BPSC 69th CCE Prelims General Studies · Question 27 of 144

Correct — B, a vector quantity. Current density is the electric current per unit area of cross-section taken at right angles to the flow, and unlike current itself it is attached to a point inside the conductor and carries a direction — that of the drift of positive charge at that point. Three standard definitions each make the direction explicit. First, the microscopic one: J = nqv_d, where n is the number of free charge carriers per unit volume, q the charge on each and v_d the drift velocity. Number density and charge are scalars, drift velocity is a vector, so J takes its direction straight from v_d. Second, the point form of Ohm's law: J = σE, where σ is the conductivity, the reciprocal of resistivity. The electric field E is a vector and in an isotropic conductor J lies parallel to it, so an equation with a vector on one side must have a vector on the other. Third, the defining relation between the two quantities: the current crossing a surface is the flux of the current density through it, I = ∫ J · dA. A dot product exists only between vectors, and it is precisely that surface integral that turns the vector J into the scalar I — which is why the derived quantity is a vector while the more familiar one is not. The SI unit follows from J = I/A and is the ampere per square metre, A m⁻², with dimensional formula [M⁰ L⁻² T⁰ A¹], so the quantity is very far from dimensionless. The numbers are large even in ordinary wiring: a 5 A current in a 1 mm² copper conductor corresponds to J = 5 × 10⁶ A m⁻², and it is this figure, not the total current, that decides whether a conductor will overheat — which is why cables are specified by cross-section in square millimetres rather than by current alone.

• (a)a scalar quantity — The intended trap, and the answer most candidates give, because electric current genuinely is a scalar and the property is assumed to carry over to its density. It does not. Current is a scalar for a specific and testable reason: currents meeting at a junction add algebraically, exactly as Kirchhoff's first law says, and never by the parallelogram law. Two wires each bringing 3 A into a node deliver 6 A out of it whatever the angle between them, which no vector would do. Current density is different in kind — it is a field quantity defined at every point of the conductor, and current densities genuinely do combine as vectors. The arrow drawn on a circuit diagram marks a sense of flow along a fixed wire, not a direction in space.
• (c)dimensionless — Straightforwardly false, and quickly disproved by writing the definition down: J = I/A gives amperes divided by square metres, so the SI unit is A m⁻² and the dimensional formula is [M⁰ L⁻² T⁰ A¹]. Any quantity that carries a unit is dimensional by definition. The genuinely dimensionless quantities in this part of physics are the ratios — relative permittivity and relative permeability, the power factor cos φ, refractive index, mechanical strain, specific gravity, the coefficient of friction and the fine-structure constant — and the useful mental check is that a dimensionless quantity is always one physical quantity divided by another of the same kind.
• (d)None of the above — Available only if the three preceding descriptions were all wrong, and one of them is exactly right. This option earns its keep on stems where every listed choice is defective, which does happen on this paper, but here it costs a mark to a candidate who is unsure between scalar and vector and picks the escape route instead of applying the definition J = nqv_d.

Electric current and current density describe the same physical flow at two different levels of detail, and telling them apart is the point of this question. Current I is a bulk, whole-circuit quantity: the net charge crossing a chosen section of the conductor per unit time, I = dq/dt, measured in amperes and completely specified by a single number. Current density J is the microscopic counterpart: a vector field defined at every point inside the conducting material, giving both how much charge is flowing there per unit area and in which direction. The link between them is a surface integral, I = ∫ J · dA, and integrating a vector field over a surface produces a scalar — which is exactly why the point quantity is a vector and the bulk quantity is not. The same pairing recurs throughout physics: velocity is a vector while speed is a scalar, and force is a vector while pressure, force per unit area, is a scalar because it acts equally in all directions in a fluid. The lesson is that dividing a vector by an area does not automatically preserve or destroy its vector character; what matters is how the quantity is defined and, decisively, how two of them add together.

