Electrons in the Bohr Model (7.2.3) | AP Physics 2: Algebra Notes

AP Syllabus focus: 'The Bohr model treats electrons as moving in circular orbits determined by electron charge, mass, and electric force with the nucleus.'

The Bohr model connects electric attraction and circular motion to describe electron behavior in a simple atom. Understanding that force balance is the core idea of this topic.

The basic picture

In the Bohr model, an atom is represented by a small, positively charged nucleus and an electron moving around it in a circular path. The model uses familiar ideas from mechanics and electrostatics rather than a more detailed atomic description. It treats the electron as a moving particle and the nucleus as the source of the inward electric attraction.

For AP Physics 2, the key point is that the electron's motion is determined by three linked features:

its negative charge
its mass
the electric attraction exerted by the nucleus

Because opposite charges attract, the nucleus pulls the electron inward. That inward pull is always directed toward the center of the orbit, so the force does not push the electron forward around the circle. Instead, it continually changes the electron's direction.

Circular motion requires an inward force

For any object to move in a circle, it must experience a centripetal force.

A particle moving in a circular path has an instantaneous velocity vector tangent to the circle, while the net centripetal force points radially inward. This geometry is the reason a centripetal force changes the direction of motion without needing to point along the direction of travel. Source

Centripetal force: The net inward force required to keep an object moving in a circular path.

In the Bohr model, the centripetal force is supplied by the electric force between the nucleus and electron. This is an important modeling idea: there are not two separate inward forces. The electric force is the real interaction, and it also plays the role of the required centripetal force.

For a hydrogen atom, the force balance can be written by setting the electric attraction equal to the force needed for circular motion.

$k\dfrac{e^2}{r^2} = \dfrac{mv^2}{r}$

$k$ = Coulomb constant, $N\cdot m^2/C^2$

$e$ = magnitude of the electron charge, C

$r$ = orbit radius, m

$m$ = electron mass, kg

$v$ = electron speed, $m/s$

This relation shows how orbit size and electron speed are connected. The left side depends on the electric interaction, while the right side depends on the mass of the electron and the motion needed for a circular path.

How charge, mass, and motion are linked

The equation reveals several useful qualitative ideas.

Effect of electric charge

The electric force depends on charge. In the Bohr model, the electron has a fixed negative charge magnitude $e$ , and the proton has a fixed positive charge magnitude $e$ . Because these charges are constant, the attraction becomes stronger mainly when the separation distance $r$ becomes smaller.

A smaller radius means a much larger electric force because the force varies as $1/r^2$ . This stronger inward pull can support tighter circular motion. If the electron is farther from the nucleus, the electric attraction is weaker, so the circular motion must adjust accordingly.

Effect of electron mass

The electron's mass matters because mass measures inertia. An object with greater mass is harder to bend into a circular path. For the same orbit radius and speed, a larger mass would require a larger centripetal force.

That means the available electric attraction must match the electron's mass. The Bohr model therefore treats the electron not just as a charged particle, but as a particle whose resistance to changes in motion affects the orbit.

Effect of speed and radius

The required centripetal force depends on $v^2/r$ . This means speed has a strong effect on circular motion:

if the electron's speed increases, the needed inward force increases
if the orbit radius increases, the needed inward force decreases
if the radius decreases, the needed inward force increases

So the orbit is not determined by charge alone. It is determined by the balance between the inward electric attraction and the motion of a massive particle moving in a circle.

Interpreting the electron's path

In this model, the electron is pictured as having a definite circular orbit at each moment.

This textbook figure illustrates that the velocity vector is tangent to the circular path while the centripetal force vector points toward the center. Interpreting the electron’s orbit this way makes it clear why an inward electric force can continuously redirect the motion while the electron’s instantaneous velocity remains tangential. Source

Its velocity is tangent to the circle, while the electric force points inward toward the nucleus. Since the force is perpendicular to the motion, it continuously redirects the electron.

This is why the electron does not simply travel in a straight line. Without the nucleus pulling inward, straight-line motion would result. With the inward force present, the path curves into a circle.

A common mistake is to think that the electron must be moving toward the nucleus because the force points toward the nucleus. In circular motion, an inward force changes direction, not necessarily speed. The electron can keep moving around the nucleus while being pulled inward at every point.

What you should be able to do

For this subsubtopic, you should be comfortable with the following tasks:

identifying the electric force as the force responsible for circular motion
recognizing that the force is attractive because the nucleus and electron have opposite charges
using $k\dfrac{e^2}{r^2} = \dfrac{mv^2}{r}$ for a hydrogen atom
reasoning qualitatively about how changes in charge, mass, speed, or radius affect the orbit

When analyzing the Bohr model, focus on force balance: the electron moves in a circular orbit because the electric attraction with the nucleus provides exactly the centripetal force required for that motion.

FAQ

Hydrogen has only one electron, so the model only has to describe the attraction between that electron and the nucleus.

With more than one electron, electron-electron repulsion becomes important, and the simple circular-orbit picture becomes much less accurate.

The nucleus is much more massive than the electron, so it accelerates far less during the interaction.

A more complete treatment has both particles orbiting a common center of mass, but for AP-level Bohr-model work, treating the nucleus as fixed is usually an excellent approximation.

It is a useful analogy, but it should not be taken too literally.

The model borrows the idea of circular motion from planetary motion, yet electrons are not tiny classical planets. The Bohr model is mainly a simplified way to connect electric force, mass, and circular motion.

Yes, the model works best for one-electron systems, not just hydrogen.

For ions such as $He^+$ or $Li^{2+}$, there is still only one electron orbiting the nucleus, so the same basic force ideas apply. The stronger positive nuclear charge changes the strength of the electric attraction.

The name comes from the fact that the force is modeled using Coulomb's law, which depends on electric charge and separation distance.

In this simplified treatment, the dominant interaction is the electric attraction between opposite charges. More advanced effects are ignored so the model can focus on the main cause of the circular motion.

Practice Questions

In the Bohr model of a hydrogen atom, what force keeps the electron in its circular orbit, and in what direction does that force act?

1 mark for identifying the force as the electric or electrostatic attraction between the proton and electron
1 mark for stating that the force acts radially inward, toward the nucleus

A hydrogen atom is modeled as an electron of mass $m$ moving with speed $v$ in a circular orbit of radius $r$ around a proton.

(a) Write an expression for the electric force between the proton and electron.

(b) Write an expression for the centripetal force needed for the electron's circular motion.

(d) If the electron speed doubles while $r$ stays the same, describe how the required centripetal force changes.

(e) If the orbit radius is halved, describe how the electric force changes.

(a) 1 mark for $F = k\dfrac{e^2}{r^2}$
(b) 1 mark for $F = \dfrac{mv^2}{r}$
(c) 1 mark for setting them equal: $k\dfrac{e^2}{r^2} = \dfrac{mv^2}{r}$
(d) 1 mark for stating that the required centripetal force becomes 4 times larger because it depends on $v^2$
(e) 1 mark for stating that the electric force becomes 4 times larger because it depends on $1/r^2$

Try All Topic Practice Questions

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.