NCERT Solutions Class 12 Physics Chapter 6 Electromagnetic Induction

Why Lenz's law has to be true - energy conservation. Imagine if the induced current supported the change instead of opposing it. Pushing a magnet towards a coil would create a current that pulled the magnet in faster - which would grow the current more - which would pull harder still, building unlimited kinetic energy from nowhere. That violates energy conservation. The minus sign in Faraday's law is nature's way of saying "no free lunch": you must do work against the induced force to drive the change.

Right-hand rule for loop current → flux. Curl the fingers of your right hand along the direction of current flow in the loop; the thumb points in the direction of the magnetic field produced inside the loop. This is the workhorse rule for translating "I need flux pointing this way" into "current must flow this way."

Common mistake. Students sometimes apply Lenz's law to the field of the magnet instead of the loop. Lenz's law speaks only about the flux through the loop and the field produced by the induced current - never about the magnet's own field. Always ask: "How is the flux through the loop changing, and what current would oppose that change?"

Real-world example - induction cooktops. A copper coil under the ceramic surface carries an alternating current at ~25 kHz, producing rapidly changing magnetic flux. The iron base of the pan acts as the second coil - induced eddy currents heat it (Joule heating). The cooktop itself stays cool because ceramic is non-magnetic. This is Lenz's law put to work: the induced current heats the metal of the pot, not the air around it.

Isoperimetric inequality. Among all simple closed curves of a given perimeter, the circle encloses the maximum area. This is why soap bubbles, planet cross-sections, and water droplets all settle into circular/spherical shapes - they minimise surface for a given volume (or maximise area for a given boundary). In part (a), this geometric fact tells us the area must grow.

Vector form of Lenz's law. If we set up a normal n on the loop (say, out of the page), the flux is Φ = B· n A. The induced emf is ε = -dΦ/dt. The positive sense of the induced current is given by the right-hand rule applied to n: curling fingers in the positive current direction makes the thumb point along n. A positive ε drives current in the positive sense; a negative ε drives it the other way.

Common mistake. Forgetting that the area is what's changing, not B. Many students assume B is given as "varying" and look for a time-dependent B(t). Here B is constant - it's the loop's geometry that drives dΦ/dt.

Real-world example - flux-change motors. Some electrical generators don't spin a coil but instead deform it (e.g., piezoelectric-actuated flexible coils). The principle is exactly part (b): change the enclosed area in a fixed field, get an induced emf without rotation. Useful for energy harvesting from vibrations.

Why the loop's number of turns doesn't appear. The problem says "a small loop" - singular. If it were a multi-turn coil with N turns, the emf would be N× larger. Always check: does the inner loop have one or many turns?

Vector view. Inside an ideal infinite solenoid, B = ₀ n I z (axial, uniform). Outside, B = 0. So a small flat loop normal to the axis intercepts the full flux BA. If you tilted the loop by angle θ from the axis, the flux would shrink to BAcosθ and the emf likewise.

Common mistake. Forgetting to convert n from turns/cm to turns/m. 15 turns/cm = 1500 turns/m, not 15. A factor-of-100 error is easy to make here.

Real-world example - transformer principle in miniature. The setup is exactly a transformer: a primary solenoid carrying a changing current, a secondary loop catching the changing flux. The induced emf in the secondary depends on the rate dI/dt - which is why transformers only work with AC, not DC. With steady DC, dI/dt = 0, and no flux change means no induced emf.

Derivation of motional emf from Lorentz force. Inside the moving conductor, free charges experience a magnetic Lorentz force F = qv × B. For v perpendicular to B, this force is along the wire and has magnitude qvB. The "equivalent electric field" inside the wire is E = vB, and the emf across length of wire is ε = E= Bv. The full energy balance: external agent does work F_ext· v = BI· v per second, and dissipated power in resistance is I² R = ε²/R. They match.

Why ε· t is invariant. The total change in flux as the loop exits is ΔΦ = B· A = B· L. The induced emf is |ε| = ΔΦ/Δ t (since the rate is steady). Therefore ε· t = ΔΦ - independent of which side faces the motion. The orientation only redistributes between "voltage" and "duration."

