NDA (II) 2018 General Ability Test · Question 68 of 146

Correct — A, depends upon the current in the wire. For a long straight conductor the magnetic field at a point outside it, a distance r from the axis, is B = μ₀I ÷ 2πr. Only two quantities appear in that expression — the current I flowing in the wire and the distance r from its axis. Double the current and the field doubles; move twice as far out and the field halves. The question fixes the distance by saying 'at a particular distance from the axis', so the one thing left that the field can depend on is the current, and that is what statement (a) says. The thickness of the wire does not enter, and neither does the temperature of the room.

• (b)depends upon the radius of the wire — For any point outside the conductor the field is exactly what it would be if the whole current ran along the axis as a thin line, so the wire's own radius drops out of the answer. The radius matters only for a point inside the conducting material, where the field grows with distance from the axis instead of falling.
• (c)depends upon the temperature of the surroundings — Temperature changes the resistance of the wire and so, in a real circuit driven by a fixed voltage, can change the current it carries. But that is an effect on the current, not on the relation between current and field, and the field itself is set by the current flowing at that moment.
• (d)None of the above — This can only be correct if all three preceding statements fail, and the first one plainly holds — the field around a current-carrying wire is proportional to the current. A 'none of the above' option is worth a moment's thought precisely because it is defeated by any single true statement above it.

A steady current produces a magnetic field around itself. For a long straight wire the field lines are concentric circles lying in planes perpendicular to the wire, and their direction is given by the right-hand thumb rule — point the thumb of the right hand along the current and the curl of the fingers gives the sense of the field. The magnitude falls off as one over the distance from the wire, which is a slower fall than the inverse-square laws of gravitation and electrostatics because the source here is a line rather than a point.

Every question of this family is answered by writing the formula down first and reading the answer off it. B = μ₀I ÷ 2πr for a straight wire; B = μ₀nI for the inside of a long solenoid, with n the turns per unit length; B = μ₀I ÷ 2R at the centre of a circular loop of radius R. Once the expression is on paper, questions about what the field does or does not depend on answer themselves — the solenoid field, for instance, does not depend on the solenoid's radius either, and NDA has asked that too. The one refinement worth carrying is that 'outside the wire' is doing quiet work in this item; inside the conducting material the radius genuinely does matter.

• For a long straight wire, B = μ₀I ÷ 2πr — proportional to the current, inversely proportional to the distance from the axis.
• The field lines are concentric circles in planes perpendicular to the wire, with their sense given by the right-hand thumb rule.
• Outside the conductor the field is the same as if the entire current were concentrated along the axis.
• Inside a long solenoid the field is uniform and equals μ₀nI, independent of the solenoid's radius.
• μ₀, the permeability of free space, is 4π × 10⁻⁷ T m A⁻¹.

• Assuming a thicker wire must produce a stronger field at the same distance outside it.
• Bringing temperature in because resistance depends on it; the question is about the field-current relation.
• Reaching for 'none of the above' without first checking whether any of the three statements is true.

NDA tests this relation almost every session — sometimes as a dependence question like this one, sometimes as a direction question using the right-hand thumb rule.

• practice — not a real PYQ

  The current in a long straight wire is doubled and the point of observation is moved to twice its original distance from the axis. The magnetic field at the new point is

  • (a)four times the original
  • (b)twice the original
  • (c)the same as the original
  • (d)one quarter of the original

  Answer(c) the same as the original — B is proportional to I and inversely proportional to r, so doubling both leaves the field unchanged.

• practice — not a real PYQ

  The magnetic field inside a long current-carrying solenoid depends on

  • (a)the radius of the solenoid
  • (b)the number of turns per unit length and the current
  • (c)the length of the solenoid only
  • (d)the temperature of the core

  Answer(b) the number of turns per unit length and the current — B = μ₀nI, in which the solenoid's radius does not appear.