PAU Physics (Andalusia) common mistakes
PAU Physics (Andalusia) · Understand the traps and how to avoid them.
When computing the gravitational field of a spherical shell or solid sphere at an interior point, the candidate applies the full enclosed mass M (or the total field g = GM/r²) instead of using only the mass inside radius r, giving a field that grows toward the centre rather than vanishing there.
The candidate memorises g = GM/r² as a universal formula and never re-derives it from Gauss's law, so the distinction between 'mass enclosed by the Gaussian surface' and 'total mass of the body' is never made explicit. The shell theorem is often quoted but not applied.
What to do instead
Before substituting numbers, always draw a Gaussian sphere of radius r and write M_enc explicitly: M_enc = M for r ≥ R, and M_enc = M·(r³/R³) for a uniform solid sphere with r < R. Then g = G·M_enc/r². Check the limiting cases g(R) and g(0) = 0 as a self-test before finalising the answer.
Treating gravitational potential V = −GM/r as if it were a vector and adding magnitudes, or reporting the potential energy of a two-body system as −GMm/r without the negative sign, and then confusing potential with field when asked for the work needed to move a mass between two points.
Students conflate the scalar potential with the vector field because both carry the same symbol family and both are introduced together; the minus sign in the potential is treated as cosmetic rather than as encoding that the force is attractive and the bound state has negative energy.
What to do instead
Write the two definitions side by side: g = −dV/dr (a vector pointing down the potential gradient) and V = −GM/r (a scalar with an arbitrary zero at infinity). For work done by an external agent, use W_ext = q·ΔV (or m·ΔV), and always compute ΔV = V_final − V_initial algebraically, keeping signs. Test with the escape-velocity case, where total energy must equal zero.
Applying Kepler's third law with the satellite mass included, or using the planet's radius instead of the orbital radius measured from the centre, when finding the period or orbital speed of a satellite.
The formula T² = 4π²r³/(GM) is often remembered loosely as 'T² ∝ r³' with r ambiguously meaning altitude or radius, and candidates who have seen the two-body version with reduced mass assume the satellite mass matters.
What to do instead
Fix the convention explicitly: r is the distance from the centre of the central body. For a satellite at altitude h above a planet of radius R, r = R + h. Derive T from equating GMm/r² = mω²r once, so it is clear the satellite mass cancels. Use the two-body form only when the problem states both masses are comparable.
Computing the net electric field at a point from several charges by adding magnitudes (E = kq₁/r₁² + kq₂/r₂²) instead of resolving components, or placing the test charge's own sign into the field expression and thereby reversing the field direction.
Coulomb's law is learned first as a scalar magnitude, and vector addition with angles is a separate skill that candidates do not automatically attach to electrostatics problems, especially when the geometry is a triangle or a line of charges.
What to do instead
Adopt a fixed routine: choose axes, compute each |E_i| = k|q_i|/r_i², determine the direction from the sign of q_i (field points away from positive, toward negative), then sum x and y components separately. Never insert the test charge into E; the test charge's sign only matters when computing force F = qE.
Mixing up electric potential (a scalar, V = kq/r) with electric field when asked for the potential at the centre of a square or the midpoint between two charges, and adding potentials with signs incorrectly (e.g. treating a negative charge's potential as positive because 'potential is always positive').
Because potential energy of a positive charge near a positive charge is positive, candidates generalise that potential itself is positive, forgetting that V has the sign of the source charge and can be negative or cancel to zero.
What to do instead
State the rule before calculating: V = Σ k·q_i/r_i with the sign of q_i carried explicitly; V is a scalar so there are no components, only algebraic sum. Practise with a symmetric configuration (two +q and two −q at the corners of a square) where the centre potential is exactly zero, and verify that the field there is not zero.
Applying the formula for the electric field of an infinite sheet (E = σ/2ε₀) or of a spherical shell to a situation where it does not apply, or forgetting that inside a conductor in electrostatic equilibrium the field is zero and the charge resides on the surface.
Sheet and shell results are memorised as 'the formula for a charged surface' without the assumptions (infinite extent, thin sheet, or spherical symmetry) being recorded alongside them. Conductor screening is treated as a qualitative remark rather than a quantitative constraint.
What to do instead
Attach the assumption to every formula in the notes: infinite sheet → uniform field both sides; spherical shell → field zero inside, kQ/r² outside; conductor → E = 0 inside, all charge on the outer surface. When a problem mentions a conductor, write 'E_inside = 0' as the first line of the solution and use it to derive surface charge distributions.
Determining the direction of the magnetic force on a moving charge or current-carrying wire by using the direction of B instead of the cross product v × B, or forgetting that a charge moving parallel to B experiences no force.
The right-hand rule is applied mechanically without the vector product being understood, so candidates point the thumb along B rather than along v, and they do not recognise the sin θ factor as the reason parallel motion gives zero force.
What to do instead
Write F = qv × B explicitly and compute the magnitude as |F| = |q|vB sin θ before applying the right-hand rule to the direction. For negative charges, reverse the result. As a check, verify that the force is always perpendicular to both v and B, so it can never do work on the charge.
Using Faraday's law with the wrong sign or omitting the number of turns, and computing the induced emf as ΔΦ/Δt with Φ = B·A even when the area is not perpendicular to B (missing the cos θ factor).
The minus sign in ε = −N·dΦ/dt is memorised as a formality rather than as Lenz's law, and flux is introduced in the simplest perpendicular case so the general dot product Φ = B·A·cos θ is not internalised.
What to do instead
Write Φ = B·A·cos θ first, then differentiate: ε = −N·dΦ/dt. Decide the sign physically with Lenz's law (the induced current opposes the change in flux) and only then assign a direction. For a rotating coil, differentiate cos θ explicitly to obtain the sinusoidal emf rather than assuming a constant value.
Describing a sound or light wave with the wrong relationship between frequency, wavelength and medium: assuming frequency changes when a wave crosses from one medium to another, or confusing the observer's motion with the source's motion in the Doppler effect.
Frequency is intuitively associated with 'how fast the wave is', so candidates expect it to change with speed. The Doppler formula is learned as a single expression without the sign convention for approaching versus receding, and without distinguishing the moving-source and moving-observer cases.
What to do instead
Anchor the invariant: frequency is set by the source and does not change on refraction; what changes is speed and therefore wavelength (v = fλ with f fixed). For the Doppler effect, write the general form f' = f·(v ± v_o)/(v ∓ v_s) and decide each sign by asking whether the relative motion increases or decreases the observed frequency. Test with the stationary case first.
In relativity and quantum problems, adding velocities with the Galilean rule (u = u' + v) instead of the relativistic velocity-addition formula, or computing the de Broglie wavelength with the classical kinetic energy p = √(2mE) when the particle is relativistic.
Classical mechanics is over-learned and the relativistic corrections look like small perturbations that can be ignored; candidates also do not check whether v is a significant fraction of c or whether the kinetic energy exceeds the rest energy before choosing a formula.
What to do instead
Before any calculation, compute the ratio v/c (or E_k/mc²). If it is not negligible, use u = (u' + v)/(1 + u'v/c²) and the relativistic energy-momentum relation E² = (pc)² + (mc²)². For de Broglie, use λ = h/p with p from the relativistic relation whenever E_k is comparable to mc², and verify the classical limit as a check.