Common mistake. Mixing up which side is the "cutting" side. The rule is: the conductor that sweeps through new field lines is the one perpendicular to the velocity. If you walk a fishing net through water, only the front edge catches fish - same idea.

Real-world example - railguns and MHD generators. Motional emf in a conductor moving through a magnetic field is the principle behind both railguns use F = BIL to launch projectiles and magnetohydrodynamic generators (a conducting fluid flows through a magnetic field, generating emf without any moving solid parts). NASA studied MHD for Mars-mission power.

Derivation from v× B. Consider a tiny element of the rod at distance r from the axis. It moves with speed v = ω r (tangential). The Lorentz "motional" emf contribution from this element is dε = (v B) dr = Bω r dr. Integrating from r = 0 to r = L: ε = ₀^L Bω r dr = Bω L²2 = 12 B ω L². Same answer, derived from the microscopic Lorentz force.

Common mistake. Using ε = B L v with v = ω L. That would give Bω L² - twice the correct answer. The mistake is to treat every part of the rod as if it moved at the tip's speed. In reality, the inner parts of the rod move slower than the outer parts, so the average speed and hence ε is half what you'd get with v_tip. The factor of 12 is essential.

Polarity. If B points out of the page and the rod rotates anticlockwise, the force on positive charges qv× B is radially outward. So positive charges pile up at the tip (ring) and electrons at the centre - the tip is at higher potential. The induced current in an external circuit would flow from ring → external load → centre.

Real-world example - homopolar generator. A disk rotating in a perpendicular magnetic field generates DC voltage between centre and edge (this exact geometry, with a disk instead of a rod). Faraday built the first one in 1831. Modern variants are used to produce huge low-voltage currents for electromagnetic pulse experiments. The formula is the same: ε = 12Bω R².

Why we only use B_H, not the full Earth field. The Earth's magnetic field has horizontal and vertical components. Only the part perpendicular to both v (vertical) and L (east-west) contributes to v×B. The vertical component is parallel to v and gives zero contribution to motional emf. The horizontal component (north-pointing) is the only useful piece.

Numerical sanity check. Earth's field is tiny ∼ 10^-4 T. A 10-m wire falling at 5 m/s gets only a mV-scale emf - small but measurable. By contrast, dropping a wire through a strong lab magnet B ∼ 1 T would give kilovolts.

Common mistake - direction of force. Students often forget that v×B gives the direction on positive charges. Free charges in metals are electrons (negative); they go the opposite way. But the convention for "which end is at higher potential" follows the positive-charge convention, so always reason with positive charges first.

Real-world example - power lines in storms. A long horizontal transmission line oscillating in the wind can pick up small AC voltages from its motion through Earth's field. This is a tiny effect - usually swamped by the lines' own normal voltage - but it's why long high-altitude balloons or tethers can generate measurable potentials. NASA's Tethered Satellite System (1996) deployed a 20-km conducting tether from the Space Shuttle and measured kV-scale emfs from v×B - until the cable broke!

What L physically means. Self-inductance measures how much flux a coil produces (within itself) per unit current. A coil with large L "resists" changes in current - it builds up a large back emf when you try to change I. That's why inductors are also called "chokes" - they choke off rapid changes.

Where does the energy go? When current is decreasing in an inductor, the back emf actually drives current to maintain the existing flow (Lenz's law). The energy stored in the magnetic field of the inductor U = 12LI² gets dumped - either into a load, or as a spark across the switch contacts, which is why opening an inductive circuit can produce arcs.

Common mistake - sign of emf. The minus sign in ε = -L dI/dt is essential for circuit analysis, but in magnitude problems like this one, just use the size. Don't try to "carry" the minus sign through; instead, reason about polarity separately using Lenz's law.

Real-world example - automotive ignition coils. A car's spark plug needs ~20,000 V to ionise the gap. An ignition coil has L ∼ 100 mH and carries about 5 A. When the points open, the current is killed in microseconds, giving dI/dt ∼ 5/10^-5 = 5× 10⁵ A/s. The induced emf is L· dI/dt = 0.1× 5× 10⁵ = 50 000 V - plenty to spark the plug. Same physics, just bigger numbers.

