Electric Fields and Field Lines (9702 A Level Physics)
Welcome to the fascinating world of Electric Fields! If you found Gravitational Fields manageable, you're already halfway there. This chapter explores the invisible influence that charged objects exert on each other. Understanding fields is crucial because it forms the foundation for later topics like capacitance and particle physics. Don't worry if the formulas seem complicated; they all follow a few core, logical principles!
Let's dive into how charges create forces without actually touching.
18.1 The Concept of the Electric Field
What is an Electric Field?
Imagine you have a magnet (or a charged balloon). You don't need to touch another object for it to feel a force—it influences the space around it. This influenced region is the field.
The syllabus defines an electric field simply and precisely:
Definition: An electric field is a region in which a charged particle experiences an electrical force.
An electric field is an example of a field of force. This means every point in that space has a specific force associated with it.
Defining Electric Field Strength (E)
To measure how strong the field is at any point, we use the concept of Electric Field Strength, \(E\).
Definition: The electric field strength \(E\) at a point is the force per unit positive charge acting on a small test charge placed at that point.
Mathematically, this is expressed as:
$$E = \frac{F}{q}$$
Where:
- \(F\) is the force experienced (in Newtons, N)
- \(q\) is the charge placed in the field (in Coulombs, C)
- \(E\) is the electric field strength (in \(\text{N C}^{-1}\))
Since Force (\(F\)) is a vector, and charge (\(q\)) is a scalar, the electric field strength (\(E\)) is also a vector quantity. It has both magnitude and direction.
If you know the field strength \(E\), you can calculate the force \(F\) acting on any charge \(q\):
$$F = qE$$
Remember: If \(q\) is positive, \(F\) is in the same direction as \(E\). If \(q\) is negative, \(F\) is in the opposite direction to \(E\).Representing Electric Fields: Field Lines (Lines of Force)
Since we can't see the field, we draw diagrams using field lines to visualize the magnitude and direction of the electric field.
Rules for Drawing Field Lines:
- Direction: Field lines always point away from positive charges and towards negative charges. (This reflects the direction a small, imaginary positive test charge would move).
- Density (Strength): The closer the lines are drawn to each other, the stronger the electric field.
- Perpendicularity: Field lines meet a conductor surface at 90 degrees (perpendicularly).
- Non-Crossing: Field lines never cross each other (because at any single point, the field can only have one direction).
Analogy: Hillside Contour Maps
Think of field lines like arrows showing the direction a tiny object would roll down a hill.
- Positive charges are like sources (mountain peaks).
- Negative charges are like sinks (valleys/lakes).
- Where the lines are dense (close together), the slope is steep (strong force/strong E).
E is force per unit positive charge. \(E = F/q\).
Field lines show direction (+ to -) and strength (density).
18.3 & 18.4 Electric Fields and Forces from Point Charges
18.3 Coulomb's Law (Force)
To determine the force between two stationary point charges, we use Coulomb's Law. This is the electrical equivalent of Newton's law of universal gravitation.
Coulomb's Law: The electrostatic force \(F\) between two point charges \(Q_1\) and \(Q_2\) is directly proportional to the product of the charges and inversely proportional to the square of their separation \(r\).
$$F = \frac{Q_1 Q_2}{4\pi \epsilon_0 r^2}$$
Where:
- \(Q_1\) and \(Q_2\) are the magnitudes of the charges (C).
- \(r\) is the distance between their centres (m).
- \(\epsilon_0\) (epsilon naught) is the permittivity of free space (a constant included in your data sheet, approximately \(8.85 \times 10^{-12} \text{ F m}^{-1}\)).
- \(4\pi\epsilon_0\) is often replaced by a single constant \(k\).
Action and Reaction
The force \(F\) calculated using Coulomb's law is the magnitude of the force acting on both charges. Remember Newton's Third Law: the force exerted by \(Q_1\) on \(Q_2\) is equal and opposite to the force exerted by \(Q_2\) on \(Q_1\).
Important Rule for Conductors:
For a point located outside a spherical conductor, you can treat all the charge on the sphere as if it were concentrated at a point charge at its centre. This simplifies calculations greatly!
18.4 Electric Field Strength (Point Charge)
If we want the field strength \(E\) created by a single charge \(Q\), we use the definition \(E = F/q\) and substitute Coulomb's law:
$$E = \frac{F}{q} = \frac{1}{q} \left( \frac{Q q}{4\pi \epsilon_0 r^2} \right)$$
The test charge \(q\) cancels out, leaving the formula for the field strength created by \(Q\):
$$E = \frac{Q}{4\pi \epsilon_0 r^2}$$
This formula only applies to point charges (or spherically symmetrical charge distributions).
Do not confuse the formulas!
- Force \(F\) (between two charges \(Q_1\), \(Q_2\)) depends on \(Q_1 Q_2\) and \(r^2\).
- Field \(E\) (created by one charge \(Q\)) depends on \(Q\) and \(r^2\).
