1. Magnetic force provides centripetal force

For an electron moving perpendicular to a uniform magnetic field $B$:

$e v B = \frac{m v^2}{r}$

So,

$mvr = eBr^2 \quad \text{(1)}$

2. Apply Bohr quantization

According to Bohr's quantization of angular momentum,

$mvr = n\frac{h}{2\pi}$

For the first excited state, $n=2$. Hence,

$mvr = 2\frac{h}{2\pi} = \frac{h}{\pi} \quad \text{(2)}$

3. Equate (1) and (2)

From (1) and (2):

$eBr^2 = \frac{h}{\pi}$

Therefore,

$r^2 = \frac{h}{\pi e B}$

$r = \sqrt{\frac{h}{\pi e B}}$

4. Match with options

This corresponds to:

$\boxed{\sqrt{\frac{h}{\pi e B}}}$

So the correct option is A.

5. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They agree.