Symmetry of mutual inductance. One of the deep results of magnetostatics is that M₁₂ = M₂₁: the flux linkage in coil 2 due to unit current in coil 1 equals the flux linkage in coil 1 due to unit current in coil 2. This is not obvious from the geometry - it follows from the structure of the vector potential and Stokes's theorem. It's why we write just one M for any pair of coils.

Coupling coefficient. M is bounded by the geometric mean of the two self-inductances: M = k √L₁ L₂, 0 ≤ k ≤ 1. Here k is the "coefficient of coupling." For a well-designed transformer with an iron core, k approaches 1 (almost all flux from primary threads the secondary). For coils in air, k is much smaller.

Common mistake. Confusing flux Φ with flux linkage NΦ. For a single-turn coil they're equal; for an N-turn coil the linkage is NΦ. In this problem "the other coil" is treated as one coil (any number of turns is absorbed into M), so ΔΦ = M Δ I gives the total linkage change directly.

Real-world example - wireless charging. Phone wireless chargers use mutual inductance. A coil in the charger pad carries AC, generating changing flux. A second coil in the phone's back catches that flux and converts it back to current. Typical M ∼ , but with kHz-range dI/dt, enough power is transferred to charge a battery - all without physical contact. Tesla envisioned this in 1891; we finally built it into consumer products around 2010.

Why only B_V matters here. The Lorentz force on charges in the wing is qv×B. With v = west and B = B_Hn + B_V-z (north + downward in northern hemisphere), the cross product gives:

• v× B_H n: west crossed with north = upward - drives charges vertically (i.e., out of the wing, but the wing has no vertical extent, so no emf between wingtips).
• v× B_V-z: west crossed with down = north - i.e., along the wing. This produces the emf.

Magnetic dip explained. Earth's magnetic field tilts down as you move from equator to poles. The "dip angle" δ is the angle between the field and the horizontal. At the magnetic equator, δ = 0 (field is horizontal). At the magnetic poles, δ = 90^∘ (field is vertical). At latitudes like Delhi (~28°N), δ ≈ 47^∘; the problem's 30^∘ corresponds to closer to the equator.

Common mistake. Using the full B instead of Bsinδ. The horizontal component of Earth's field, combined with horizontal flight velocity, gives a Lorentz force pointing vertically - irrelevant for emf between two horizontally-separated points on the wing.

Real-world example - aircraft navigation. 3.1 V across a wing is too small to power anything, but it's enough to confuse sensitive avionics if not shielded. More importantly, the SAME effect plays havoc with long-haul HVDC submarine cables, where ocean currents flowing past stationary cables (or cables vibrating in currents) can generate measurable potentials. Engineers must account for this in any long-distance DC transmission design.

Why the loop heats up at all. A stationary loop in a stationary field has no induced current. But here the field is being actively changed by an external agent. From the loop's point of view, the flux Φ = BA is decreasing, so an emf appears, drives a current, and the current dissipates power as heat. Every joule of heat in the loop is matched by an extra joule of work the external agent has to do.

Energy conservation in detail. When the electromagnet's current is reduced, the energy stored in the electromagnet's field U = 12L_magI_mag² decreases. Most of that energy returns to the source through the supply leads (as the magnet's back emf). A tiny fraction "leaks" into the loop via mutual coupling: it appears as heat in the loop's resistance. The "tiny fraction" is set by the geometry - the mutual inductance between magnet and loop.

Common mistake. Saying "the magnetic field provides the energy." Magnetic fields don't do work on charges (force is always perpendicular to velocity), so they can't be the energy source. The energy traces back to whatever is changing the field - typically the battery feeding the electromagnet.

Real-world example - eddy-current braking. Same principle, larger scale. A rotating metal disk passes between the poles of a strong magnet (or vice versa). Changing flux induces eddy currents in the disk; the disk's resistance dissipates the kinetic energy as heat. Used in train brakes, gym treadmills, and roller-coaster safety stops. The energy source is the rotational kinetic energy of the disk - which is exactly what we want to remove.

Why two terms add. Even if the loop stayed still, the flux would change due to ∂ B/∂ t. Even if B were time-independent, the moving loop would experience a flux change because it walks into regions of different B. Both effects produce real emf - and they add directly because flux is a scalar.

