Physics (9630) | Electric field strength

Electric Field Strength (\(E\)): Comprehensive Study Notes

Hello future physicist! This chapter is all about understanding the invisible influence that charged objects exert on the space around them. If you nailed the concepts of gravitational fields, this section will feel very familiar. We are shifting from calculating forces between two charges (Coulomb's Law) to describing the potential for force at a single point in space—this is the essence of the Electric Field Strength.

Don't worry if this seems tricky at first; we will break it down into manageable chunks using clear definitions and helpful analogies!

1. The Concept of an Electric Field

Imagine a charged object, like a positively charged sphere. How does it exert a force on another charge that isn't even touching it? It does this by creating an Electric Field in the surrounding space.

• Definition: An electric field is a region around a charged object in which another charged object experiences an electrostatic force.
• Analogy: Just as the Earth creates a gravitational field that pulls apples down, a charge creates an electric field that pushes or pulls other charges.

Visualising Electric Fields: Field Lines

We represent electric fields using Electric Field Lines (sometimes called Lines of Force). These lines tell us two crucial things about the field at any point:

1. Direction: The direction of the line shows the direction a positive test charge would move if placed at that point.
   • Lines point away from positive charges (+).
   • Lines point towards negative charges (-).
2. Strength: The density of the lines indicates the strength of the field.
   • Where lines are closer together, the field is stronger.
   • Where lines are further apart, the field is weaker.

Key Takeaway: Electric field lines always run from positive charges to negative charges, and they never cross!

2. Defining Electric Field Strength (\(E\))

Electric Field Strength, \(E\), quantifies how powerful the field is at a specific location. It is fundamentally defined using the idea of a test charge.

The Fundamental Definition

The electric field strength \(E\) at a point is the force (\(F\)) per unit positive charge (\(Q\)) experienced by a small test charge placed at that point.

$$E = \frac{F}{Q}$$

• \(F\) is the electrostatic force experienced (measured in Newtons, N).
• \(Q\) is the magnitude of the test charge (measured in Coulombs, C).
• \(E\) is the Electric Field Strength.

Units and Nature of \(E\)

Since \(E\) is force divided by charge, its SI unit is Newtons per Coulomb (\(N\, C^{-1}\)).

The electric field strength \(E\) is a vector quantity. This means it has both magnitude (strength) and direction (the direction of force on a positive charge).

Memory Aid: Remember the equation \(F = QE\). If you know two variables, you can find the third!

3. Electric Field Strength in a Uniform Field

A uniform electric field is one where the field strength \(E\) is constant in magnitude and direction throughout the region. The best example of this is the field between two parallel metal plates connected to a voltage source.

Relation to Potential Difference (\(V\))

In a uniform field, there is a direct link between the field strength \(E\), the potential difference (\(V\)) between the plates, and the separation distance (\(d\)) between them.

$$E = \frac{V}{d}$$

• \(V\) is the potential difference (voltage) between the plates (Volts, V).
• \(d\) is the separation of the plates (metres, m).

This gives us a second, equivalent unit for \(E\): Volts per metre (\(V\, m^{-1}\)).

The Work Done Derivation (Connecting the Formulas)

This relationship comes directly from the definition of potential difference (work done per unit charge, \(V = W/Q\)) and work done (\(W = Fd\)):

1. Work done (\(W\)) to move a charge \(Q\) across a distance \(d\) in a field that exerts force \(F\):
$$W = Fd$$

2. Work done (\(W\)) related to potential difference (\(\Delta V\)):
$$W = Q\Delta V$$

3. Equating these two expressions for work done gives us the relationship provided in the syllabus:
$$\mathbf{Fd = Q\Delta V}$$

4. Rearranging this:
$$\frac{F}{Q} = \frac{\Delta V}{d}$$

5. Since \(E = F/Q\), we prove that:
$$\mathbf{E = \frac{\Delta V}{d}}$$

Key Takeaway: The uniform field formula \(E=V/d\) is often more useful in lab settings because voltage and distance are easy to measure.

4. Electric Field Strength in a Radial Field (Point Charges)

Unlike the field between parallel plates, the field around a single point charge is radial (it spreads outwards) and non-uniform (its strength changes with distance).

For a field generated by a point charge \(Q\), the strength \(E\) decreases rapidly as you move away from the charge. This follows an inverse square law, similar to gravity and Coulomb's Law.

The Radial Field Formula

Starting from Coulomb's Law (Force $F$ between two charges $Q$ and $q$ separated by distance $r$):

$$F = \frac{1}{4\pi\epsilon_0} \frac{Qq}{r^2}$$

Since the electric field strength is \(E = F/q\), we divide the force by the test charge \(q\):

$$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$

• \(Q\) is the charge generating the field (Coulombs, C).
• \(r\) is the distance from the centre of the charge (metres, m).
• \(\epsilon_0\) is the permittivity of free space (a constant found in your data booklet).

Note: For a charged sphere, the charge \(Q\) is considered to act from the centre of the sphere, just like mass in a gravitational field.

Common Mistake to Avoid: Don't forget that \(E \propto 1/r^2\). If you double the distance \(r\), the field strength \(E\) drops by a factor of four (\(2^2\)).

5. Trajectory of Charged Particles in Uniform Fields

This section links your mechanics knowledge (especially Section 3.2.4: Projectile Motion) with electric fields.

The Scenario

Consider a charged particle (like an electron or proton) moving horizontally (initially at right angles) into the uniform field between two parallel plates.

Step-by-Step Analysis

1. Force: Because the field is uniform, the force \(F = QE\) acting on the particle is constant in magnitude and direction.

2. Acceleration: Since \(F = ma\), a constant force results in a constant acceleration, \(a = F/m\), acting parallel to the field lines.

3. Motion Components:

• Horizontal Motion: Since the force (and acceleration) is strictly vertical, there is no force or acceleration horizontally. The horizontal velocity remains constant.
• Vertical Motion: The particle experiences constant acceleration vertically.

4. Trajectory: Constant horizontal velocity combined with constant vertical acceleration results in a parabolic path.

Analogy: This motion is exactly the same as a bowling ball rolling horizontally off a cliff (constant horizontal speed, constant vertical acceleration due to gravity \(g\)). In this case, the electric force \(F=QE\) replaces the gravitational force \(W=mg\).

Did you know? This principle is used in devices like Cathode Ray Oscilloscopes (CROs) where electric fields deflect electron beams to trace images on a screen.

6. Comparing Gravitational and Electrostatic Forces

The syllabus requires a comparison of the magnitude of gravitational and electrostatic forces, especially between subatomic particles (like electrons and protons).

Chapter Summary: Electric Field Strength

We defined the Electric Field Strength \(E\) in two primary ways, depending on the situation:

1. Fundamental Definition (Any Field): \(E = F/Q\) (Force per unit charge).
2. Uniform Field (Parallel Plates): \(E = V/d\) (Voltage per distance).
3. Radial Field (Point Charge): \(E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}\) (Inverse square law dependency).

Master these three formulas and understand the concept of field lines, and you'll be well-prepared for your exams! You've got this!