallel to B", θ = 90° ⇒ τ = mB (max). If "plane is perpendicular to B", θ = 0° ⇒ τ = 0 (but this is stable equilibrium).
(e) Force between parallel wires
F/L = μ₀I₁I₂/(2πd). Same direction currents → attract. Opposite → repel. For grids of wires, add pairwise force contributions vectorially.
(f) Galvanometer conversion
For ammeter of full-scale I: S = I_g G/(I − I_g). For voltmeter of full-scale V: R_s = V/I_g − G. Ideal ammeter: R → 0 (shunt very small); ideal voltmeter: R → ∞ (series R very large).
(g) Cyclotron numericals
f = qB/(2πm). K_max = q²B²R²/(2m). Use q and m of the specific ion (proton, deuteron, α, etc.). To double K_max at fixed B ⇒ double R only if same particle.
(h) Helical path pitch
If v is at angle θ to B: r = mv sinθ/(qB) (uses perpendicular component); pitch = v cosθ · T = 2πm v cosθ/(qB).
⚖️ Constant-vs-Changes Tables
(a) Particle in a magnetic field — as B is changed
| Quantity | Behaviour |
|---|---|
| Speed |v| | Unchanged (B does no work) |
| KE | Unchanged |
| Radius r = mv/qB | ↓ as B ↑ (inversely proportional) |
| Period T = 2πm/qB | ↓ as B ↑ (inversely proportional) |
| Frequency | ↑ as B ↑ |
(b) Solenoid — as n or I changes
| Change | Effect on B = μ₀nI |
|---|---|
| Double n (turns/m) | B doubles |
| Double I | B doubles |
| Double length (same total N) | n halves ⇒ B halves |
| Change radius | No effect (inside long solenoid) |
(c) Cyclotron — as B or v changes
| Change | Effect |
|---|---|
| Increase v (same B) | r increases · T unchanged · K_max reached later |
| Increase B (same v) | r decreases · T decreases · f increases |
| Increase R (dee size) | K_max increases as R² |
📈 Graphs to Know
| Graph | Shape | Meaning |
|---|---|---|
| B vs r for straight wire | Hyperbola (B ∝ 1/r) for r > wire radius | μ₀I/(2πr) outside · linear inside for thick wire |
| B vs r for thick current cylinder | Linear rise inside (0 → μ₀I/2πR at r = R), then 1/r decay outside | Kink at r = R |
| B vs x on axis of loop | Peaks at centre (x = 0), falls to zero as x → ∞ | Decays as 1/x³ for x ≫ R (dipole) |
| B along axis of solenoid | Nearly uniform μ₀nI inside, drops to μ₀nI/2 at each end, → 0 far outside | Constant plateau interior |
| B for toroid | Zero outside AND in central hole; ≠ 0 only inside the ring | 1/r inside the ring |
| τ vs θ (loop in B) | Sinusoid — max at θ = 90°, zero at 0 and 180° | Slope at θ = 0 gives restoring const |
| U vs θ (loop in B) | Cosine: min at θ = 0 (stable), max at θ = π (unstable) | U = −mB cosθ |
🧷 Units & Dimensions
| Quantity | SI unit | Dimensional formula |
|---|---|---|
| Magnetic field B | tesla (T) = kg·A⁻¹·s⁻² = Wb·m⁻² | [M T⁻² A⁻¹] |
| Magnetic flux Φ | weber (Wb) = T·m² = V·s | [M L² T⁻² A⁻¹] |
| Magnetic moment m | A·m² | [A L²] |
| Permeability μ₀ | T·m·A⁻¹ = N·A⁻² | [M L T⁻² A⁻²] |
| Current sensitivity (θ/I) | rad·A⁻¹ | [A⁻¹] |
| Voltage sensitivity (θ/V) | rad·V⁻¹ | [M⁻¹ L⁻² T³ A] |
| Torsional constant k | N·m·rad⁻¹ | [M L² T⁻²] |
| Cyclotron frequency | Hz | [T⁻¹] |
🔢 Standard Values
