Electric Field of a Point Charge: Your Comprehensive Study Guide (9702 A-Level Physics)
Hello future physicist! This chapter moves us beyond simple circuits and asks us to understand the fundamental force that governs electricity: the electric field. This concept might seem abstract, but it's essential. Just like we used gravitational fields to explain why things fall, we use electric fields to explain how charges interact without physically touching. Understanding point charges is the building block for all advanced electrostatics. Let's dive in!
Section 1: The Force - Coulomb's Law
The very first thing we need to know is how two charges actually interact. This is governed by Coulomb's Law, which describes the force between two point charges.
1.1 What is a Point Charge?
In physics, a point charge is an idealized model where a charged object is so small that its dimensions are negligible compared to the distance between it and other charges.
Think of it like treating the Earth as a point mass when calculating the gravitational force between the Earth and the Moon—it simplifies the math tremendously.
Syllabus Requirement (18.3.1): For a point outside a uniform spherical conductor, you can treat the total charge on the sphere as if it were concentrated as a point charge right at the sphere's centre.
1.2 Stating Coulomb's Law (The Equation for Force)
The force \(F\) between two point charges, \(Q_1\) and \(Q_2\), separated by a distance \(r\) in a vacuum (or free space), is:
$$F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2}$$
- \(F\): The electric force (in Newtons, N). This is a vector quantity, meaning it has magnitude and direction.
- \(Q_1\) and \(Q_2\): The magnitudes of the charges (in Coulombs, C).
- \(r\): The distance between the charges (in metres, m).
- \(\epsilon_0\): The permittivity of free space (a constant found in your data booklet). It determines the strength of the electric field in a vacuum.
- \(\frac{1}{4 \pi \epsilon_0}\): This whole term is often combined into a single constant, \(k\), the Coulomb constant.
1.3 Key Features of the Coulomb Force
- Inverse Square Law: Notice the \(r^2\) term. If you double the distance (\(r\)), the force decreases by a factor of four (\(1/2^2\)).
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Attraction or Repulsion:
- Like charges (both positive or both negative) repel.
- Unlike charges (one positive, one negative) attract.
Analogy: Coulomb's Law is mathematically identical to Newton's Law of Gravitation, \(F = G \frac{m_1 m_2}{r^2}\). The difference is that gravity is only attractive (due to mass), while the electric force can be attractive or repulsive (due to charge).
Force ($F$) is proportional to the product of charges ($Q_1 Q_2$) and inversely proportional to the distance squared ($1/r^2$). Force is a vector.
Section 2: Electric Field Strength (\(E\))
While Coulomb's Law tells us the force between two specific charges, the Electric Field Strength (\(E\)) tells us about the environment created by a single source charge (\(Q\)).
2.1 Defining Electric Field Strength (18.1.1)
The electric field strength (\(E\)) at a point is formally defined as:
The force per unit positive charge acting on a small test charge placed at that point.
Mathematically, this definition gives us the core relationship:
$$E = \frac{F}{q}$$
Where \(q\) is the small positive test charge.
- Units of \(E\): Newtons per Coulomb (\(N C^{-1}\)).
- \(E\) is a vector quantity.
2.2 Electric Field Strength due to a Point Charge (\(Q\))
By combining Coulomb's Law (\(F\)) with the definition of \(E = F/q\), we can find the magnitude of the field created by a source charge \(Q\):
$$E = \frac{Q}{4 \pi \epsilon_0 r^2}$$
Notice the crucial similarities and differences with force F:
- It still follows the inverse square law (\(1/r^2\)).
- It only depends on the source charge \(Q\) and the distance \(r\), not on the test charge.
2.3 Representing Fields: Field Lines (18.1.3)
We use electric field lines (or lines of force) to visually represent the electric field.
Rules for Drawing Field Lines:
- They start on positive (+) charges and end on negative (-) charges (or extend to infinity).
- The direction of the lines shows the direction of the force acting on a small positive test charge.
- The closer the lines are drawn together, the stronger the electric field.
- Lines never cross.
- Lines must meet the surface of a conductor at a 90° angle (perpendicularly).
When calculating force ($F$), you use two charges ($Q_1 Q_2$). When calculating field strength ($E$) due to a single point source, you use only one charge ($Q$). Be careful not to mix them up!
Key Takeaway: Electric Field Strength ($E$) tells you the force per unit charge. For a point charge, $E$ decreases rapidly as $1/r^2$.
Section 3: Electric Potential (\(V\))
To understand potential, we switch from thinking about forces (vectors) to thinking about energy and work (scalars). This makes calculation much simpler when dealing with multiple charges!
3.1 Defining Electric Potential (18.5.1)
The Electric Potential (\(V\)) at a point is formally defined as:
The work done per unit positive charge in bringing a small test charge from infinity to that point.
$$V = \frac{W}{q}$$
- Units of \(V\): Joules per Coulomb (\(J C^{-1}\)), which is known as the Volt (V).
