Q4. Assertion: a long straight lead joined to a circular arc, and pointing straight at the arc’s centre, adds nothing to the field at that centre. Reason: for every element of such a lead, dl and r are parallel. Are both true, and does the reason explain the assertion?
Both are true, and the reason explains the assertion: with dl parallel to r, dl × r = 0, so each element of the lead gives no field at the centre (NCERT Example 4.5, page 117).
Q5. Two concentric coplanar loops have radii 10 cm and 20 cm and carry 2.0 A and 4.0 A. Find the field at their common centre when the currents go (a) the same way, (b) opposite ways.
Each gives μI/(2R) = 2π × 10−7 × 20 ≈ 1.26 × 10−5 T, since I/R = 20 A/m for both. (a) Same way: they add, about 2.5 × 10−5 T. (b) Opposite ways: they cancel, giving zero (NCERT Eq. (4.12), page 116).
Q6. A wire carrying a current I is bent into a quarter circle of radius R. Find the field at the centre of the arc.
A quarter circle is a quarter of a loop, φ = π/2, so B = μIφ/(4πR) = μI/(8R) (reasoned from NCERT Example 4.5, page 117).
Q7. How far along the axis from the centre of a loop does the field fall to one-eighth of its value at the centre?
B/B0 = R3/(x2 + R2)3/2 = 1/8 when (x2 + R2)3/2 = 8R3, that is x2 + R2 = 4R2, so x = √3 R ≈ 1.73R (reasoned from NCERT Eqs. (4.11)–(4.12)).
Q8. A current enters a uniform circular ring at one point and leaves at another, through straight leads that point at the centre. Find the field at the centre.
Zero. The two arcs are in parallel, and a uniform wire’s resistance is proportional to its length, so Il = Il. An arc of length l gives μIl/(4πR) at the centre, so the two fields are equal in size. The arcs carry current round the centre in opposite senses, so the fields cancel; the leads add nothing (reasoned from NCERT Example 4.5, page 117).