18.2 Uniform Electric Fields
The fields discussed so far decrease with distance (\(E \propto 1/r^2\)). However, it is possible to create a region where the field strength is constant in both magnitude and direction. This is a uniform electric field.
How do we create one? By using two large, parallel metal plates separated by a small distance, charged to a potential difference (\(V\)).
Field Strength and Potential Difference
In a uniform field, the field lines are parallel and equally spaced. The magnitude of the electric field strength \(E\) is related to the potential difference (\(\Delta V\)) across the plates and their separation (\(\Delta d\)).
$$E = \frac{\Delta V}{\Delta d}$$
Where:
- \(\Delta V\) is the potential difference (Voltage, V)
- \(\Delta d\) is the plate separation (distance, m)
- \(E\) is the electric field strength.
This equation shows us an alternative unit for \(E\): Volts per metre (\(V m^{-1}\)).
Did you know? \(1 \text{ N C}^{-1}\) is physically the same as \(1 \text{ V m}^{-1}\)!
Motion of Charged Particles in a Uniform Field
A uniform electric field applies a constant force \(F = qE\) to any charged particle placed inside it.
Since the force is constant, the particle experiences a constant acceleration (\(a = F/m\)).
Step-by-Step Motion Description:
- A particle (e.g., an electron) enters the field moving horizontally (velocity \(v_x\)).
- The electric force \(F = qE\) acts vertically (parallel to the E-field lines if positive, opposite if negative).
- This setup is exactly like a projectile thrown horizontally in a gravitational field!
- The particle maintains constant horizontal velocity \(v_x\).
- The particle experiences constant vertical acceleration \(a_y\).
The result is that the particle follows a parabolic path (a curve).
Example: This principle is used in devices like cathode ray oscilloscopes (CROs) where charged beams are deflected by parallel plates.
The field strength \(E\) between parallel plates is constant and related to the voltage gradient: \(E = \Delta V / \Delta d\). Charged particles move in a parabola, just like objects in a uniform gravitational field.
18.5 Electric Potential (V) and Potential Energy (Ep)
Just like gravitational fields have Gravitational Potential Energy (GPE), electric fields involve Electric Potential Energy and Electric Potential. This is where the concept of "work done" comes in.
Defining Electric Potential (V)
Electric potential is defined relative to infinity. We assume that at infinity, the electric potential is zero (\(V = 0\)).
Definition: The Electric Potential (\(V\)) at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point.
$$V = \frac{W}{q}$$
The unit for potential is the Volt (V), where \(1 \text{ V} = 1 \text{ J C}^{-1}\).
Potential (\(V\)) is a scalar quantity, making calculations involving multiple charges much easier than calculating vector fields (\(E\)).
Electric Potential Due to a Point Charge (Q)
For a single point charge \(Q\), the potential \(V\) at a distance \(r\) is:
$$V = \frac{Q}{4\pi \epsilon_0 r}$$
Notice a crucial difference from the field strength formula: potential \(V\) is proportional to \(1/r\), not \(1/r^2\).
Analogy Trick: Think of a Point Charge:
\(\mathbf{F}\) (Force) requires 2 charges and is $1/\mathbf{r^2}$.
\(\mathbf{E}\) (Field Strength) requires 1 charge and is $1/\mathbf{r^2}$.
\(\mathbf{V}\) (Potential) requires 1 charge and is $1/\mathbf{r}$. (It’s easier—no square!)
Electric Potential Energy (Ep)
The Electric Potential Energy (\(E_p\)) of two charges \(Q\) and \(q\) separated by distance \(r\) is the work done to bring those two charges from infinity to that separation.
Since \(V = E_p / q\), we can rearrange for \(E_p\):
$$E_p = qV$$
Substituting the expression for \(V\) due to charge \(Q\):
$$E_p = \frac{Qq}{4\pi \epsilon_0 r}$$
If the charges are like charges (both positive or both negative), \(E_p\) is positive (repulsion). If they are unlike charges, \(E_p\) is negative (attraction).
The Relationship Between E and V: Potential Gradient
There is a powerful link between the scalar Electric Potential \(V\) and the vector Electric Field Strength \(E\).
The Electric Field Strength at any point is equal to the negative of the potential gradient at that point.
$$E = -\frac{\Delta V}{\Delta r} = - \text{(potential gradient)}$$
- \(\Delta V\) is the change in potential.
- \(\Delta r\) is the distance moved.
What does the negative sign mean?
The electric field vector \(E\) always points in the direction where the electric potential (\(V\)) is decreasing most rapidly. In other words, a positive test charge naturally wants to move towards lower potential (just like a ball falls naturally towards lower GPE).
- Field is defined by force: \(F = qE\).
- Point charges exert \(F \propto 1/r^2\) (Coulomb's Law).
- Point charges create \(E \propto 1/r^2\).
- Uniform fields use \(E = V/d\).
- Potential \(V\) is work done per unit charge from infinity, and \(V \propto 1/r\).