Electric fields and field lines

1. Overview

An electric field is a physical phenomenon that mediates the interaction between stationary electric charges. It is defined as a field of force, which is a specific region in space where an object possessing a particular property (in this case, electric charge) experiences a non-contact force.

The concept of a field allows us to explain "action at a distance." Rather than one charge directly "reaching out" to touch another, we understand that a charge modifies the properties of the space surrounding it. Any other charge placed within this modified space interacts with the field at that specific location, resulting in an electrostatic force. This topic establishes the quantitative framework for measuring the strength of these fields and the visual conventions used to map them across space.

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Key Definitions

• Electric Field: A region of space where a stationary charge experiences an electric force.
• Electric Field Strength ($E$): The force per unit positive charge acting on a stationary point charge.
• Field of Force: A region of space where an object experiences a force due to a specific property (such as mass, charge, or magnetic velocity).
• Test Charge: A theoretical, infinitesimal positive point charge used to determine the direction and magnitude of an electric field without significantly altering the field being measured.

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Content

3.1 The Concept of a Field of Force

In physics, a field is a mapping of a physical quantity at every point in space. A field of force specifically describes a region where a force is exerted on an object.

• In a gravitational field, the property is mass.
• In an electric field, the property is electric charge.

The electric field is a vector field. This means that at every single point in the region, the field has both a magnitude (how strong the force is) and a direction (which way the force acts).

3.2 Defining Electric Field Strength ($E$)

To define the strength of an electric field at a specific point, we use a standard reference: the unit positive charge. By defining it this way, we ensure that the value of $E$ is a property of the field itself, independent of the size of the charge used to measure it.

The mathematical definition is: $$E = \frac{F}{q}$$

Where:

• $E$ is the electric field strength, measured in Newtons per Coulomb ($\text{N C}^{-1}$).
• $F$ is the electrostatic force acting on the charge, measured in Newtons ($\text{N}$).
• $q$ is the magnitude of the charge experiencing the force, measured in Coulombs ($\text{C}$).

Important Note on Units: While $\text{N C}^{-1}$ is the standard unit derived from this definition, you will also encounter Volts per metre ($\text{V m}^{-1}$) in later topics. These units are dimensionally equivalent ($1 \text{ N C}^{-1} = 1 \text{ V m}^{-1}$).

3.3 The Fundamental Force Equation

By rearranging the definition of field strength, we obtain the formula for the force exerted on any charge $q$ placed within an electric field $E$:

$$F = qE$$

This equation is central to predicting the motion of subatomic particles (like electrons and protons) in accelerators or vacuum tubes.

Vector Direction and Charge Sign: The direction of the force $\mathbf{F}$ relative to the field $\mathbf{E}$ depends entirely on the sign of the charge $q$:

1. Positive Charge ($+q$): The force acts in the same direction as the electric field lines.
2. Negative Charge ($-q$): The force acts in the opposite direction to the electric field lines.

3.4 Representing Electric Fields: Field Lines

Electric field lines (sometimes called "lines of force") are a visual representation of the field's geometry. They are not physical objects but a map of the forces in a region.

Standard Conventions for Drawing Field Lines:

1. Directionality: Arrows must always point away from positive charges and towards negative charges. This represents the path a positive test charge would take.
2. Surface Interaction: Field lines must always meet the surface of a conductor at right angles ($90^\circ$).
3. Line Density: The relative closeness of the lines indicates the magnitude of the field strength. Closer lines represent a stronger field; lines further apart represent a weaker field.
4. Continuity and Intersection: Field lines are smooth, continuous curves that never cross. If they crossed, a single point would have two different field directions, which is physically impossible.
5. Equilibrium: In a static situation, there are no field lines inside a hollow conductor; the field is zero.

Common Field Configurations:

• Isolated Point Charges (Radial Fields):

  • For a positive point charge, lines radiate outwards symmetrically in three dimensions.
  • For a negative point charge, lines radiate inwards symmetrically.
  • The field is non-uniform because the line density decreases as $1/r^2$ as you move away from the center.

• Uniform Electric Field (Parallel Plates):

  • Created by two parallel metal plates with opposite charges.
  • Between the plates, the lines are parallel and equally spaced.
  • This indicates the field strength $E$ is constant in both magnitude and direction at all points between the plates.
  • Note: At the edges of the plates, the lines curve outwards; this is known as the "fringing effect."

• Electric Dipole (Equal and Opposite Charges):

  • Lines curve from the positive charge to the negative charge.
  • The field is strongest on the direct line between the two charges.

• Two Like Charges (e.g., Two Positive Charges):

  • Lines from both charges curve away from each other.
  • Exactly midway between two identical like charges, there is a neutral point where the net electric field strength is zero ($E = 0$). No field lines pass through this point.

3.5 Motion of Charged Particles in Fields

While the definition of $E$ specifies a "stationary" charge, we often use $F = qE$ to calculate the acceleration of moving particles.

• Since $F = ma$ and $F = qE$, the acceleration $a$ of a particle of mass $m$ is: $$a = \frac{qE}{m}$$
• In a uniform field, the acceleration is constant.
• If a particle enters a uniform field perpendicular to the field lines (e.g., an electron flying horizontally between vertical plates), it will follow a parabolic path, similar to a projectile in a gravitational field.

