Current density is

Question

Current density is

UnlockIAS · Accepted Answer

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.

Why the other options are wrong — (a) 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) 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) 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.

Concept: 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.

State PCS 2023 Q27 — Science & Technology PYQ with Answer

State PCS 2023 Q27 — Science & Technology PYQ with Answer