Welcome to Electric Potential: The Physics of Stored Energy!
Hello! This chapter is where we move beyond thinking about electric forces (vectors) and start thinking about energy (scalars). This makes calculations much simpler and gives us a deeper understanding of how charges behave in fields.
Don't worry if 'potential' sounds abstract—it's just a way of measuring the potential energy stored per unit charge, similar to how altitude determines gravitational potential. By the end of these notes, you'll be able to define, calculate, and use electric potential in the context of point charges and fields.
1. Revisiting the Basics: Potential Difference
Before diving into electric potential, let's quickly remember the concept we learned in D.C. circuits (Topic 9): Potential Difference (p.d.).
Potential difference (or voltage, $V$) between two points is the energy transferred when unit charge passes between those two points.
This definition is still useful, but in field theory, we need a standard reference point for potential itself, not just the difference between two points.
Key Formula Reminder (AS Level Prerequisite)
The work done (\(W\)) to move a charge (\(Q\)) across a potential difference (\(V\)) is:
$$W = QV$$
Therefore, potential difference is defined as:
$$V = \frac{W}{Q}$$
Units: The unit for potential difference (and electric potential) is the volt (V), which is $1 \text{ Joule per Coulomb} (J \ C^{-1})$.
Quick Takeaway: Potential difference is the energy change per unit charge moving between two points.
2. Defining Electric Potential (\(V\))
In field theory, potential is measured relative to a fixed zero point, just like gravitational potential. This reference point is called infinity.
Definition: Electric Potential at a Point
The electric potential ($V$) at a point in an electric field is defined as:
The work done per unit positive charge in bringing a small test charge from infinity to that point.
- Work Done: Energy is needed to push the test charge against the electric field.
- Per Unit Positive Charge: We divide the work done by the charge magnitude to get a property of the field itself (potential, V), not dependent on the specific test charge.
- From Infinity: Infinity is the location where the force from the charge causing the field is zero. By convention, potential at infinity is zero (\(V_\infty = 0\)).
Analogy: The Gravitational Hill
Think of a gravitational field:
- If you lift a rock from the ground (our "zero potential" reference), you do work, and the rock gains Gravitational Potential Energy ($E_p$).
- If we replace the ground with "infinity" (a place so far away the Earth's gravity has no effect), lifting the rock from there to a point near Earth requires work, defining the gravitational potential at that point.
For electric fields, potential is always a scalar quantity, meaning it has magnitude but no direction. This is a huge advantage over dealing with electric field strength ($E$) which is a vector.
Quick Review Box: Potential vs. Potential Difference
Potential (V): Work done moving charge from infinity to a point.
Potential Difference (\(\Delta V\)): Work done moving charge between two points (A and B).
3. Electric Potential due to a Point Charge
The Formula for a Point Charge
For an isolated point charge $Q$ in free space (or vacuum), the electric potential $V$ at a distance $r$ from the charge is given by:
$$V = \frac{Q}{4\pi\epsilon_0 r}$$Where:
- \(Q\) is the magnitude and sign of the source charge (in Coulombs, C).
- \(r\) is the distance from the charge (in metres, m).
- \(\epsilon_0\) is the permittivity of free space. The term \(1/(4\pi\epsilon_0)\) is a constant (often denoted as \(k\)) supplied in your data sheet.
Understanding the Sign of V
The sign of $V$ is determined by the sign of the source charge $Q$.
- Positive Charge (+Q): \(V\) is always positive. Work must be done by an external agent to bring a positive test charge from infinity (repulsion).
- Negative Charge (-Q): \(V\) is always negative. The field does the work (attraction), so the external agent releases energy.
Think of it like money: A positive potential is like a large debt you have to pay (work done) to escape infinity; a negative potential is like finding money (field does work) as you approach.
Dependence on Distance (\(V \propto 1/r\))
The potential $V$ decreases as the distance $r$ increases. Since $V$ is proportional to $1/r$:
- The graph of $V$ against $r$ is a curve (not linear).
- $V$ approaches zero as $r$ approaches infinity.
- Compare this to electric field strength $E$, which varies as \(1/r^2\). Potential falls off more slowly than the field strength.
Did you know?
Since potential is a scalar, if you have multiple charges, you simply add up the potential created by each charge (with their appropriate signs) to find the total potential at that point. No complicated vector addition needed!
Key Takeaway: Electric potential due to a point charge is calculated using \(V = \frac{Q}{4\pi\epsilon_0 r}\). It's a scalar value, and its sign depends on the charge creating the field.
4. Relationship between Electric Field and Potential Gradient
Electric field strength ($E$, a vector) and electric potential ($V$, a scalar) are fundamentally linked.
The Concept of Potential Gradient
The potential gradient is simply the rate at which the electric potential changes with distance in a particular direction.