Generalized Faraday's law. For a moving loop in a time-varying field, ε = -∮ ∂ A∂ t· d+ ∮(v× B)· d, where A is the vector potential. The first term captures ∂ B/∂ t; the second is the motional emf and corresponds to our "motion-through-gradient" piece. Together they reproduce ε = -dΦ/dt for any sufficiently nice situation.

Common mistake. Adding the wrong signs. Both ∂ B/∂ x and v need to be tracked carefully. Here ∂ B/∂ x < 0 (field decreases as x grows), and v > 0, so the motion contribution is negative - i.e., a flux decrease. The time contribution is also negative. The two combine constructively.

Real-world example - railway and pipeline corrosion sensors. A coil carried over a rail with a slight non-uniformity in the rail's magnetisation will produce a small emf - the spatial-gradient term in our equation. Eddy-current sensors detect cracks, voids, or material thinning by exactly this mechanism: a moving probe scans the surface, and any local anomaly in the magnetic environment shows up as an emf spike. Same physics as our textbook problem, but with industrial sensitivity.

Why Q is independent of speed. The induced emf ε gets bigger if you snatch the coil out faster dΦ/dt is larger. But then the current I = ε/R also gets bigger - and the duration of the pulse gets correspondingly shorter. The product ∫ I dt (the total charge) only depends on the total change in flux linkage, not on the rate. This is exactly what ballistic galvanometers exploit: they measure the integrated charge pulse, which directly gives NΔΦ.

Ballistic galvanometer. A long-period moving-coil galvanometer that responds to the integral of current rather than its instantaneous value. The maximum deflection angle is proportional to the total charge that flowed (provided the pulse is much shorter than the galvanometer's period). Standard 19th-century instrument for measuring magnetic flux - still used pedagogically.

Common mistake. Forgetting the factor N (turns count). The flux per turn is BA; the flux linkage which is what ε responds to is N times that. With N = 25 turns, the measured charge is 25× larger than it would be for a single-turn coil. Always check whether the "area" or "flux" given is per turn or total.

Real-world example - magnetometers. Modern Hall-effect and fluxgate magnetometers replaced the ballistic galvanometer for measuring fields, but the basic principle (relating a measured electrical signal to a magnetic flux) is unchanged. The Earth's field is mapped this way, and MRI machines calibrate their fields using closely related coil-based techniques.

The big idea - rails as a 'simple generator'. This problem is the cleanest illustration of how mechanical work becomes electrical energy. An external agent pushes the rod. The rod's motion through the field creates an emf. The emf drives a current. The current in the field experiences a force that opposes the motion (Lenz). The agent must do work against this force; that work shows up as heat in the resistance. P_agent = P_heat, exactly - no magic, no extra energy from nowhere.

Microscopic picture of part (c). When the rod first starts moving, the Lorentz force qv× B pushes the (positive) charges to one end. They pile up. The pile-up creates an electric field that pushes back in the opposite direction. Equilibrium is reached when these two forces balance - the pile-up is exactly large enough that eE = evB, i.e., E = vB, and the potential difference EL = BvL = ε. This is a classic example of how an "emf source" works at the microscopic level.

Common mistake. Confusing K open with K closed for the equilibrium of forces. With K open: net charge piles up at the ends (static); electrons inside the rod feel zero net force in steady state. With K closed: there is steady current flow; the Lorentz force on individual electrons isn't fully balanced (otherwise no drift), but there's a steady balance between the Lorentz force, the field from end-charges, AND the small electric field that drives the current. Subtle.

Generator vs. motor. Same hardware, opposite direction of power flow. In a generator (this problem) you push the rod and extract electrical energy. In a motor, you pass current through the rod and extract mechanical motion. The two are connected by the same equations: emf in one direction = force in the other. Every electrical generator is a motor running backwards, and vice versa.

Real-world example - eddy-current dynamometer. Some dynamometers use exactly this setup to measure engine power: the engine drives a conducting disk between magnets; the eddy current dissipates known power that can be measured, giving an absolute measurement of the engine's mechanical output. The "rod on rails" generalises to "disk in a slot" and is heavily used in mechanical engineering testbeds.