A $$12.5$$ eV electron beam is used to bombard gaseous hydrogen at room temperature. The number of spectral lines emitted will be:

$$\text{Energy levels of a hydrogen atom: } E_n = -\frac{13.6}{n^2}\text{ eV}$$

$$E_1 = -13.6\text{ eV}, \quad E_2 = -\frac{13.6}{4} = -3.4\text{ eV}, \quad E_3 = -\frac{13.6}{9} \approx -1.51\text{ eV}, \quad E_4 = -\frac{13.6}{16} \approx -0.85\text{ eV}$$

$$\text{Maximum energy of the excited state from the ground state } (n=1)\text{: } E_{\text{max}} = E_1 + \Delta E = -13.6 + 12.5 = -1.1\text{ eV}$$

$$\text{Since } E_3 = -1.51\text{ eV} > -1.1\text{ eV} \text{ and } E_4 = -0.85\text{ eV} < -1.1\text{ eV, electrons can only be excited up to } n = 3.$$

$$\text{Number of emitted spectral lines: } N = \frac{n(n - 1)}{2} = \frac{3(3 - 1)}{2} = 3$$