Moving Charges and Magnetism: NEET notes

1. Magnetic field and the Lorentz force

NCERT §4.1, §4.2.1, §4.2.2

• In 1820 Oersted saw a compass needle swing when a current flowed in a nearby wire. The needle sets itself along the tangent to a circle centred on the wire; reversing the current reverses it, and a bigger current or a closer needle deflects it more.
• Iron filings round a current-carrying wire settle into concentric circles. Oersted's conclusion: moving charges, that is currents, set up a magnetic field in the space around them.
• Drawing convention: a dot (⊙) is a current or field coming out of the page, a cross (⊗) one going into it, like the tip and the tail feathers of an arrow.
• The magnetic field B is a vector field, defined at every point, and it obeys superposition: the fields of several sources add as vectors, just as electric fields do.
• A charge q moving with velocity v through fields E and B feels the Lorentz force F = q[E + v × B]. The electric part does not care about motion; the magnetic part does.
• The magnetic force q(v × B) is zero for a charge at rest and zero when v is parallel or antiparallel to B. It is perpendicular to both v and B, with direction from the right-hand (screw) rule, and it reverses for a negative charge.
• Its size is qvB sin θ, where θ is the angle between v and B. This defines the unit: 1 tesla (T) is the field that pushes 1 N on 1 C moving at 1 m/s at right angles to it, so 1 T = 1 N s C⁻¹ m⁻¹.
• The tesla is large. The non-SI gauss is 10⁻⁴ T, and the earth's field is about 3.6 × 10⁻⁵ T.

2. Force on a current-carrying conductor

NCERT §4.2.3

• A straight rod of length l and cross-section A with n carriers per unit volume holds nlA carriers, each drifting at v_d. Adding their forces in a field B gives F = (nlA)q v_d × B.
• Since nqv_d A is the current I, the force becomes F = I l × B, where the vector l has the rod's length and points along the current. The current itself is a scalar; the direction sits on l.
• Size: F = IlB sin θ, with θ the angle between the wire and B. A wire along the field feels nothing; one at right angles feels the most, IlB.
• B in this formula is the external field, not the field of the wire itself. For a wire of any shape, add I dl × B over small straight pieces.
• Example 4.1: a 200 g, 1.5 m wire carrying 2 A floats in a horizontal field when IlB = mg, so B = (0.2 × 9.8)/(2 × 1.5) = 0.65 T. Only m/l matters, and the earth's field (about 4 × 10⁻⁵ T) is too small to count.
• Example 4.2: with B along +y and a particle moving along +x, v × B points along +z. A proton is pushed along +z and an electron along −z.

3. Motion in a magnetic field

NCERT §4.3

• The magnetic force is always perpendicular to the velocity, so it does no work. The speed and kinetic energy of the charge stay fixed; only the direction of motion changes. An electric force, by contrast, can change the energy.
• When v is perpendicular to a uniform B, qvB supplies the centripetal force mv²/r, so the charge moves in a circle of radius r = mv/(qB) in the plane normal to B. More momentum means a bigger circle.
• The angular frequency is ω = 2πν = qB/m and the period T = 2πm/(qB). Neither depends on the speed or the energy: a faster charge runs a bigger circle in the same time. ν = qB/2πm is called the cyclotron frequency.
• If v has a component v∥ along B, that part is untouched while the perpendicular part still turns in a circle. The path is a helix whose radius uses v⊥ and whose pitch (advance per turn) is p = v∥T = 2πm v∥/(qB).
• Example 4.3: an electron (m = 9 × 10⁻³¹ kg) at 3 × 10⁷ m/s in 6 × 10⁻⁴ T moves on a circle of r = mv/(qB) = 0.28 m, at ν = v/2πr ≈ 17 MHz, with energy ½mv² ≈ 4 × 10⁻¹⁶ J ≈ 2.5 keV.
• A positive and a negative charge entering the same field with the same velocity curve in opposite senses.

