1. Potential energy of the electron

For an electron revolving around a nucleus of charge $+Ze$, the electrostatic potential energy is

$U = -\frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}$

2. Use centripetal force = electrostatic force

The electrostatic attraction provides the centripetal force:

$\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2}$

Multiplying by $r$,

$mv^2 = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}$

3. Kinetic energy of the electron

$K = \frac{1}{2}mv^2 = \frac{1}{2}\left(\frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}\right)$

So,

$K = \frac{1}{8\pi\varepsilon_0}\frac{Ze^2}{r}$

Comparing with

$U = -\frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}$

we get

$K = -\frac{U}{2}$

4. Total energy

$E = K + U = -\frac{U}{2} + U = \frac{U}{2}$

Hence,

$2E = U$

5. Check options

• A: $2E = 3U$ ❌
• B: $E = U$ ❌
• C: $2E = U$ ✅
• D: $E = 2U$ ❌

Therefore, the correct option is C.