Almost every NEET paper carries three to five questions from this unit, and they cluster around a small set of standard results — field of a wire and loop, radius of a circular path in a magnetic field, torque on a current loop, and the classification of magnetic materials. Getting the geometry and the direction conventions right is what separates a quick mark from a lost one.
Magnetic Field Produced by a Current
A steady current sets up a magnetic field in the space around it. The basic law describing the contribution of a tiny piece of a current-carrying conductor is the Biot–Savart law. For a small element of length $d\vec{l}$ carrying current $I$, the field at a point whose position vector from the element is $\vec{r}$ is
$$d\vec{B}=\frac{\mu_0}{4\pi},\frac{I,d\vec{l}\times \hat{r}}{r^{2}}, \qquad dB=\frac{\mu_0}{4\pi}\frac{I,dl\sin\theta}{r^{2}}$$
where $\mu_0 = 4\pi\times10^{-7}$ T m A⁻¹ is the permeability of free space and $\theta$ is the angle between the element and the line joining it to the point. Note the vector product: the field is perpendicular to both the current element and the position vector, so field lines curl around the wire (right-hand thumb rule — thumb along current, curled fingers give the sense of $\vec B$). Also note that $dB=0$ directly along the line of the element ($\theta = 0$ or $180°$), which is why a point on the extension of a straight wire experiences no field from that wire.
Standard results worth memorising:
- Infinitely long straight wire, perpendicular distance $a$: $B=\dfrac{\mu_0 I}{2\pi a}$, with circular field lines.
- Finite straight wire: $B=\dfrac{\mu_0 I}{4\pi a}(\sin\theta_1+\sin\theta_2)$, the angles measured from the perpendicular foot to the two ends.
- Centre of a circular coil of $N$ turns, radius $R$: $B=\dfrac{\mu_0 N I}{2R}$.
- Circular arc subtending angle $\phi$ (in radians) at the centre: $B=\dfrac{\mu_0 I\phi}{4\pi R}$.
- On the axis of a circular loop, at distance $x$ from centre: $B=\dfrac{\mu_0 I R^{2}}{2,(R^{2}+x^{2})^{3/2}}$.
Ampere's circuital law gives a shortcut whenever the situation is highly symmetric: the line integral of $\vec B$ around any closed loop equals $\mu_0$ times the net current threading that loop, $\oint \vec B\cdot d\vec l=\mu_0 I_{\text{enc}}$. Applying it to a long solenoid with $n$ turns per unit length gives a uniform interior field $B=\mu_0 n I$ (and $B\approx \mu_0 n I/2$ at either open end), while outside it is nearly zero. For a toroid of $N$ total turns and mean radius $r$, $B=\dfrac{\mu_0 N I}{2\pi r}$, with no field in the hollow interior or outside the ring. For a thick straight cylindrical conductor of radius $R$ with uniform current density, the field inside varies as $B=\dfrac{\mu_0 I r}{2\pi R^{2}}$ and outside falls as $1/r$.
Force on Moving Charges and on Currents
A charge $q$ moving with velocity $\vec v$ in a magnetic field feels $\vec F=q(\vec v\times\vec B)$; including the electric field gives the Lorentz force $\vec F=q(\vec E+\vec v\times\vec B)$. The magnetic part is always perpendicular to the velocity, so it does no work — it changes direction, never speed or kinetic energy.
- If $\vec v \perp \vec B$, the path is a circle with $r=\dfrac{mv}{qB}=\dfrac{p}{qB}$ and period $T=\dfrac{2\pi m}{qB}$, which is independent of speed and radius.
- If $\vec v$ makes an angle with $\vec B$, the parallel component is untouched and the path is a helix of pitch $= v\cos\theta \times T$.
- In a velocity selector, crossed $\vec E$ and $\vec B$ allow only particles with $v=E/B$ to pass undeflected.
- A cyclotron exploits the speed-independent period: the resonance condition is $\nu=\dfrac{qB}{2\pi m}$ and the maximum kinetic energy attainable at exit radius $R$ is $\dfrac{q^{2}B^{2}R^{2}}{2m}$. It cannot usefully accelerate electrons (they become relativistic too quickly) and cannot accelerate neutral particles.
For a conductor, summing the force over all the moving charges gives $\vec F=I(\vec L\times\vec B)$, magnitude $BIL\sin\theta$. A key consequence: for any shaped wire in a uniform field, only the straight-line vector from entry point to exit point matters, so a closed current loop in a uniform field experiences zero net force.
Two long parallel wires a distance $d$ apart exert on each other a force per unit length
$$\frac{F}{l}=\frac{\mu_0 I_1 I_2}{2\pi d}$$
attractive when the currents are parallel and repulsive when antiparallel. (This is the historical basis of the definition of the ampere.)
Current Loop as a Magnetic Dipole; Galvanometer
A plane coil of $N$ turns, area $A$, carrying current $I$ behaves like a magnetic dipole of moment $\vec m = NI\vec A$ (SI unit A m², direction given by the right-hand rule, along the normal to the plane). In a uniform field it feels no net force but a torque
$$\vec\tau=\vec m\times\vec B,\qquad \tau=mB\sin\theta$$
which is maximum when the plane of the coil contains $\vec B$ ($\theta = 90°$) and zero when $\vec m \parallel \vec B$. The associated potential energy is $U=-\vec m\cdot\vec B$, minimum ($-mB$) in the stable aligned position and maximum ($+mB$) in the unstable antiparallel position.