In a uniform field (like between parallel plates, where $E$ is constant), the field strength $E$ is related to the potential difference $\Delta V$ over a distance $\Delta d$:
$$E = \frac{\Delta V}{\Delta d}$$(Recall this formula from Uniform Electric Fields, 18.2)
The General Relationship (\(E\) and Potential Gradient)
The syllabus requires us to understand that the electric field at a point is equal to the negative of the potential gradient at that point.
$$E = - \frac{\Delta V}{\Delta d}$$Let's break down this powerful statement:
- Magnitude (\(E = \text{Gradient}\)): The magnitude of the electric field strength is equal to the gradient (steepness) of the potential vs. distance graph. Where potential changes quickly, the field is strong.
- Direction (The Negative Sign): The negative sign tells us that the electric field ($E$) always points in the direction of decreasing potential.
Analogy: Contour Maps
Imagine a topographical map showing height (which is like gravitational potential).
- The steepest downhill path is equivalent to the direction of the gravitational field strength ($g$).
- In an electric field, $E$ is like the steepest downhill path on the potential map. Positive charges naturally 'fall' towards lower potential.
Equipotential Surfaces
An equipotential surface (or line in 2D diagrams) is a surface on which every point has the same electric potential.
- Since $V$ is constant on this surface, no work is done moving a charge along it (because $\Delta V = 0$).
- Because $E = -\Delta V / \Delta d$, the electric field lines must be perpendicular to the equipotential surfaces at all points. (If $E$ had a component parallel to the surface, work would be done, and it wouldn't be equipotential!)
Common Mistake Alert!
Do not confuse $E = \frac{V}{d}$ (for uniform fields) with the general relationship $E = -\frac{\Delta V}{\Delta d}$ (which relates the field to the local gradient of potential). In non-uniform fields (like that around a point charge), $E$ is constantly changing.
Key Takeaway: The electric field strength is the negative of the potential gradient. Field lines always cross equipotential lines at $90^\circ$ and point towards lower potential.
5. Electric Potential Energy (\(E_p\))
We defined potential $V$ as the work done per unit charge. If we multiply $V$ by a specific charge $q$, we get the total energy associated with placing that charge at that point in the field. This is the Electric Potential Energy ($E_p$).
$E_p$ for a Single Charge in a Field
If a charge $q$ is placed at a point where the potential is $V$:
$$E_p = qV$$Since $V$ is measured in J/C, and $q$ in C, $E_p$ is measured in Joules (J).
$E_p$ of Two Point Charges
If we have two point charges, $Q$ (source) and $q$ (test), separated by a distance $r$, the potential energy stored in this pair (or system) is found by substituting the potential formula ($V = Q/(4\pi\epsilon_0 r)$) into the energy definition ($E_p = qV$).
This gives the formula for the Electric Potential Energy of two point charges:
$$E_p = \frac{Qq}{4\pi\epsilon_0 r}$$Interpreting the Sign of Potential Energy
Understanding the sign of $E_p$ is vital:
- Charges have the same sign (QQ > 0): \(E_p\) is positive. This represents a repulsive force. Energy must be put into the system (work done) to bring them together from infinity.
- Charges have opposite signs (QQ < 0): \(E_p\) is negative. This represents an attractive force. The system has a 'lower' energy state than when the charges were at infinity. Energy is released as they come together.
Tip: A system always wants to reach the lowest energy state possible (like objects falling). A negative potential energy means the system is stable and attractive.
Work and Energy Conservation
When a charge moves from a point of potential $V_A$ to $V_B$:
The work done by the field is equal to the decrease in potential energy.
The work done by an external force is equal to the increase in potential energy.
Work done by external force $W = E_{p, \text{final}} - E_{p, \text{initial}} = q(V_B - V_A)$.
Example: Calculating Minimum Energy
If a particle of mass \(m\) and charge \(q\) is accelerated from rest by an electric field, the work done by the field (loss of $E_p$) is converted into kinetic energy ($E_k$):
$$W = \Delta E_p = \Delta E_k$$ $$q \Delta V = \frac{1}{2} m v^2$$This calculation is fundamental in particle physics, often using the unit electronvolt (eV), which is the kinetic energy gained by an electron (or any charge \(e\)) accelerated through a potential difference of 1 Volt.
Key Takeaway: Electric Potential Energy is the energy stored in a system of charges, calculated as \(E_p = qV\). The potential energy for two point charges is \(E_p = \frac{Qq}{4\pi\epsilon_0 r}\).
Chapter Summary: Electric Potential
You have successfully moved from the complex world of vector forces to the simpler, scalar concept of energy in fields.
The most important concepts to recall for examination success are:
- Definition: Electric potential $V$ is work done per unit positive charge moving from infinity to a point.
- Point Charge Potential: \(V = \frac{Q}{4\pi\epsilon_0 r}\) (scalar, sign depends on Q).
- Field Link: Electric field $E$ is the negative potential gradient ($E = -\Delta V / \Delta d$). Field lines point where $V$ decreases.
- Potential Energy: The energy stored in two charges $Q$ and $q$ is \(E_p = \frac{Qq}{4\pi\epsilon_0 r}\).
Keep practicing those calculations, and remember: understanding potential is essential for mastering capacitors and magnetic fields later on! Good luck!