NCERT Solutions Class 12 Physics Chapter 4 Moving Charges and Magnetism

Where the formula comes from. Start with the Biot-Savart law for a current element I d: dB = ₀4π I d×rr². For every element on the circular loop, d⊥ r and the distance to the centre is exactly the radius r. So |dB| = ₀ I/4/r². Integrating d once round the loop gives 2π r. Putting it together: B = ₀ I4π r²· 2π r = ₀ I2r. The directions of dB from every element point the same way (along the axis), so they add, not cancel.

Direction by the right-hand rule. Wrap the right-hand fingers along the direction of current flow; the thumb points along B. If viewed from the side along which B emerges, the current flows counter-clockwise.

Order-of-magnitude check. Earth's magnetic field is roughly 5× 10^-5 T. Our coil produces about 3× 10^-4 T - roughly six times Earth's field, easily measurable with a simple compass. That's the whole point of multi-turn coils: each turn is essentially a free amplifier.

Common mistake. Forgetting the factor of N. A single loop with 0.4 A would give a field of just 3.14× 10^-6 T - a hundred times smaller. Always multiply by the number of turns for a tightly-wound coil.

Derivation via Ampere's law. Draw an Amperian loop - a circle of radius a centered on the wire, lying in a plane perpendicular to it. By symmetry, B has the same magnitude everywhere on this loop and runs tangent to it. So ∮ B· d= B (2π a) = ₀ I_enc = ₀ I. Solving for B reproduces our formula. The 1/a falloff slower than the 1/r² of a point source is a hallmark of a one-dimensional source.

Vector form. B = ₀ I2π a ϕ, where ϕ is the azimuthal unit vector around the wire (curled fingers, current along thumb).

Comparison with Earth's field. The Earth's field is about 3× 10^-5 T - roughly the same as our answer! That's why nearby power lines or appliance wiring can briefly throw off an old magnetic compass.

Real-world. The "infinite wire" idealisation works very well when the distance from the wire is much smaller than the wire's length. Power transmission lines, household wiring, and electromagnets all use this formula in the field engineer's toolbox.

Visualising the geometry. Set up axes: let x point east, y point north, z point up. The current flows in the -y direction (north to south). The test point is at r = 2.5 x m (east of the wire).

Vector form of the field at the test point: B = ₀ I2π a (I × r), where I is the unit vector along the current and r is the unit vector from the wire to the field point. Here I = -y (south) and r = +x (east), so I × r = (-y)×(x) = -(y×x) = -(-z) = +z. The algebra delivers +z - upward - in perfect agreement with the right-hand-thumb rule applied above. Whenever the cross-product and the curl-the-fingers rule give different answers, recheck the cyclic ordering x×y = z - both methods must agree.

Common mistake. Confusing current direction with field direction. The two are perpendicular to each other in a long straight wire: current goes along the wire, field curls around it.

Real-world. A field of 4 is small but measurable. A power line carrying 50 A at 2.5 m distance produces a field about a tenth of Earth's. Electricity-board safety codes give detailed minimum-distance rules to limit exposure for residential structures.

Why "horizontal"? The field at a point not on the axis of a straight wire is always perpendicular to the radius vector AND perpendicular to the current. Since the wire is horizontal (east-west) and the radius from wire to point is vertical (downward), the field - perpendicular to both - must lie horizontally in the north-south direction.

Practical implication. A field of 1.2× 10^-5 T is about 1/4 of Earth's magnetic field. A magnetic compass placed under such an overhead line will deflect noticeably, and old-fashioned magnetic-needle navigation instruments would mis-read in the vicinity. Engineers carefully route compass-sensitive equipment away from high-current cables.

Alternative method via Biot-Savart integration. Treat the wire as infinite and integrate dB = ₀4π I d×rr² from -∞ to +∞ along the wire. The result reduces to ₀ I/2π a - same as the Amperian shortcut.

Common mistake. Choosing the wrong side. Always pin down whether the field point is above, below, east, or west of the wire BEFORE applying the right-hand rule. Sketching it on paper rarely fails.

Vector form. dFdL = I L×B. The direction of the force is perpendicular to BOTH the wire and the field. Use the right-hand rule: fingers along L (current), curl toward B; thumb shows dF/dL.

Why the sinθ factor? Only the component of B perpendicular to the wire pushes on the current. The parallel component is wasted - moving charges in the wire don't feel a Lorentz force from a field parallel to their motion. So effective field is B_⊥ = Bsinθ, giving F/L = IB_⊥ = BIsinθ.

Sanity check at limits.