This torque is the working principle of the moving coil galvanometer: a coil in a radial field produced by a horseshoe magnet with a soft-iron core deflects until the magnetic torque is balanced by the restoring torque $k\phi$ of a spring, giving
$$\phi=\frac{NAB}{k},I \quad\Rightarrow\quad \text{current sensitivity }=\frac{NAB}{k},\qquad \text{voltage sensitivity}=\frac{NAB}{kR}.$$
Because deflection is directly proportional to current, the scale is linear. A galvanometer is converted into an ammeter by connecting a low resistance (shunt) in parallel, and into a voltmeter by a high resistance in series. Remember: an ideal ammeter has zero resistance and goes in series; an ideal voltmeter has infinite resistance and goes in parallel. Increasing the number of turns raises current sensitivity but not necessarily voltage sensitivity, since $R$ also rises.
The Bar Magnet, Earth's Magnetism
A bar magnet is equivalent to a magnetic dipole of moment $m$ pointing from its south pole to its north pole. Its field at a distance $d$ (much larger than its length) is
- Axial (end-on): $B=\dfrac{\mu_0}{4\pi}\dfrac{2m}{d^{3}}$
- Equatorial (broadside-on): $B=\dfrac{\mu_0}{4\pi}\dfrac{m}{d^{3}}$, and directed opposite to $\vec m$.
Magnetic field lines form closed loops (there are no isolated magnetic poles — the magnetic flux through any closed surface is zero, i.e. Gauss's law for magnetism gives $\oint \vec B \cdot d\vec S = 0$). Cutting a magnet in half produces two shorter magnets, each with both poles, each with half the original moment if the mass is halved.
A magnet free to rotate in a horizontal field $B$ oscillates with period $T=2\pi\sqrt{I/mB}$, where $I$ is the moment of inertia — the standard method for comparing field strengths or magnetic moments.
The Earth's field resembles that of a giant dipole tilted with respect to the rotation axis; its magnitude is a few times $10^{-5}$ T. Three elements of Earth's magnetism describe it at a place:
- Magnetic declination — the angle in the horizontal plane between geographic north and magnetic north.
- Angle of dip (inclination) $\delta$ — the angle the total field makes with the horizontal.
- Horizontal component $B_H$ — with $B_H=B\cos\delta$, $B_V=B\sin\delta$, so $\tan\delta = B_V/B_H$. At the magnetic poles $\delta = 90°$ (the field is purely vertical) and at the magnetic equator $\delta = 0°$ (purely horizontal). A dip circle measures $\delta$ directly, while a deflection magnetometer combined with an oscillation magnetometer is used to determine $B_H$ and, from it, the magnetic moment of a test magnet.
Magnetic Properties of Materials
Every material responds to an external magnetic field to some extent, characterised by the magnetic susceptibility $\chi_m = M/H$ (intensity of magnetisation per unit applied field $H$) and the relative permeability $\mu_r = 1+\chi_m$, with $\mu = \mu_0\mu_r$.
- Diamagnetic materials (bismuth, copper, water, most inert gases) have small negative $\chi_m$ (typically $-10^{-5}$), are weakly repelled from a strong field, and their magnetisation is temperature-independent. The effect arises from an induced opposition set up in the orbital motion of electrons and exists in every material, but is masked wherever the next two effects operate.
- Paramagnetic materials (aluminium, sodium, oxygen, platinum) have small positive $\chi_m$ (of order $10^{-3}$ to $10^{-5}$), are weakly attracted into a field, and possess a net atomic magnetic moment due to unpaired electrons that partially aligns with the field. Their susceptibility falls with rising temperature according to Curie's law, $\chi_m \propto 1/T$, because thermal agitation disrupts alignment.
- Ferromagnetic materials (iron, cobalt, nickel and their alloys) have very large positive $\chi_m$ (hundreds to thousands) due to spontaneous alignment of atomic moments within small regions called domains. Above a material-specific Curie temperature $T_C$, thermal motion destroys this domain order and the material turns paramagnetic.
Plotting the magnetisation $M$ (or $B$) against $H$ for a ferromagnet traces out a hysteresis loop rather than retracing the same curve: $M$ lags behind $H$. The value of $B$ remaining when $H$ is reduced to zero is the retentivity (residual magnetism), and the reverse field needed to bring $B$ to zero is the coercivity. Materials with a fat loop, high retentivity and high coercivity (like steel) make good permanent magnets; materials with a thin, easily reversed loop and low retentivity (like soft iron) are preferred for electromagnets and transformer cores, where the field must be switched on and off (or reversed) repeatedly with minimal energy loss — the area of the loop is the energy dissipated as heat per unit volume per cycle.
Common Mistakes and Exam Traps
- Mixing up the right-hand rules. The right-hand thumb rule (curl fingers from current to find $\vec{B}$) locates the field due to a current, while $\vec{F} = q\vec{v}\times\vec{B}$ or $\vec{F} = I\vec{L}\times\vec{B}$ needs the right-hand palm/slap rule (or equivalently Fleming's left-hand rule for the force on a current) — applying the field rule to a force question is a very common mix-up.
- Forgetting the sign of the charge in $\vec{F} = q\vec{v}\times\vec{B}$. For a negative charge, the force is opposite to what $\vec{v}\times\vec{B}$ alone gives; questions on electron beams in a magnetic field are frequently missed purely on this sign.
- Treating diamagnetism as if it behaves like paramagnetism. Diamagnetic susceptibility is essentially independent of temperature, whereas paramagnetic susceptibility falls as $1/T$ (Curie's law) — assuming all magnetic susceptibilities weaken with heating is wrong for diamagnets.
- Assuming cyclotron period depends on speed or radius. $T = 2\pi m/qB$ is independent of both; what changes as the particle speeds up is only the radius of its circular path, which is why the cyclotron can keep the accelerating frequency fixed.