Question

If the total energy of an electron in an orbit is positive, then

• A. electron will revolve in a circular orbit
• B. electron will revolve in an elliptical orbit
• C. electron will not follow a closed orbit
• D. electron will fall into the nucleus

Hint:

The sign of the total energy determines the type of orbit in a central force system (like gravity or electrostatics): - \(E<0\): Bound, closed orbit (ellipse or circle). - \(E = 0\): Marginally unbound, parabolic trajectory. - \(E>0\): Unbound, hyperbolic trajectory.

Answer 1

The Correct Option is C

The total energy of an electron orbiting a nucleus is the sum of its kinetic energy and its potential energy.
Total Energy \( E = K.E. + P.E. \).
The kinetic energy (\(K.E. = \frac{1}{2}mv^2\)) is always positive.
The electrostatic potential energy (\(P.E. = -\frac{kZe^2}{r}\)) of an electron in the field of a positive nucleus is always negative.
For an electron to be in a bound state, meaning it is trapped in an orbit around the nucleus, its total energy must be negative.
In a bound state, the potential energy's magnitude is greater than the kinetic energy, resulting in a negative total energy. This corresponds to circular or elliptical orbits.
If the total energy of the electron is zero (\(E=0\)), the electron is just able to escape the nucleus's pull and will follow a parabolic path to infinity.
If the total energy of the electron is positive (\(E>0\)), the electron has more than enough kinetic energy to overcome the potential energy holding it to the nucleus.
In this case, the electron is not in a bound state. It will follow an open, hyperbolic path, approaching the nucleus once and then flying away, never to return.
Therefore, if the total energy is positive, the electron will not follow a closed orbit.