| Constant / value | Value |
|---|---|
| μ₀ (permeability of vacuum) | 4π × 10⁻⁷ T·m·A⁻¹ |
| μ₀/(4π) | 10⁻⁷ T·m·A⁻¹ |
| Electron charge e | 1.6 × 10⁻¹⁹ C |
| Electron mass m_e | 9.1 × 10⁻³¹ kg |
| Proton mass m_p | 1.67 × 10⁻²⁷ kg (≈ 1836 m_e) |
| Bohr magneton μ_B | 9.27 × 10⁻²⁴ A·m² |
| Gyromagnetic ratio e/2m_e | ≈ 8.8 × 10¹⁰ C·kg⁻¹ |
| Earth's magnetic field (typical) | ~ 30–60 μT (3–6 × 10⁻⁵ T) |
| Force per m between 1 A parallel wires (1 m apart) | 2 × 10⁻⁷ N·m⁻¹ |
| Cyclotron freq of proton in 1 T | ≈ 15.2 MHz |
⚡ Shortcuts & Sign Rules
- Right-hand thumb rule for B from wire: thumb along I → fingers curl in direction of B.
- Right-hand rule for F on +ve charge: palm-slap rule — fingers point along v, curl into B, thumb gives F. Reverse for electron.
- Fleming's left-hand rule for force on wire (Force / Field / Current — three mutually ⟂ fingers).
- Same currents attract, opposite repel — opposite of electrostatics.
- Solenoid interior: field is ~ uniform ⇒ constant B inside a good approximation.
- Field at centre of loop = μ₀I/(2R). Field at distance a from straight wire = μ₀I/(2πa). Note 2R vs 2πR.
- α vs proton same K: radius equal (√4 mass, but ½ charge). Same v: r_α/r_p = 2. Same p: r_α/r_p = ½.
- Torque max when loop plane ∥ B; zero when plane ⟂ B.
- Stable eqm when m ∥ B (θ = 0); unstable when m ∥ −B (θ = π).
⚠️ Trap Points / Common Mistakes
2R vs 2πR — loop centre uses 2R, straight wire uses 2πa. The single most common slip.
Magnetic force does NO work. Never write "work = Fd" for a pure magnetic force.
T is independent of v (and r). Fast particles just make bigger circles in the same time.
Net force on a closed loop in uniform B = 0. Only torque exists. In non-uniform B, net force appears.
Current sensitivity vs voltage sensitivity. Adding turns raises CS AND R — so VS may not rise.
Sign flip for electron. Direction of F for +ve charge — then reverse for e⁻.
Cyclotron can't accelerate neutrons (no q) or electrons (relativistic mass rise breaks resonance).
Solenoid end field = μ₀nI/2 (not μ₀nI). Full field only deep inside.
Toroid outside and in the empty central hole → B = 0. Only inside the ring is B ≠ 0.
Torque max when plane ∥ B, not when plane ⟂ B. The plane ⟂ B is stable eqm (τ = 0).
Ammeter — shunt in PARALLEL (small R). Voltmeter — R in SERIES (large R). Don't swap.
Ampere's law is always true — but only useful when symmetry lets you pull B out of the integral.
🧭 Direction Drill — Right-Hand & Fleming Rules Zero-error subject
The three rules — which to use when
Rule 1
Right-hand curl rule
Use for: B produced by a current (wire, loop, solenoid). Thumb = current direction (I) · fingers curl = direction of B.
Rule 2
Fleming's left-hand rule (FBI)
Use for: force on a current-carrying wire in a field. ForeFinger = B · Middle = I (current) · Thumb = F (force). Order: F-B-I.
Rule 3
Fleming's right-hand rule
Use for: induced current in a moving conductor (EMI). Same layout: ForeFinger = B · Thumb = v (motion) · Middle = induced I.
