Question

A proton is placed midway between points $A$ and $B$. The potential at point $A$ is $-20\mathrm{V}$, and the potential at point $B$ $+20\mathrm{V}$. The potential at the midpoint is $0\mathrm{V}$. The proton will a) remain at rest. b) move toward point $B$ with constant velocity. c) accelerate toward point $A$. d) accelerate toward point $B$. e) move toward point $A$ with constant velocity.

Short answer

Answer: c) accelerate toward point A

Step-by-step solution

Define the electric potential values

Point A has an electric potential of $-20V$, while point B has an electric potential of $+20V$. The proton is placed midway between these points, where the electric potential is $0V$.

Determine the electric field direction

The presence of electric potential between points A and B indicates that there is an electric field between them. The electric field lines move from high potential to low potential. In this case, it means that the electric field points from point B towards point A.

Identify the electric force acting on the proton

A proton is a positively charged particle. In an electric field, its motion will be determined by the electric force acting on it. Since the electric field goes from point B to point A (from high to low potential), the electric force on the proton will follow the same direction. The proton will, therefore, be attracted towards point A.

Check the given options according to the determined behavior of the proton

Based on the electric force, we can rule out options a), b), and e), as the proton will not remain at rest, and will not move towards point B in any way. The remaining options are c) accelerate toward point A and d) accelerate toward point B.

Determine whether the proton accelerates or moves with a constant velocity

The electric field between points A and B remains constant throughout (since the potential difference of $\pm 20V$ doesn't change). This constant electric field means that a constant electric force will act on the proton. According to Newton's second law, a constant force acting on a particle causes it to accelerate in the direction of the force. Therefore, the proton will accelerate.

Select the correct option based on the proton's behavior

Based on our analysis, the proton will accelerate towards point A due to the electric force acting on it. The correct answer is: c) accelerate toward point A.

Key concepts

Electric Potential

Electric potential is a fundamental concept in electrostatics. It is the amount of electric potential energy per unit charge at a specific point in an electric field. Think of it like a landscape of energies that charged particles move through.

• Points with higher electric potential have more potential energy for positive charges.
• Negative charges experience the opposite direction of energy flow compared to positive charges.

In the exercise, point A has an electric potential of $-20\text{V}$ and point B has $+20\text{V}$. The midpoint between them, where the proton is placed, has a potential of $0\text{V}$. This means that the energy landscape at the midpoint is neutral, but there is a slope of electric potential from B to A, causing movement.

Electric Field

An electric field is a region around a charged particle where other charges feel a force. It is depicted as a series of lines pointing from higher potential towards lower potential.

• This direction is critical because it tells us how positive charges, like protons, will behave: they move in the direction of the electric field.
• Conversely, negative charges move against the electric field.

In the given exercise, the electric field points from point B (where potential is $+20\text{V}$) to point A (where potential is $-20\text{V}$). Because electric fields direct from high to low potential, the field direction is crucial in determining the movement of the proton.

Electric Force

Electric force arises from the interaction of electric fields and charges. The force experienced by a charge in an electric field is proportional to both the charge and the field's strength.

• This force is given by the formula $F=qE$, where $q$ is the charge and $E$ is the electric field.
• For a proton, which carries a positive charge, this force will act in the same direction as the electric field.

In the context of the exercise, the electric force on the proton is directed from point B to point A, due to the electric field between these points.

Proton Behavior

Protons are positively charged particles, and their behavior in an electric field can be directly influenced by the electric force present. A proton will move in the direction of electric field lines if free to do so.

• Protons move from points of higher potential to lower potential due to the attractive force exerted by the electric field.

In the described situation, as the proton is at the midpoint between points A and B, and given the directions of the potential and field, it will naturally move towards point A, where potential is lower.

Newton's Second Law

Newton's Second Law states that an object will accelerate if a net force is applied to it. It relates force, mass, and acceleration with the equation $F=ma$, where $F$ is the force applied, $m$ is the mass of the object, and $a$ is the acceleration.

• In an electric field, a constant electric force results in continuous acceleration.
• For the proton in the exercise, there is a constant force exerted by the electric field.

Thus, according to Newton's Second Law, the proton will accelerate towards point A where the force directs. This consistent application of force ensures the proton's acceleration rather than constant velocity, pointing to the fact that the proton won’t stop until acted upon by another force.