The safe route to the answer is not to ask whether current density 'has a direction' — that intuition misleads on this exact topic — but to write the defining relation and read the answer off it. J = nqv_d puts a drift velocity on the right-hand side, and a velocity is a vector, so J is one; J = σE puts an electric field there and gives the same verdict. The discriminating fact, and the one BPSC is testing, is the deliberate asymmetry between the two neighbouring quantities: current is a scalar, current density is a vector, and a candidate who knows only the first half of that sentence will answer (a). The proper test of vector character is the addition rule, not the presence of a direction. Electric current fails that test — Kirchhoff's junction rule adds currents algebraically regardless of the geometry of the wires — while current densities at a point add head to tail like any other vectors. Option (c) can be eliminated even faster than either of these, purely on units, since anything measured in A m⁻² has dimensions by definition. Two related traps sit nearby and are worth clearing at the same time: electric current has a sense but is scalar, whereas pressure and work also look directional in ordinary language and are scalar too; and the presence of an arrow in a diagram is a notation, not a proof of vector status.

• Current density is defined as J = I/A for uniform flow through a section normal to it, and in general by I = ∫ J · dA. Its SI unit is the ampere per square metre (A m⁻²) and its dimensional formula is [M⁰ L⁻² T⁰ A¹].
• The microscopic definition is J = nqv_d. In copper the free-electron density n is about 8.5 × 10²⁸ per cubic metre, which makes the drift velocity for an ordinary household current of the order of a millimetre per second — even though the electrical signal itself travels down the wire at close to the speed of light as an electromagnetic disturbance.
• The point form of Ohm's law is J = σE, with conductivity σ = 1/ρ. Copper's resistivity is about 1.68 × 10⁻⁸ ohm-metre at 20 °C and silver's is marginally lower, which is why copper is the standard conductor and silver the laboratory best.
• Electric current is a scalar despite having a sense of flow, because currents at a junction obey Kirchhoff's first law and add algebraically, ΣI = 0, rather than by the parallelogram law. Charge, potential, potential difference, resistance, capacitance, power and magnetic flux are likewise scalars.
• The vectors in this chapter are electric field, drift velocity, current density, magnetic field and magnetic dipole moment. A convenient sanity check on any 'dimensionless' option is that dimensionless quantities are ratios of like quantities — relative permittivity, power factor, refractive index, strain, the coefficient of friction.

• Transferring the scalar character of electric current to current density — current is scalar, its density is a vector, and the question exists only to test that distinction
• Treating 'has a direction' as proof of vector status; the real test is whether two of them add by the parallelogram law, which currents at a junction do not
• Marking a quantity dimensionless without writing down its unit — A m⁻² settles option (c) in a single step

BPSC asks this as a two-word stem with a three-way classification — scalar, vector or dimensionless — so it is pure definition recall, decided in seconds by anyone who can write J = nqv_d, and it is the same instinct the Commission showed on the 71st CCE in 2025 when it tested an electrical quantity through its defining relation rather than through any phenomenon. UPSC has never named current density. It asks the classification in its general form, giving four quantities and asking which is a vector (1997), or it hides the same conduction relations inside a numerical item on resistivity, length and cross-sectional area (2001) — so the BPSC candidate needs the definition and the UPSC candidate needs to use it.

• practice — not a real PYQ

  The SI unit of current density is

  • (a)ampere
  • (b)ampere per square metre
  • (c)ampere-metre
  • (d)coulomb per second

  Answer(b) ampere per square metre — current density is current per unit area of cross-section, J = I/A, so its unit is A m⁻². The ampere and the coulomb per second are the same thing and are the unit of current itself, while the ampere-metre is not a unit of any standard quantity in this chapter.

• practice — not a real PYQ

  Electric current is treated as a scalar quantity mainly because

  • (a)it has no magnitude of its own
  • (b)currents meeting at a junction add algebraically and not by the parallelogram law
  • (c)it can never be negative
  • (d)it flows only through solid conductors

  Answer(b) currents meeting at a junction add algebraically and not by the parallelogram law — that is precisely Kirchhoff's first law, ΣI = 0, and failing the vector addition test is what makes current a scalar despite having a definite sense of flow. Current certainly has magnitude, it can be negative relative to a chosen direction, and it flows in electrolytes and plasmas as well as solids.