Rule 4
Right-hand slap / palm rule
Use for: force on a positive charge moving in B. Fingers point in v · curl into B · thumb gives F. For negative charge, reverse the answer.
10 quick drill Qs — 30 seconds each
- Current flows north (→) in a wire. Where does B point directly above the wire?
Answer
West (out of page if looking down). Curl right-hand fingers: thumb north → above the wire, fingers point west. - Positive charge moves east (→); B points north. Force direction?
Answer
Vertically up (out of ground). F = qv×B. Fingers east, curl to north, thumb up. - Same as Q2 but negative charge — force?
Answer
Vertically down. Negative charge → reverse the answer. - Circular loop lies flat, current flows clockwise when viewed from above. B at centre points?
Answer
Downward (into the ground). Curl fingers clockwise → thumb points down. - Two parallel wires · both carry current north. Force on each?
Answer
Attract each other. Same-direction currents attract. - An electron enters a B field pointing right (→) with velocity into the page (⊗). Force?
Answer
Upward. For +q: v×B = (⊗)×(→) = ↓, but electron is negative → force is up. - Straight wire vertical, current up. B at a point to your right?
Answer
Points away from you (into page). Thumb up, fingers curl → at right side, B goes into page. - Rectangular loop in a horizontal B field, loop plane is horizontal. Torque?
Answer
Zero. Loop plane ∥ B means m ⟂ B — wait: plane ∥ B means area vector m ⟂ B → sinθ = 1 → torque is MAX, not zero. Trap. τ = 0 only when loop plane ⟂ B (m ∥ B). - Solenoid current flows counter-clockwise when you look at its right end. Which end is N pole?
Answer
Left end is N. Curl right-hand fingers with current at right end (counter-clockwise) → thumb points LEFT → left = N. - Cyclotron: proton is accelerating. Radius changes but period is …?
Answer
Constant. T = 2πm/qB — independent of v and r. This is why the D-gap oscillator frequency is fixed.
Torque angle trap
Formula: τ = NIAB sinθ, where θ = angle between area vector m and B.
- Loop plane ∥ B (m ⟂ B, θ = 90°) → sinθ = 1 → τ MAX
- Loop plane ⟂ B (m ∥ B, θ = 0°) → sinθ = 0 → τ = ZERO — stable equilibrium
- m antiparallel to B (θ = 180°) → sinθ = 0 → τ = 0 but unstable equilibrium
🧠 Mnemonics
Loop vs wire coefficient — "Loop is 2R (round trip 2 arms), wire is 2πR (full round)." Loops have a diameter; wires have a circumference.
Cyclotron independence — "Time same, radius grows." T = 2πm/qB has no v or r in it.
Torque max/zero — "Plane parallel = torque maximum · Plane perpendicular = torque zero." Or: "m along B = zero, m ⟂ B = max."
Parallel wires — "Same-way = friends (attract); opposite-way = enemies (repel)."
Ammeter vs voltmeter — "Ammeter loves company (parallel shunt), Voltmeter goes solo (series R)."
Radius formulas — "Momentum divided by charge-B" (r = p/qB). Then swap p in for √(2mK) or √(2mqV).
Cyclotron limits — "No neutron (no charge), no electron (goes relativistic)."
Right-hand pointer — "Thumb = I, curl = B" (fields). "Fingers = v, curl into B, thumb = F" (forces on +q).
🔗 Cross-Chapter Links
- Electric Charges & Fields — Lorentz force generalises Coulomb; velocity selector uses both E and B.
- Current Electricity — galvanometer conversion uses Ohm's law + Kirchhoff for ammeter/voltmeter design.
- Magnetism & Matter (Ch 5) — magnetic dipole moment concept feeds directly into it; Bohr magneton is the bridge.
- EMI (Ch 6) — flux Φ = B·A comes from here; motional EMF ε = Bvl is a Lorentz-force application.
- Alternating Current — AC generator's rotating coil in B is a direct application of τ = NIAB sinθ.