• If θ = 0 wire parallel to B: sin 0 = 0, no force. Makes sense - no perpendicular component of B.
• If θ = 90^∘ wire perpendicular to B: sin 90^∘ = 1, force is maximum, f = BI = 0.15× 8 = 1.2 N/m. Half this at our 30^∘ tilt - consistent with sin 30^∘ = 1/2.

Real-world. This force underpins every electric motor on Earth. In a DC motor, current flows through coils sitting in a permanent-magnet field; the resulting force on the wires creates the torque that spins the rotor. Larger current, stronger field, or longer wire (more turns) ⇒ more torque.

Direction analysis. The force F = IL×B is perpendicular to L AND B. Since B is along the solenoid axis and L is perpendicular to that axis, F lies perpendicular to both - i.e., it pushes the wire sideways in the plane perpendicular to the wire, but still inside the cylindrical solenoid.

Why is B uniform inside the solenoid? A long solenoid produces an essentially uniform axial field inside it, with magnitude B = ₀ nI_s where n is turns per metre of the solenoid and I_s its own current. The field outside is approximately zero. So the wire experiences a constant force everywhere along its length - making the calculation simple.

Alternative method. Express the force per unit length first, f = BI = 2.7 N/m. Then multiply by L = 0.030 m to get F = 0.081 N. Useful when comparing forces on wires of different lengths in the same field.

Common mistake. Plugging length in centimetres instead of metres. L must be in SI metres for the formula to yield F in newtons.

Real-world. Solenoids inside loudspeakers, MRI machines, and particle accelerators provide nearly uniform fields exactly so that test currents (or charges) feel predictable forces. This is also how a "voice-coil" actuator drives speaker cones - a wire (the voice coil) sits in the radial field of a permanent magnet and feels axial forces.

Why "attractive" for parallel currents? Each wire's field circles around it (right-hand rule). At wire A's location, B's field points perpendicular to A and to the line joining the wires. When you cross A's current direction with this field IL×B, the resulting force vector points toward B. Reverse one current and the force reverses too.

Why this defined the ampere (historically). The pre-2019 SI definition of one ampere was: "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2× 10^-7 newton per metre of length." Plugging I_A = I_B = 1 A and d = 1 m into F/L = ₀ I_A I_B/2π d gives exactly 2× 10^-7 N/m - so this constraint pinned ₀ = 4π× 10^-7 exactly. After the 2019 SI redefinition, the ampere is defined via the elementary charge e instead, and ₀ is now an experimental quantity very close to but no longer exactly 4π× 10^-7.

Common mistake. Forgetting to multiply by length L. The formula ₀ I_A I_B/2π d gives force per unit LENGTH, not total force. For a 10 cm section, multiply by 0.10 m.

Newton's third law. Wire A feels a force F toward B; wire B feels the SAME magnitude force toward A. The pair forms a balanced action-reaction pair, just like gravity between two masses.

Why does the diameter NOT appear in the formula? For an ideal (long) solenoid, the interior field is independent of the radius of the windings - it depends only on turns per unit length n and current I. This is a remarkable result: a thin solenoid and a fat one produce the same axial field, provided both have the same n and I. The diameter matters only near the ends (where the field starts to "leak out") or if you go off-axis.

Derivation via Ampere's law. Draw a rectangular Amperian loop with one side of length inside the solenoid (parallel to the axis) and the other side outside where B ≈ 0. The line integral of B· d collapses to B on the inside side, and the enclosed current is nI. Setting them equal: B = ₀ n I ⇒ B = ₀ n I.

Why "near the centre"? The end-effects of a finite solenoid drop the field at each end to about half its central value. Only deep inside (within a few diameters of the centre) does the full ₀ n I formula apply. For our solenoid, length/diameter = 80/1.8 ≈ 44, so the "long" approximation is excellent.

Real-world. Solenoids power electromagnets (for cranes, MRI, particle accelerators), inductors in circuits, relays, and electric door locks. The trick to a strong field: maximise nI (more turns per metre, more current).

Vector form and direction. τ = m×B. The torque vector is perpendicular to BOTH m and B. It always points in the direction that, by right-hand rule, would rotate m toward B (i.e., it tries to align them). Equilibrium occurs when m ∥ B, θ = 0, torque zero.

Why sinθ? When the coil's normal is parallel to B θ = 0, the coil is already in the lowest-energy orientation - no torque. When perpendicular θ = 90^∘, the lever arm is maximum and so is the torque. The sinθ interpolates smoothly.

Why doesn't the SHAPE of the loop matter? The formula τ = NIABsinθ holds for ANY planar loop, not just squares. The forces on individual sides combine into a couple whose torque depends only on the enclosed area A, the current I, and the field orientation. Replace the square by a circle of the same area, the torque is the same.