4. Biot–Savart law

NCERT §4.4

• Every known magnetic field comes from currents (moving charges) or from the intrinsic magnetic moments of particles. The Biot–Savart law gives the field of a small piece of current.
• A current element I dl produces at a point a displacement r away the field dB = (μ₀/4π) I dl × r/r³. Its size is dB = (μ₀/4π) I dl sin θ/r², with θ the angle between dl and r.
• dB is perpendicular to the plane containing dl and r; its sense follows the right-hand screw rule for dl × r.
• μ₀ is the permeability of free space, with μ₀/4π = 10⁻⁷ T m A⁻¹ (μ₀ = 4π × 10⁻⁷ T m A⁻¹ in SI).
• Like Coulomb's law, it is long range (1/r²) and obeys superposition, and the field is linear in its source. Unlike it, the source I dl is a vector, the field is perpendicular to r rather than along it, and there is an angle factor sin θ.
• On the line of the element itself (θ = 0) the element makes no field at all.
• The constants are linked to the speed of light: ε₀μ₀ = 1/c², with c = 3 × 10⁸ m/s. Fixing μ₀ fixes ε₀.
• Example 4.4: a 1 cm element carrying 10 A along x, seen from 0.5 m along y (θ = 90°), gives dB = 10⁻⁷ × 10 × 10⁻²/0.25 = 4 × 10⁻⁸ T, pointing along +z.

5. Field on the axis of a circular loop

NCERT §4.5

• For a loop of radius R carrying I, every element is perpendicular to the line joining it to a point on the axis, so |dl × r| = r dl with r² = x² + R².
• Each element's field has a part along the axis and a part across it. The cross parts from opposite elements cancel, so only the axial parts add.
• Summing over the whole loop (total length 2πR) gives B = μ₀IR²/[2(x² + R²)^(3/2)] along the axis, at a distance x from the centre.
• At the centre (x = 0) this becomes B = μ₀I/2R. For N closely wound turns, multiply by N: B = μ₀NI/2R.
• The field lines of a loop are closed curves. Right-hand thumb rule for a loop: curl the fingers along the current, and the thumb points along the field through the loop. One face acts like a north pole, the other like a south pole.
• Example 4.5: straight leads make no field at a centre in line with them (dl × r = 0). A semicircle gives half a full loop's field, μ₀I/4R; for 12 A and R = 2.0 cm that is 1.9 × 10⁻⁴ T. Bending the arc the other way reverses the field but keeps its size.
• Example 4.6: a 100-turn coil of radius 10 cm with 1 A has B = μ₀NI/2R = 6.28 × 10⁻⁴ T at its centre.

6. Ampere's circuital law

NCERT §4.6

• Take any open surface with a boundary loop. Ampere's law says the line integral of B round the boundary equals μ₀ times the net current through the surface: ∮B·dl = μ₀I.
• Sign convention: curl the right-hand fingers in the sense the loop is traversed; the thumb shows which way a current counts as positive.
• When a loop can be chosen so that B is tangential and constant along part of length L, and normal to the loop or zero elsewhere, the law reduces to BL = μ₀Iₑ, with Iₑ the enclosed current.
• Long straight wire: a circle of radius r round it gives B × 2πr = μ₀I, so B = μ₀I/2πr. The field is the same all round the circle (cylindrical symmetry), tangential to it, and falls as 1/r.
• The field lines round a wire are closed circles, unlike electric field lines, which start and end on charges. Right-hand rule for a straight wire: thumb along the current, fingers curl the way B goes.
• Example 4.7: for a thick wire of radius a with current spread uniformly, B = μ₀I/2πr outside (r > a) and B = (μ₀I/2πa²)r inside (r < a). B grows linearly inside, peaks at the surface, then falls as 1/r.
• Ampere's law carries the same physics as the Biot–Savart law, the way Gauss's law does for Coulomb's law. It holds for steady currents and any loop, but it only gives B easily when there is enough symmetry; it cannot give μ₀I/2R at the centre of a loop.

7. The solenoid

NCERT §4.7

• A solenoid is a long insulated wire wound as a closely spaced helix; a long solenoid is one whose length is much more than its radius. Each turn acts as a circular loop.
• Between neighbouring turns the fields cancel. Inside, the field is strong, uniform and along the axis; just outside it is weak, and for a very long solenoid it is taken as zero.
• A rectangular Amperian loop with one side h inside along the axis and one outside gives Bh = μ₀(nh)I, so B = μ₀nI, with n counting the turns in each metre of winding.
• The inside field does not depend on the radius or on the position inside, only on n and I. Its direction follows the right-hand rule.
• A solenoid is the usual way to make a uniform field; a soft iron core inside makes it much larger.
• Example 4.8: 500 turns on 0.5 m (n = 1000 per metre) with 5 A: B = 4π × 10⁻⁷ × 10³ × 5 = 6.28 × 10⁻³ T. The ratio of length to radius is 50, so the long-solenoid formula is fair.