- Circular motion (mechanics) — Lorentz-force → circular motion uses centripetal force = qvB.
🎯 High-Yield Top 15 Master these first
- Radius: r = mv/qB = p/qB = √(2mK)/qB = (1/B)√(2mV/q).
- Period T = 2πm/qB, f = qB/(2πm) — independent of v and r.
- B at centre of loop = μ₀NI/(2R). B due to straight wire = μ₀I/(2πa).
- Arc of angle φ: B = μ₀Iφ/(4πR). Semicircle: μ₀I/(4R).
- Solenoid inside: B = μ₀nI · At the end: μ₀nI/2 · Outside: ≈ 0.
- Toroid: B = μ₀NI/(2πr) inside the ring · zero outside and in the central hole.
- Force on wire: F = IL × B, magnitude BIL sinθ. Net force on closed loop in uniform B = 0.
- Parallel wires: F/L = μ₀I₁I₂/(2πd). Same direction ⇒ attract; opposite ⇒ repel.
- Torque on loop: τ = NIAB sinθ = m × B. Max when plane ∥ B; zero when plane ⟂ B.
- PE: U = −mB cosθ. Stable at θ = 0, unstable at θ = π.
- Velocity selector: v = E/B (undeviated for any q, m).
- Helical pitch: 2πm v cosθ/(qB). Radius uses v sinθ.
- Cyclotron K_max = q²B²R²/(2m). Cannot accelerate neutrons or electrons.
- Bohr magneton = 9.27 × 10⁻²⁴ A·m². Gyromagnetic ratio e/2m ≈ 8.8 × 10¹⁰ C/kg.
- Galvanometer: ammeter shunt S = I_g·G/(I − I_g) [parallel] · voltmeter series R = V/I_g − G.
📝 Graded Self-Test — 15 NEET MCQs
Q1. A proton and an α-particle enter a magnetic field with the same kinetic energy. The ratio r_α : r_p is:
(a) 1 : 1 (b) 2 : 1 (c) √2 : 1 (d) 1 : 2
Answer
(a) 1 : 1. r = √(2mK)/qB. Ratio = √(m_α/m_p) × (q_p/q_α) = √4 × (1/2) = 1.Q2. A charged particle moving in a magnetic field experiences no force. This means:
(a) B = 0 (b) v = 0 (c) v ∥ B (d) any of these
Answer
(d) any of these. F = qv × B is zero if B = 0, v = 0, or v ∥ B (θ = 0 or π).Q3. A circular loop of radius 0.1 m carries 2 A. B at centre?
(a) 1.26 × 10⁻⁵ T (b) 4π × 10⁻⁵ T (c) 2π × 10⁻⁵ T (d) 2 × 10⁻⁵ T
Answer
(a) B = μ₀I/(2R) = (4π × 10⁻⁷)(2)/(2 × 0.1) = 4π × 10⁻⁶ ≈ 1.26 × 10⁻⁵ T.Q4. A long solenoid has 500 turns per metre and carries 4 A. B inside:
(a) 2.51 × 10⁻³ T (b) 6.28 × 10⁻³ T (c) 8π × 10⁻⁴ T (d) 1.6π × 10⁻³ T
Answer
(a) B = μ₀nI = (4π × 10⁻⁷)(500)(4) = 8π × 10⁻⁴ ≈ 2.51 × 10⁻³ T.Q5. A circular coil of 100 turns, area 0.05 m², carries 2 A. Placed in a 0.4 T field. Max torque:
(a) 4 N·m (b) 40 N·m (c) 0.4 N·m (d) 400 N·m
Answer
(a) τ_max = NIAB = 100 × 2 × 0.05 × 0.4 = 4 N·m.Q6. The magnetic moment of the coil in Q5:
(a) 10 A·m² (b) 100 A·m² (c) 200 A·m² (d) 0.1 A·m²
Answer
(a) m = NIA = 100 × 2 × 0.05 = 10 A·m².Q7. Two long parallel wires 20 cm apart carry 10 A each in the same direction. Force per metre on each:
(a) 10⁻⁴ N/m attract (b) 10⁻⁴ N/m repel (c) 2 × 10⁻⁵ N/m attract (d) 10⁻⁵ N/m attract
Answer
(a) F/L = μ₀I₁I₂/(2πd) = (2 × 10⁻⁷)(10)(10)/(0.20) = 10⁻⁴ N/m. Same direction ⇒ attract.Q8. Cyclotron with B = 1 T, dee radius 0.5 m. Max KE of proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C)?