- \(V\) is a scalar quantity. It has magnitude but no direction, so you just add potentials together directly!
Analogy: Height in a Gravitational Field
Potential is like height (altitude). If you are at a certain height (potential), you have the ability to do work (fall). If you bring a charge \(q\) to a point with potential \(V\), you give it potential energy, $E_p = qV$.
3.2 Electric Potential due to a Point Charge (\(Q\))
For a source charge \(Q\), the electric potential \(V\) at a distance \(r\) is:
$$V = \frac{Q}{4 \pi \epsilon_0 r}$$
Crucial Comparison:
- Potential \(V\) depends on \(1/r\) (inversely proportional to distance).
- Field Strength \(E\) depends on \(1/r^2\) (inversely proportional to distance squared).
This means that \(V\) drops off much slower than \(E\) as you move away from the charge!
3.3 Potential and Sign Convention
Potential is a scalar, but the sign of the charge matters greatly:
- Positive charge \(Q\): Creates a positive potential. You must do work against the repulsion to bring a positive test charge toward it from infinity.
- Negative charge \(Q\): Creates a negative potential. Work is done *by* the field as the positive test charge is attracted from infinity.
3.4 The Potential Gradient and Field Strength (18.5.2)
Electric field strength ($E$) and potential ($V$) are related: $E$ is the rate at which $V$ changes with distance. This is called the Potential Gradient.
$$E = - \frac{\Delta V}{\Delta d}$$
- \(\frac{\Delta V}{\Delta d}\) is the potential gradient (change in potential over change in distance).
- The negative sign means the electric field \(E\) points in the direction of decreasing potential.
- This relationship gives $E$ a second unit: Volts per metre (\(V m^{-1}\)). Thus, \(1 N C^{-1} = 1 V m^{-1}\).
Did you know? This relationship is very powerful. If you map out the potential across a region, you can instantly determine the field strength simply by looking at how steep the potential gradient is!
Potential ($V$) is work per unit charge. It is a scalar, depends on $1/r$, and its sign depends on the source charge. The field ($E$) is the negative potential gradient.
Section 4: Electric Potential Energy (\(E_p\))
If potential \(V\) is energy per unit charge, then the Electric Potential Energy (\(E_p\)) is the total energy stored when two (or more) charges are held in proximity.
4.1 The Calculation of Potential Energy (18.5.4)
If a charge \(q\) is placed at a point where the potential created by charge \(Q\) is \(V\), the potential energy \(E_p\) of the system is:
$$E_p = qV$$
Substituting the equation for \(V\) of a point charge, we get the potential energy for two charges, \(Q\) and \(q\), separated by distance \(r\):
$$E_p = \frac{Q q}{4 \pi \epsilon_0 r}$$
Units of \(E_p\): Joules (J). \(E_p\) is a scalar quantity.
4.2 Interpreting the Sign of \(E_p\)
The sign of the potential energy tells you about the stability of the configuration:
- Positive \(E_p\): If \(Q\) and \(q\) have the same sign (like charges). This means the charges repel, and you had to do positive work to force them together. The system is unstable and will release energy if they move apart.
- Negative \(E_p\): If \(Q\) and \(q\) have opposite signs (unlike charges). This means the charges attract, and energy was released as they came together (work was done by the field). This is a stable, bound system.
Memory Aid: A configuration is stable if its energy is negative, just like the gravitational potential energy of an object sitting on the ground is negative relative to infinity.
Potential $V$ describes the location (like height), while Potential Energy $E_p$ describes the interaction (like stored gravitational energy).
Summary Table of Electric Field Quantities (Point Charges)
| Quantity | Definition | Equation (Point Charge) | Dependency on $r$ | Vector or Scalar |
|---|---|---|---|---|
| Force (\(F\)) | The interaction between two charges. | \(F \propto \frac{Q_1 Q_2}{r^2}\) | \(1/r^2\) | Vector |
| Field Strength (\(E\)) | Force per unit positive charge. | \(E \propto \frac{Q}{r^2}\) | \(1/r^2\) | Vector |
| Potential (\(V\)) | Work done per unit positive charge (from infinity). | \(V \propto \frac{Q}{r}\) | \(1/r\) | Scalar |
| Potential Energy (\(E_p\)) | Stored energy of a charge configuration. | \(E_p \propto \frac{Q q}{r}\) | \(1/r\) | Scalar |
Final Encouragement
Don't worry if the concepts of field strength and potential feel similar but different! The trick is always remembering the denominator: $r^2$ for force/field (the vector concepts), and $r$ for potential/energy (the scalar concepts). Practice using the definitions precisely, and you'll master this topic! Good luck!