The correct relation(s) among total energy, kinetic energy and potent…

The correct answer is Option (1) → (A), (C) and (D) only

In Bohr's model the electrostatic attraction of the nucleus provides the centripetal force on the electron:

    \( \displaystyle \frac{m v^{2}}{r} = \frac{k e^{2}}{r^{2}} \;\Rightarrow\; \text{K.E.} = \frac{1}{2} m v^{2} = \frac{k e^{2}}{2r} \)

    \( \displaystyle \text{P.E.} = -\frac{k e^{2}}{r} \)

    \( \displaystyle \text{T.E.} = \text{K.E.} + \text{P.E.} = -\frac{k e^{2}}{2r} \)

So the three are in the ratio \( \text{K.E.} : \text{P.E.} : \text{T.E.} = 1 : -2 : -1 \).

(A) T.E. = −K.E. — Correct. Both have the size \( ke^{2}/2r \), and the total energy is negative.

(B) K.E. = P.E. — Incorrect. The kinetic energy is positive and the potential energy negative, and the potential energy is also twice as large in size.

(C) K.E. = \( -\displaystyle\frac{\text{P.E.}}{2} \) — Correct. \( -\text{P.E.}/2 = ke^{2}/2r \), which is the kinetic energy.

(D) T.E. = \( \displaystyle\frac{\text{P.E.}}{2} \) — Correct. \( \text{P.E.}/2 = -ke^{2}/2r \), which is the total energy.

(E) T.E. = P.E. — Incorrect. The total energy is only half the potential energy, because the positive kinetic energy is added to it.

NCERT (2026-27):
Electron Orbits – “… The kinetic energy (K) and electrostatic potential energy (U) of the electron in hydrogen atom are

    \( \displaystyle K = \frac{1}{2}mv^2 = \frac{e^2}{8\pi\varepsilon_0 r} \quad \text{and} \quad U = -\frac{e^2}{4\pi\varepsilon_0 r} \)

(The negative sign in U signifies that the electrostatic force is in the –r direction.) Thus the total energy E of the electron in a hydrogen atom is

    \( \displaystyle E = K + U = \frac{e^2}{8\pi\varepsilon_0 r} - \frac{e^2}{4\pi\varepsilon_0 r} = -\frac{e^2}{8\pi\varepsilon_0 r} \qquad (12.4) \)”