Real-world. This is the foundation of the moving-coil galvanometer (Q 4.10 next) and DC motors. The coil rotates in the field; a clever commutator reverses the current every half-turn so the torque always drives rotation in the same direction.

Why two different "sensitivities"? A galvanometer measures current via the deflection of a coil. But often we connect it across a voltage source (e.g., to make a voltmeter), and we care about deflection per volt, not per amp. The two quantities are related by Ohm's law: S^(V) = S^(I)R.

Engineering takeaway. To make a galvanometer more current-sensitive, you can: (i) add more turns N, (ii) use a larger coil area A, (iii) use a stronger field B, or (iv) use a weaker spring (smaller k). But options (i) and (iii) increase the resistance R too, which trades off voltage sensitivity. Real meter design is a balance.

Common mistake. Forgetting that R enters voltage sensitivity. Many students compute only the current-sensitivity ratio (1.4) and incorrectly report it for both parts. The voltage-sensitivity is (NAB/kR), not (NAB/k).

Why B₂ > B₁ but A₂ < A₁? Increasing B shrinks the coil for the same torque. Manufacturers play with this tradeoff - small, strong-field coils for portable instruments; large, weaker-field coils for precision lab work. In modern digital multimeters the moving coil is replaced by silicon sensors, but the underlying physics remains the same.

Why CIRCLE and not some other curve? The force F = qv×B has two crucial properties:

• Always perpendicular to v ⇒ no change in speed (no work done).
• For v⊥B, magnitude is constant: |F| = qvB with v constant.

The r = mv/(qB) formula. This is called the gyroradius or Larmor radius - fundamental in plasma physics, accelerator physics, and ionosphere studies. Lighter or slower particles spiral tighter; stronger fields squeeze the radius. For an electron at v ∼ 10⁶ m/s in Earth's field, r is centimetres - the basis for the cathode-ray tubes in old TVs.

Common mistake. Forgetting that the speed (not just velocity) stays constant. Some students think the electron decelerates as it curves - wrong. The magnetic force does no work; it only steers.

Real-world. The Northern Lights (aurora) form when solar-wind electrons spiral down Earth's magnetic field lines and crash into upper-atmosphere atoms. Cyclotrons and mass spectrometers exploit the same equation to sort charged particles by their mass-to-charge ratio.

Why the speed cancels. A faster particle has higher momentum mv, giving a bigger radius r = mv/(qB). The circumference 2π r is also bigger by the same factor. But the speed itself is higher in the same proportion. So time-to-go-around (circumference ÷ speed) is unchanged. It's a beautiful balance - and it's what makes the cyclotron work.

The cyclotron. Ernest Lawrence built the first cyclotron in 1932 using exactly this insight: ions of a fixed q/m circle at a fixed frequency in a magnet, regardless of their speed. By applying an oscillating electric field at this same frequency across a gap, each crossing accelerates the ion further out, into ever-larger circles, until it has enough energy to be ejected. Won him the 1939 Nobel Prize.

Limits - when does speed start to matter? At relativistic speeds (v close to the speed of light), the mass effectively increases: m → γ m. The cyclotron frequency drops as energy rises. This is why high-energy accelerators use "synchrotrons" - devices that vary the field strength and oscillator frequency to compensate.

Sanity check. Modern radio operates between 10⁵ and 10¹⁰ Hz. Our cyclotron frequency ∼ 17 MHz sits in the radio band - exactly the regime where solar electrons interacting with Earth's field generate natural "whistler" radio emissions detectable from the ground.

Why doesn't shape matter? Consider any planar loop carrying current I in field B. Slice it into thin strips parallel to one side. Each strip is a rectangle whose torque calculation is straightforward. Adding up all the rectangles' contributions, the cross-terms cancel and only the total enclosed area survives. The same conclusion follows from the more advanced result τ = m×B with m = IA - and A is the vector with magnitude equal to the loop's enclosed area, pointing along the normal (right-hand rule). The vector A depends only on the loop's projection, not on its zig-zags.

Counter torque vs. restoring torque. The field exerts a torque ON the coil that tries to rotate it. To hold the coil stationary, you must apply an EQUAL magnitude torque in the opposite direction. Net torque = zero ⇒ no angular acceleration.

Numerical sanity. τ ∼ 3 N m is sizable - like the torque you exert turning a screwdriver. Strong magnets can grip current loops with quite formidable force. This is exactly how an electric motor delivers its torque.

Going further. If you wanted to find the WORK done in rotating the loop from ₁ to ₂: W = _1²τ dθ = -mB(cos₂ - cos₁). The potential energy of the loop is U = -m·B = -mBcosθ, minimised when m∥B. This is why current loops naturally orient with their normals along the field - like compass needles aligning with Earth's field.