Q4. Assertion: a short current element produces no magnetic field at points on the line along which it lies. Reason: the field of the element is proportional to sin θ, where θ is the angle between dl and r. Are both true, and does the reason explain the assertion?
Both are true, and the reason explains the assertion: on that line θ = 0 or 180°, so sin θ = 0 (NCERT page 114).
Q5. The element of NCERT Example 4.4 (Δx = 1 cm along +x at the origin, 10 A) is used again. Find the field at the point (0.5 m, 0.5 m, 0).
r = √(0.5 + 0.5) ≈ 0.707 m, so r = 0.50 m, and θ = 45°. dB = 10−7 × 10 × 0.01 × sin 45°/0.50 ≈ 1.4 × 10−8 T. Its direction is that of î × (î + ĵ) = k̂, the +z direction (NCERT Eqs. [4.7(a)]–[4.7(b)], pages 113–114).
Q6. Which of these is true of the Biot–Savart law but not of Coulomb’s law: (a) the field falls off as 1/r, (b) the field obeys superposition, (c) the source is a vector, (d) the field depends on the distance only?
(c). Both laws are inverse square and both obey superposition; the magnetic field has a vector source, I dl, and depends also on the angle θ (NCERT page 114).
Q7. Use the Biot–Savart law to find the SI unit and the dimensions of μ.
μ = 4π r dB/(I dl sin θ), so its unit is T m/(A m) = T m A−1. With 1 T = 1 N A−1 m−1, this is N A−2, with dimensions [M L T−2 A−2] (NCERT table, page 134).