GATE PH common mistakes
GATE PH · Understand the traps and how to avoid them.
Evaluates every residue with the simple-pole formula lim(z→a)(z−a)f(z), even when the denominator has a repeated root (e.g. a double pole at z = 0 from a 1/z² factor), and reports the resulting wrong contour integral.
The simple-pole residue formula is the one most often memorized, and after partial fractions or a quick mental factorisation the multiplicity of a root is easy to overlook. Jordan's-lemma problems often hide a higher-order pole at the origin.
What to do instead
Before integrating, factor the denominator completely and write each pole with its multiplicity m. For m ≥ 2 use [1/(m−1)!] d^(m−1)/dz^(m−1) [(z−a)^m f(z)] evaluated at a, and cross-check by expanding f in a Laurent series around that pole. Also confirm the closing half-plane matches the sign of the exponent before applying Jordan's lemma.
Writes H = T + V and equates it with the conserved energy for a system with an explicitly time-dependent constraint, such as a bead on a wire rotating at constant angular speed, and then claims energy is conserved.
In most textbook Lagrangian problems the transformation is scleronomic (no explicit t), so T is quadratic in the velocities and H equals E. That habit is carried over to rheonomic constraints, where ∂L/∂t ≠ 0 and the two quantities differ.
What to do instead
Always compute H by the Legendre transform H = Σ p_i q̇_i − L and check ∂H/∂t and ∂L/∂t separately. If ∂L/∂t = 0, H is conserved but generally H ≠ E when the kinetic energy is not a homogeneous quadratic in the velocities; only then decide what is actually constant.
Uses E = −∇V and scalar-potential boundary conditions in a problem where the magnetic field is time-varying, dropping the −∂A/∂t term, and therefore predicts a zero induced field or a wrong sign for the induced EMF.
Electrostatics dominates the early part of the syllabus and the relation E = −∇V is used so often that the full expression E = −∇V − ∂A/∂t is not recalled under time pressure, especially in Faraday-law and gauge problems.
What to do instead
Before using any potential, ask whether ∂B/∂t is zero. If not, use the full E = −∇V − ∂A/∂t, or bypass potentials with ∇×E = −∂B/∂t and the flux rule. Choose the gauge explicitly and state it; a Coulomb-gauge and Lorentz-gauge computation must give the same physical field.
Applies non-degenerate perturbation theory to a level that is degenerate in the unperturbed Hamiltonian, for example the first excited level of the 2D harmonic oscillator or a p-level in a crystal field, and quotes a single first-order shift.
The formula E_n^(1) = ⟨n|V|n⟩ is easy to apply mechanically and gives a plausible-looking number, so the preliminary check of the degeneracy of the unperturbed level is skipped, particularly when the perturbation looks small.
What to do instead
First list the unperturbed energies and count the degeneracy of the level you are correcting. If the degeneracy is greater than one, form the matrix of V in that degenerate subspace, diagonalise it, and report the eigenvalues, noting that the good basis states are the eigenvectors, not the original ones.
Applies the equipartition theorem to all degrees of freedom of a molecule at a temperature where some modes are frozen out, and so overestimates the heat capacity (for example giving 7/2 R or 4R for a diatomic gas at room temperature).
Equipartition is learned as a universal recipe of (1/2)kT per quadratic degree of freedom, and the quantum condition that a mode contributes only when kT is comparable to its level spacing ħω is treated as an afterthought.
What to do instead
For each mode compute the characteristic temperature Θ = ħω/k (or the rotational temperature) and compare it with T. Include only modes with T ≳ Θ, and above that use the full Einstein or Debye expression rather than the classical limit. Sanity-check the result against the Dulong–Petit value 3R per mole of atoms at high T.
Determines the ground-state term of a partially filled shell by maximising J, or applies Hund's rules in the order S → J → L, and so picks the wrong term and the wrong fine-structure ordering.
Hund's three rules are often memorised as an unordered list, and the rule for J reverses direction between less-than-half-filled and more-than-half-filled shells, which is exactly the case most exam questions choose.
What to do instead
Apply in strict order: maximum S first, then maximum L among those, then minimum J if the shell is less than half filled and maximum J if it is more than half filled. Verify the result by checking that the term is allowed by the Pauli principle for the given configuration, and cross-check with the Landé interval rule.
Counts atoms per conventional cubic cell wrongly for the diamond, zincblende, or hexagonal structures and reports an incorrect number density, cohesive energy per atom, or packing fraction.
The FCC count of 4 atoms per cell is well drilled, but diamond adds two interpenetrating FCC sublattices (8 atoms), and the hexagonal close-packed cell contains 6 atoms. The easy-to-remember numbers are applied to the wrong structure.
What to do instead
Count explicitly by site type: corners (1/8), faces (1/2), edges (1/4), body (1). For diamond record 8 atoms per conventional cube (2 per primitive cell), for HCP 6 per hexagonal cell, and always state whether n is per conventional or primitive cell before computing n = ρN_A/M.
Treats the Q-value of an endoergic reaction as the beam kinetic energy needed to make it proceed, rather than applying the recoil correction, so the computed threshold energy is too low.
Q-value and threshold energy are both energies in MeV and are conflated; the need for the target to absorb momentum (so that the centre-of-mass must have kinetic energy) is a subtlety that vanishes in the centre-of-mass frame and is therefore easy to miss.
What to do instead
Use E_th(lab) = −Q (1 + m_projectile/M_target) for a fixed target, or equivalently compute s = (m_a + m_A)² + 2 m_A T_a and require s above the sum of final masses squared. Check the frame explicitly in the question before substituting numbers.
Computes the small-signal gain of a BJT or MOSFET stage by assuming a convenient value of g_m or r_π without first solving the DC bias network and confirming that the device is in the forward-active (or saturation, for MOS) region.
Small-signal models look self-contained once g_m is written down, and the algebra of a bias network is tedious; under time pressure g_m is taken from a remembered current or from the data-sheet β, and a device that is actually cut off or saturated is analysed as a linear amplifier.
What to do instead
Do DC analysis first: find the quiescent I_C or I_D, check V_CE > V_CE(sat) or V_DS > V_GS − V_th, then set g_m = I_C/V_T (≈ 38 I_C) or g_m = 2√(K I_D). Only after the operating point is verified should the small-signal equivalent circuit be drawn.
Leaves the General Aptitude section to the final minutes of the paper and attempts it with whatever time is left, so the 15 marks of comparatively easy numerical-ability and data-interpretation questions are lost.
Physics candidates mentally rank GA as 'not physics' and plan to return to it, but the physics sections expand to fill the available time, and GA's short questions then get rushed or abandoned. The scoring weight per minute is actually highest in GA.
What to do instead
Schedule a fixed block of about 20–25 minutes for the 10 GA questions early in the paper, with a hard 90-second cap per question on the first pass. Flag rather than fight a long data-interpretation set, and return only after the physics single-mark questions are secured.