(a) ≈ 12 MeV (b) ≈ 6 MeV (c) ≈ 120 keV (d) ≈ 1.2 GeV
Answer
(a) K_max = q²B²R²/(2m) = (1.6e-19)²(1)²(0.5)²/(2 × 1.67e-27) ≈ 1.92 × 10⁻¹² J ≈ 12 MeV.Q9. An electron and a proton enter a magnetic field perpendicular to their motion with the same speed. Ratio of periods T_e : T_p:
(a) m_e / m_p (b) m_p / m_e (c) 1 : 1 (d) √(m_e/m_p)
Answer
(a) T = 2πm/qB, both have same |q|, so T ∝ m ⇒ T_e/T_p = m_e/m_p ≈ 1/1836.Q10. Galvanometer of G = 100 Ω, full-scale I_g = 10 mA. Convert to ammeter for 5 A. Shunt:
(a) 0.02 Ω (b) 0.2 Ω (c) 2 Ω (d) 20 Ω
Answer
(b) S = I_g G/(I − I_g) = (10⁻²)(100)/(5 − 10⁻²) ≈ 1/5 = 0.2 Ω.Q11. A circular loop of area A carries current I. If bent into two loops each of area A/2 (semicircles superposed), the magnetic moment:
(a) halves (b) doubles (c) unchanged (d) becomes zero
Answer
(c) m = NIA. Now N = 2, A_each = A/2, so m = 2 × I × (A/2) = IA. Unchanged.Q12. A charged particle enters a magnetic field at 60° to B with speed v. The pitch of the helix is:
(a) 2πm v cos60°/(qB) (b) 2πm v sin60°/(qB) (c) 2πm v/(qB) (d) πm v/(qB)
Answer
(a) pitch = v_∥ · T = v cos60° · 2πm/(qB).Q13. A rectangular loop lies in a magnetic field with its plane parallel to B. Then:
(a) net force zero, torque zero (b) net force zero, torque max (c) net force max, torque zero (d) both max
Answer
(b) Uniform B: net force on any closed loop = 0. Plane ∥ B ⇒ m ⟂ B ⇒ θ = 90° ⇒ τ = NIAB (maximum).Q14. A toroid has 500 total turns and mean radius 10 cm. Current 2 A. Field inside the ring:
(a) 2 × 10⁻³ T (b) 4π × 10⁻³ T (c) 10⁻³ T (d) 5 × 10⁻⁴ T
Answer
(a) B = μ₀NI/(2πr) = (4π × 10⁻⁷)(500)(2)/(2π × 0.10) = 2 × 10⁻³ T.Q15. Assertion: A cyclotron cannot accelerate an electron. Reason: The relativistic mass increase causes the cyclotron frequency to fall out of resonance with the AC voltage.
(a) Both true, R explains A (b) Both true, R doesn't explain (c) A true, R false (d) A false, R true
Answer
(a) Both true and connected. At even modest energies (few keV), the electron's relativistic mass grows enough that ω_c = qB/m keeps decreasing, breaking phase-lock with the fixed AC frequency.Card built 2026-07-26 for NEET drill practice · Chapter 4 · Moving Charges and Magnetism · Strictly NCERT-scoped · Weightage rankings reflect 10–15 years of NEET / AIPMT PYQ frequency.