8. Force between parallel currents

NCERT §4.8

• Wire a makes a field B_a = μ₀I_a/2πd all along a parallel wire b a distance d away. Wire b, carrying I_b, feels a sideways force I_b L B_a on a length L.
• So the force per unit length is f = μ₀I_aI_b/2πd, equal in size on both wires and opposite in direction, as Newton's third law needs for steady currents.
• Currents in the same direction attract; currents in opposite directions repel. This is the reverse of charges, where like repels like.
• Definition of the ampere: the steady current that, flowing in two very long, straight, thin parallel wires 1 m apart in vacuum, makes each feel 2 × 10⁻⁷ N per metre. This definition dates from 1946.
• In practice the earth's field and stray fields must be removed, long wires are replaced by coils, and a current balance measures the force. One coulomb is then the charge carried past a point in 1 s by 1 A.
• Example 4.9: in a horizontal earth's field of 3.0 × 10⁻⁵ T pointing south to north, a wire with 1 A running east to west feels IB = 3 × 10⁻⁵ N per metre (downwards), far more than 2 × 10⁻⁷ N/m; running south to north it feels nothing.

9. Torque on a current loop

NCERT §4.9.1

• A rectangular loop in a uniform field feels no net force, but it can feel a torque, just as an electric dipole does in a uniform electric field.
• With B in the plane of the loop, the two sides across the field feel equal and opposite forces IbB, a distance a apart. The torque is τ = IabB = IAB, with A = ab.
• When the normal to the loop makes an angle θ with B, the forces on the side arms cancel along the axis, and the couple on the other two arms has a lever arm a sin θ. So τ = IAB sin θ.
• Define the magnetic moment m = IA, pointing along the area vector given by the right-hand thumb rule. Then τ = m × B, the twin of τ = p × E. For N turns, m = NIA.
• The unit of m is A m² (also J/T), with dimensions [L²A].
• The torque vanishes when m is parallel or antiparallel to B. Parallel is stable (a small turn is undone); antiparallel is unstable (a small turn grows). That is why a small magnet lines up with a field.
• Example 4.10: a 100-turn coil, radius 10 cm, 3.2 A has B = 2 × 10⁻³ T at its centre and m = NIπr² = 10 A m². In a 2 T field, τ = 0 when m is along B and 20 N m after a quarter turn; with moment of inertia 0.1 kg m² it reaches ω = 20 s⁻¹ after turning 90°.
• Example 4.11: a flat loop on a table cannot be spun about the vertical by any uniform field, since τ = IA × B always lies in the plane of the loop. A free loop settles with A along B, where the total flux through it is greatest; a flexible loop pulls into a circle, the shape enclosing the most area.

10. Current loop as a magnetic dipole

NCERT §4.9.2

• Far along the axis (x ≫ R), the loop's field μ₀IR²/2(x² + R²)^(3/2) becomes B ≈ μ₀IR²/2x³ = μ₀m/2πx³ = (μ₀/4π)(2m/x³), with m = IπR².
• This matches the axial field of an electric dipole, (1/4πε₀)(2p/x³), if p → m, E → B and 1/ε₀ → μ₀.
• In the plane of the loop, far away, B ≈ (μ₀/4π)(m/x³), matching the equatorial field of an electric dipole. Both results become exact for a point magnetic dipole.
• Any flat current loop is equivalent to a magnetic dipole of moment m = IA.
• An electric dipole is made of two monopoles (charges). A magnetic dipole, a current loop, is itself the simplest unit: magnetic monopoles have never been found.
• Because a loop makes a dipole field and feels torque like a compass needle, Ampere proposed that all magnetism comes from circulating currents. Electrons and protons, however, also carry an intrinsic magnetic moment not explained by any circulating current.

