Electric Potential and Work Done on a Charge

Hello! In our last lesson, we established the concepts of the electric field ($\vec{E}$) and electric potential ($V$), and we learned how to calculate them for systems of point charges. We saw that potential is a scalar property of space, making it much simpler to calculate than the vector-based electric field.

Today, we will connect this abstract concept of potential to the concrete, physical quantities of work and energy. This is the heart of why potential is so useful: it's a direct measure of the energy landscape that a charge experiences. By the end of this lesson, you will be able to relate the electric potential difference between two points to the work done in moving a charge between them, fulfilling your goal of understanding electricity in terms of "work done". This forms the theoretical basis for everything from accelerating particles to powering a simple light bulb.

1. From Electric Force to Potential Energy

Let's start with a familiar concept from mechanics: work. Work is done when a force causes displacement. For a charge $q$ in an electric field $\vec{E}$, it experiences an electric force ${\vec{F}}_E=q\vec{E}$. If this charge moves from point A to point B, the work done by the electric field is the line integral of this force along the path of motion:

$W_{A\to B}={\int }_A^B{\vec{F}}_E\cdot d\vec{l}$

Your mathematical background will recognize this as a standard work calculation. A key property of the electrostatic force is that it is a conservative force. This means the work done by the field in moving a charge between two points is independent of the path taken. This is a powerful feature, analogous to the gravitational force.

Because the force is conservative, we can define a quantity called electric potential energy, denoted by $U$. The change in potential energy, $\Delta U=U_B-U_A$, is defined as the negative of the work done by the conservative force.

$\Delta U=-W_{A\to B}=-{\int }_A^B{\vec{F}}_E\cdot d\vec{l}$

This negative sign is fundamental. Think of gravity: when a ball falls, the gravitational field does positive work, but the ball's gravitational potential energy decreases. The system naturally moves to a state of lower potential energy, and the field does work in the process.

The following reading formalizes this relationship.

The image below provides a concise mathematical summary of this derivation.

2. Relating Potential Energy to Electric Potential (Voltage)

In the previous lesson, we defined electric potential $V$ as electric potential energy per unit charge:
$V=\frac{U}{q}$
From this, the change in potential energy $\Delta U$ when a charge $q$ moves between two points with a potential difference $\Delta V=V_{\text{final}}-V_{\text{initial}}$ is simply:
$\Delta U=q\Delta V$
This potential difference, $\Delta V$, is what we commonly call voltage.

3. The Work-Potential Relationship

We can now combine the two key equations from the previous sections:

1. $W_{\text{field}}=-\Delta U$
2. $\Delta U=q\Delta V$

This gives us the central relationship for this lesson:
$W_{\text{field}}=-q\Delta V$
This equation states that the work done by the electric field on a charge $q$ as it moves through a potential difference $\Delta V$ is $-q\Delta V$.

It's also crucial to distinguish this from the work done by an external agent to move the charge. If we move a charge slowly (i.e., with no change in kinetic energy) against the electric field, we must apply a force that opposes the electric force. The work we do, $W_{\text{ext}}$, is therefore the negative of the work done by the field.

$W_{\text{ext}}=-W_{\text{field}}=-(-\Delta U)=+\Delta U=+q\Delta V$

• Work done by the field ($W_{\text{field}}$): Positive when a charge moves naturally (e.g., a positive charge moving to a lower potential). The field is "helping" the motion. Potential energy decreases.
• Work done by an external agent ($W_{\text{ext}}$): Positive when a charge is forced to move against its natural inclination (e.g., pushing a positive charge to a higher potential). You are "fighting" the field. Potential energy increases.

The following video walks through these concepts, derives the work formula, and provides several examples.

4. Conservation of Energy

The final piece of the puzzle is to connect work to kinetic energy ($K$). The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy:
$W_{\text{net}}=\Delta K$
If the only force acting on our charge is the electric force, then $W_{\text{net}}=W_{\text{field}}$. We can then write:
$\Delta K=W_{\text{field}}=-\Delta U$
Rearranging this gives the law of conservation of energy for a charge in an electric field:
$\Delta K+\Delta U=0$
This means any change in kinetic energy is perfectly balanced by an opposite change in potential energy. A charge that is free to move in an electric field will accelerate, converting potential energy into kinetic energy, just as a falling object converts gravitational potential energy into kinetic energy.

This allows us to calculate the final speed of a particle if we know the potential difference it has been accelerated through.

Conclusion

Today we have forged the crucial link between electric potential and the mechanical concepts of work and energy. This connection elevates potential from an abstract mathematical tool to a powerful physical concept that describes the energy landscape for charged particles.

Key Takeaways:

• The work done by the electric field when a charge $q$ moves through a potential difference $\Delta V$ is $W_{\text{field}}=-q\Delta V$.
• The work done by an external agent to move the charge against the field is $W_{\text{ext}}=+q\Delta V$.
• For a charge moving freely in an electric field, energy is conserved: $\Delta K+\Delta U=0$. A decrease in potential energy results in an equal increase in kinetic energy.
• Positive charges naturally accelerate from high potential to low potential.
• Negative charges naturally accelerate from low potential to high potential.

Preview of the Next Lesson:
We have now established the foundations of electrostatics. In the next lesson, we will shift our focus to magnetism. We will start by investigating the source of magnetic fields: moving electric charges, or currents. We will introduce Ampère's Law, which is for magnetism what Coulomb's Law is for electricity, and use it to describe the magnetic field around a current-carrying wire.