3.6 Worked Examples

Worked example 1 — Force on an Ion

A magnesium ion ($Mg^{2+}$) is placed in a region of space where the electric field strength is $4.5 \times 10^5 \text{ N C}^{-1}$ acting vertically upwards. Calculate the magnitude and direction of the electric force acting on the ion. (Data: elementary charge $e = 1.60 \times 10^{-19} \text{ C}$)

• Step 1: Calculate the charge of the ion. The ion has a charge of $+2e$. $$q = 2 \times (1.60 \times 10^{-19} \text{ C}) = 3.20 \times 10^{-19} \text{ C}$$
• Step 2: Apply the force equation. $$F = qE$$ $$F = (3.20 \times 10^{-19} \text{ C}) \times (4.5 \times 10^5 \text{ N C}^{-1})$$
• Step 3: Calculate the magnitude. $$F = 1.44 \times 10^{-13} \text{ N}$$
• Step 4: Determine the direction. Since the ion is positively charged, the force acts in the same direction as the field. Answer: $1.4 \times 10^{-13} \text{ N}$ acting vertically upwards. (Rounded to 2 s.f. to match the input data).

Worked example 2 — Acceleration of an Electron

An electron is situated in a uniform electric field of strength $2.0 \times 10^4 \text{ N C}^{-1}$. Determine the initial acceleration of the electron. (Data: $e = 1.60 \times 10^{-19} \text{ C}$, mass of electron $m_e = 9.11 \times 10^{-31} \text{ kg}$)

• Step 1: Calculate the force on the electron. $$F = qE = (1.60 \times 10^{-19} \text{ C}) \times (2.0 \times 10^4 \text{ N C}^{-1})$$ $$F = 3.2 \times 10^{-15} \text{ N}$$
• Step 2: Use Newton's Second Law to find acceleration. $$a = \frac{F}{m}$$ $$a = \frac{3.2 \times 10^{-15} \text{ N}}{9.11 \times 10^{-31} \text{ kg}}$$
• Step 3: Final Calculation. $$a = 3.5126... \times 10^{15} \text{ m s}^{-2}$$ Answer: $3.5 \times 10^{15} \text{ m s}^{-2}$ (in the direction opposite to the field).

Worked example 3 — Field Strength from Force

A small sphere with a charge of $-4.0 \text{ nC}$ experiences an electrostatic force of $1.2 \times 10^{-3} \text{ N}$ directed towards the East. Calculate the electric field strength at the location of the sphere.

• Step 1: Convert units to SI. $$q = -4.0 \text{ nC} = -4.0 \times 10^{-9} \text{ C}$$
• Step 2: Use the definition of $E$. $$E = \frac{F}{q}$$ $$E = \frac{1.2 \times 10^{-3} \text{ N}}{4.0 \times 10^{-9} \text{ C}}$$
• Step 3: Calculate magnitude. $$E = 3.0 \times 10^5 \text{ N C}^{-1}$$
• Step 4: Determine direction. Because the charge is negative, the force direction (East) is opposite to the field direction. Therefore, the field must point West. Answer: $3.0 \times 10^5 \text{ N C}^{-1}$ directed towards the West.

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Key Equations

Equation	Description	Status
$E = \frac{F}{q}$	Definition of electric field strength (Force per unit positive charge).	Memorise
$F = qE$	Force on a charge $q$ in an electric field $E$.	Memorise
$a = \frac{qE}{m}$	Acceleration of a charged particle in a uniform field.	Derive from $F=ma$
$1 \text{ N C}^{-1} = 1 \text{ V m}^{-1}$	Equivalence of units for field strength.	Understand

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Common Mistakes to Avoid

• ❌ Wrong: Defining Electric Field Strength as "the force on a charge."
• ✓ Right: You must state "the force per unit positive charge." The word "unit" is essential for the mark.
• ❌ Wrong: Drawing field lines for a negative charge pointing outwards.
• ✓ Right: Field lines always point towards a negative charge (the direction a positive test charge would move).
• ❌ Wrong: Forgetting to convert units like microCoulombs ($\mu\text{C}$) or nanoCoulombs ($\text{nC}$) to Coulombs ($\text{C}$).
• ✓ Right: Always use base SI units ($10^{-6}$ for $\mu$, $10^{-9}$ for $n$) before calculating.
• ❌ Wrong: Assuming the force on an electron is in the direction of the field lines.
• ✓ Right: Electrons are negative; they experience a force opposite to the field lines (towards the positive potential).
• ❌ Wrong: Drawing field lines that are "lazy" and don't hit the conductor at $90^\circ$.
• ✓ Right: Use a ruler for the parts of the lines near the surface to ensure they look perpendicular.

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Exam Tips

1. The "Stationary" Requirement: When defining the electric field or field strength, always include the word stationary. While the field exists regardless of motion, the standard definition specifies a stationary charge to avoid complications from magnetic forces (which only act on moving charges).
2. Uniform Field Characteristics: If an exam question mentions "parallel plates," immediately think "uniform field." This means the field strength $E$ is the same everywhere between the plates. If you are asked to draw it, ensure your lines are perfectly parallel and equally spaced.
3. Comparison with Gravity: You may be asked to compare electric fields to gravitational fields.
   • Similarity: Both are fields of force and follow inverse-square laws (for point sources).
   • Difference: Gravitational fields are always attractive (mass is only positive), whereas electric fields can be attractive or repulsive (charge can be positive or negative).
4. Vector Addition: If a point is affected by two different charges, the total electric field $E_{net}$ is the vector sum of the individual fields. You cannot simply add the magnitudes unless they are pointing in the exact same direction.
5. Significant Figures: Cambridge 9702 is strict on significant figures. Look at the data provided in the question. If the field is $3.0 \times 10^3$ (2 s.f.) and the charge is $1.60 \times 10^{-19}$ (3 s.f.), your final answer should be given to 2 significant figures.
6. Visualisation Practicals: Be familiar with how fields are visualised in a lab, such as using semolina grains suspended in insulating oil (like castor oil). The grains become polarised and align themselves along the field lines, making the patterns visible.