11. The moving coil galvanometer

NCERT §4.10

• A moving coil galvanometer (MCG) has a many-turn coil free to turn about a fixed axis in a radial magnetic field. A soft iron core makes the field radial and stronger.
• Because the field is radial, the plane of the coil is always along B and the torque is τ = NIAB. A spring gives a restoring torque kφ, where k is the torsional constant (restoring torque per unit twist).
• At equilibrium kφ = NIAB, so the deflection φ = (NAB/k)I is proportional to the current. A pointer on the spring reads φ off a scale.
• As a null detector (as in a Wheatstone bridge) the zero is at the centre of the scale and the pointer swings either way with the current's direction.
• A bare galvanometer is a poor ammeter: it reaches full scale at currents of the order of μA, and its large resistance would change the current it is meant to measure.
• Ammeter: add a small shunt rₛ in parallel. The combination has resistance R_G rₛ/(R_G + rₛ) ≈ rₛ, and most of the current goes through the shunt.
• Voltmeter: add a large resistance R in series, so the meter's resistance is R_G + R ≈ R, large, and it draws very little current from the circuit it is connected across.
• Current sensitivity φ/I = NAB/k; voltage sensitivity φ/V = NAB/(kR). Doubling N doubles the current sensitivity, but it also roughly doubles the coil's resistance, so the voltage sensitivity may not change.
• Example 4.12: with a 3 Ω circuit on 3 V, a 60 Ω galvanometer as the meter gives I = 3/63 = 0.048 A; with a 0.02 Ω shunt the total is about 3.02 Ω and I = 0.99 A; an ideal ammeter would read 1.00 A.

Must-know facts

1. Magnetic force F = qvB sin θ: zero for a charge at rest or moving along B, largest at right angles.
2. The magnetic force is perpendicular to v, so it does no work: the speed and kinetic energy never change.
3. Circle in a uniform field: r = mv/qB. The period T = 2πm/qB does not depend on the speed.
4. Helix when v has a part along B: pitch p = v∥ × 2πm/qB.
5. Force on a wire: F = I l × B; zero when the wire lies along B.
6. Biot–Savart: dB = (μ₀/4π) I dl sin θ/r²; no field along the line of the element.
7. μ₀ = 4π × 10⁻⁷ T m A⁻¹, and μ₀ε₀ = 1/c².
8. Centre of a coil: B = μ₀NI/2R. On its axis: B = μ₀NIR²/2(x² + R²)^(3/2).
9. Long straight wire: B = μ₀I/2πr, field lines are closed circles.
10. Thick wire with uniform current: B ∝ r inside, B ∝ 1/r outside, largest at the surface.
11. Long solenoid: B = μ₀nI inside, uniform, independent of radius; about zero outside.
12. Parallel wires: f = μ₀I₁I₂/2πd per metre; same direction attract, opposite repel.
13. 1 A is the current giving 2 × 10⁻⁷ N per metre between wires 1 m apart.
14. Current loop: m = NIA; τ = m × B; net force zero in a uniform field.
15. Stable equilibrium when m is along B, unstable when opposite.
16. Far field of a loop: (μ₀/4π)(2m/x³) on the axis, half that in the plane of the loop.
17. MCG: kφ = NIAB in a radial field; current sensitivity NAB/k.
18. Ammeter = galvanometer + small shunt in parallel; voltmeter = galvanometer + large resistance in series.

Common traps

Formulas

Key terms

Magnetic field (B)

A vector field set up by currents and moving charges, measured by the force it puts on a moving charge; unit tesla.

Tesla

The field that exerts 1 N on 1 C moving at 1 m/s at right angles to it; 1 gauss = 10⁻⁴ T.

Lorentz force

The total force q(E + v × B) on a charge in electric and magnetic fields.

Cyclotron frequency

The rate qB/2πm at which a charge circles in a uniform magnetic field, the same at every speed.

Pitch

The distance a charge moves along B during one turn of its helical path.

Current element

A tiny length dl of wire with its current, I dl, treated as a vector source of field.

Permeability of free space (μ₀)

The constant in the Biot–Savart law, 4π × 10⁻⁷ T m A⁻¹.

Amperian loop

A closed path chosen so that B is tangential and constant, or normal, or zero along its parts, making Ampere's law easy to apply.

Solenoid

A long, closely wound helical coil whose inside field is uniform and along its axis.

Ampere (unit)

The steady current that gives 2 × 10⁻⁷ N per metre between two long parallel wires 1 m apart in vacuum.

Magnetic moment (m)

NIA for a current loop, along its area vector; unit A m².

Magnetic dipole

A current loop seen from far away; its field has the same form as an electric dipole's.

Radial field

A field whose lines point along radii of the coil's axis, so the coil's plane always lies along B.

Torsional constant (k)

The restoring torque of the galvanometer spring per unit angle of twist.

Shunt

A small resistance put in parallel with a galvanometer so it can measure large currents as an ammeter.

Current sensitivity

Deflection per unit current, φ/I = NAB/k.

Voltage sensitivity

Deflection per unit voltage, φ/V = NAB/(kR).