Q.(a) Using Biot-Savart Law deduce the relation for the magnetic field at a point due to an infinitely long straight conductor carrying current. OR

(a) Integrating the Biot-Savart contribution from every current element of an infinite straight wire gives $B=\mu_0I/(2\pi a)$; (b) the hydrogen spectrum consists of five named series (Lyman, Balmer, Paschen, Brackett, Pfund), each given by the Rydberg-type formula with a different fixed lower level. Both alternatives answered below.

(a) Magnetic field due to an infinitely long straight current-carrying conductor (Biot-Savart law)

Biot-Savart law: the magnetic field due to a current element $Id\vec l$ at a point located by position vector $\vec r$ from the element is

$d\vec B=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l\times\hat r}{r^2}$

Setup: consider an infinitely long straight wire carrying current $I$, and a point $P$ at perpendicular distance $a$ from the wire. Let a current element $Idl$ be located at a distance $l$ along the wire from the foot of the perpendicular, making an angle $\theta$ (between $d\vec l$ and $\vec r$, the line joining the element to $P$).

The magnitude of the field due to this element:

$dB=\dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$

All such $d\vec B$ contributions from every element along the wire point in the same direction at $P$ (circling the wire, by symmetry), so the total field is the scalar integral. Using the substitution $l=a\cot\theta$ (so $r=a/\sin\theta$), and integrating over the entire (infinite) length of the wire, corresponding to $\theta$ from $0$ to $\pi$ (or equivalently the angle from the foot of the perpendicular ranging from $-\pi/2$ to $+\pi/2$):

$B=\dfrac{\mu_0I}{4\pi a}\displaystyle\int_0^\pi\sin\theta\,d\theta=\dfrac{\mu_0I}{4\pi a}\left[-\cos\theta\right]_0^\pi=\dfrac{\mu_0I}{4\pi a}(1-(-1))=\dfrac{\mu_0I}{2\pi a}$

So the field at perpendicular distance $a$ from an infinitely long straight wire:

$B=\dfrac{\mu_0I}{2\pi a}$

directed in circles around the wire (right-hand rule), consistent with the result obtained independently from Ampere's circuital law.

(b) Spectral series of the hydrogen atom

The wavelengths of spectral lines emitted by a hydrogen atom, as the electron transitions from a higher orbit $n_2$ to a lower orbit $n_1$, are given by the Rydberg formula:

$\dfrac1\lambda=R\left(\dfrac1{n_1^2}-\dfrac1{n_2^2}\right),\qquad n_2>n_1$ …