Figure shows a wire ring of radius a that is perpendicular to the general direction of a radially sysmmetric, diverging magnetic field. The magnetic field at the ring is everywhere of the same magnitude B, and its direction at the ring everywhere makes an angle `theta` with a normal to the plane of the ring. thge twisted lead wires have no effect on the problem. Find the magnitude of the force the field exerts on the ring if the ring carries a current i.

Explanation

For the first order reaction `A`, half life `t_(50)(1) = (0.693)/(k_(1))` and for the second order reaction `B`, half life `t_(50) (2) = (1)/(k_(2)a)`
But `t_(50)(1) = t_(50) (2)` (given)
`:. (0.693)/(k_(1)) = (1)/(k_(2)a)`
`:. (k_(1))/(k_(2)a) = 0.693` ...(i)
a. Rate at the start of reaction `A, r_(1) = k_(1)a` ...first order and aret at the start of reaction `B, r_(2) = k_(2)a^(2)` ...second order
Hence `(r_(1))/(r_(2)) = (k_(1)a)/(k_(2)a^(2)) = (k_(1))/(k_(2)a)= 0.693` by Eq. (i)
Rate after one half life when the concentration of the reactant is reduced to `(a//2)`
for reaction `A,r'_(1) = k_(1)((a)/(2))`
and for reaction `B, r'_(2) = k_(2)((a)/(2))^(2) = (k_(2)a^(2))/(4)`
Hence, `(r'_(1))/(r'_(2)) = (2k_(1))/(k_(2)a) = 2 xx 0.693 = 1.386`