An electron in a singly ionized helium atom (He ^+ ) has a...

The potential energy $U$ of an electron in a hydrogen-like ion is given by:
$U = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{Z e^2}{r}$

For a singly ionized helium atom (He$^+$), the atomic number $Z = 2$.
The given potential energy is $U = -6.8$ eV. Converting this to Joules:
$U = -6.8 \times 1.6 \times 10^{-19}$ J

Equating the two expressions for potential energy:
$-6.8 \times 1.6 \times 10^{-19} = -\dfrac{9 \times 10^9 \times 2 \times (1.6 \times 10^{-19})^2}{r}$

Dividing both sides by $-1.6 \times 10^{-19}$:
$6.8 = \dfrac{9 \times 10^9 \times 2 \times 1.6 \times 10^{-19}}{r}$

$6.8 = \dfrac{28.8 \times 10^{-10}}{r}$

$r = \dfrac{28.8 \times 10^{-10}}{6.8}$

$r \approx 4.235 \times 10^{-10}$ m

This is approximately $4.2 \times 10^{-10}$ m.

Answer: $4.2 \times 10^{-10}$ m