OCR Specification focus:
‘Use E = (1/(4πɛ0)) · (Q / r²) to find the electric field strength due to a point charge.’
A point charge creates a region of influence around itself, producing an electric field whose strength depends on charge magnitude and distance. Understanding this relationship is essential for analysing charge interactions.
The Electric Field of a Point Charge
A point charge is an idealised model in which all the electric charge is assumed to be concentrated at a single, dimensionless point. This model is valid when the physical size of the charged object is negligible compared with the distances being examined.
Electric fields describe how a charge affects other charges at a distance, and OCR requires you to use the equation for the electric field strength due to a point charge. The concept of field strength plays a central role in electric field theory because it allows predictions of force without requiring direct contact between objects.
Electric Field Strength
When discussing the field due to a single isolated charge, the term electric field strength must be clearly understood.
Electric field strength: The force per unit positive charge experienced by a small test charge placed at a point in the field.
Electric field strength provides a quantitative measure of how strongly a charge influences its surroundings. Importantly, the test charge used in this definition must be sufficiently small so that it does not alter the field produced by the source charge.
The electric field around a point charge is radial, meaning it acts directly away from a positive charge and directly towards a negative charge.
Radial electric field lines for a positive point charge, with outward arrows indicating direction and decreasing line density illustrating the inverse-square weakening of field strength. Source.
This field weakens with distance because the influence of the source charge spreads out over a larger spherical surface area.
Mathematical Expression for Field Strength
OCR requires students to apply the expression for the magnitude of the electric field strength created by a point charge at a distance r.
EQUATION
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Electric field strength of a point charge (E) = (1 / (4πɛ₀)) · (Q / r²)
E = Electric field strength in newtons per coulomb (N C⁻¹)
Q = Charge creating the field in coulombs (C)
r = Distance from the charge in metres (m)
ɛ₀ = Permittivity of free space in farads per metre (F m⁻¹)
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This equation shows that electric field strength is directly proportional to the magnitude of the source charge and inversely proportional to the square of the distance from it. The constant (1 / 4πɛ₀) appears frequently in electrostatics and reflects the way electric fields behave in vacuum.
A key implication of this relationship is that electric fields follow an inverse-square law, mirroring the behaviour of gravitational fields around point masses. The inverse-square dependence arises because the field spreads out uniformly over spherical surfaces whose areas increase as r².
Electric field lines and equipotential circles for a point charge, highlighting spherical symmetry and how field strength diminishes with distance. Note: Equipotential detail extends slightly beyond syllabus requirements but directly supports understanding of the inverse-square behaviour. Source.
Direction of the Electric Field
Although the equation above provides only the magnitude, the electric field is a vector quantity. Its direction depends entirely on the sign of the source charge.
Radial electric field lines for a negative point charge, with inward arrows indicating attraction of a positive test charge and the same inverse-square weakening as distance increases. Source.
Around a positive point charge, field lines radiate outwards.
Around a negative point charge, field lines converge inwards.
These field lines provide intuitive visual cues about how a positive test charge moves within the field. They also represent the decreasing field strength: lines become more widely spaced as the distance from the charge increases.
Features of Radial Fields
A radial field has several important characteristics:
Spherical symmetry: At any given distance from the charge, the field strength is identical in all directions.
Field decreases with distance: The inverse-square law ensures the field is strongest close to the charge.
No abrupt boundaries: The field extends infinitely, though it becomes extremely weak at large distances.
Applicable to isolated charges and small charged objects: Provided the object is small relative to the separation, it can be treated as a point charge.
These features make the point-charge model particularly useful in many physical situations, especially when interacting charged particles are separated by large distances relative to their size.
The Role of Permittivity
The constant permittivity of free space, ɛ₀, determines how strongly electric fields are established in vacuum.
Permittivity of free space: A physical constant that quantifies how electric fields interact with the vacuum, influencing the strength of electrostatic interactions.
A larger permittivity would weaken electric fields; a smaller permittivity would strengthen them. In media other than vacuum, a different permittivity applies, but for this subsubtopic we restrict attention to fields in free space, as required by OCR.
The presence of ɛ₀ in the field-strength equation ensures that the behaviour of electric fields aligns with experimental evidence. Its value regulates how strongly the source charge can exert influence across space.
Using the Field Strength Equation
Understanding how to use the point-charge field-strength equation is important for analysing interactions between charges in various contexts. Students should be able to:
Identify the source charge creating the electric field
Determine the distance r from the source charge to the point of interest
Use the equation to calculate the field magnitude
Interpret the direction based on the sign of the source charge
Recognise that electric field strength is a vector quantity
These skills prepare students to explore more complex electric field situations encountered later in the course.
FAQ
Very close to a point charge, the electric field becomes extremely large because the field strength increases rapidly as r approaches zero.
In practice, no physical object is a perfect point, so at very small distances the point-charge model breaks down and the real distribution of charge must be considered.
As distance increases, the field spreads over the surface of an expanding sphere.
A sphere’s surface area increases as 4 pi r squared, so the same total field influence is distributed across a larger area.
This geometric spreading naturally produces the inverse-square relationship for the field strength of any spherically symmetric point source.
In the idealised model, the point charge occupies no physical space, so the concept of an “inside” is undefined.
The model instead treats the field as existing everywhere except at the position of the charge, where the expression predicts an infinite value.
Real charges have finite size, so physical theories avoid this divergence by modelling charge distributions rather than true mathematical points.
The field magnitude is reduced in a medium because the permittivity is greater than epsilon0.
A higher permittivity weakens the electric field for the same charge and distance.
Students should note, however, that the subsubtopic equation assumes a vacuum, so calculations involving media require replacing epsilon0 with the medium’s permittivity.
Yes, if two point charges are positioned so their electric fields oppose each other, a point of zero net field can form.
This occurs when the magnitudes of the opposing fields are equal.
For charges of different signs, this point lies between them; for charges of the same sign, it lies on the line extending beyond the smaller-magnitude charge.
Practice Questions
Question 1 (2 marks)
A point charge creates an electric field in the space around it.
State the equation used to calculate the electric field strength at a distance r from a point charge Q, and identify one factor that causes the field strength to decrease with distance.
Mark scheme:
• Correct equation: E = (1 / (4 pi epsilon0)) multiplied by (Q / r squared). (1 mark)
• Correct factor: field strength decreases because r increases (inverse-square relationship). (1 mark)
Question 2 (5 marks)
A small positively charged particle is placed at a distance of 0.20 m from an isolated point charge.
(a) Explain what is meant by electric field strength at the position of the small charge.
(b) State and use the expression for the electric field strength due to a point charge to describe how the magnitude of the field changes as the distance from the source charge increases.
(c) Explain how the direction of the electric field depends on the sign of the source charge.
Mark scheme:
(a)
• Electric field strength defined as force per unit positive charge at a point. (1 mark)
(b)
• Correct expression stated: E = (1 / (4 pi epsilon0)) multiplied by (Q / r squared). (1 mark)
• Field decreases with increasing distance because it follows an inverse-square law. (1 mark)
(c)
• Direction is away from a positive source charge. (1 mark)
• Direction is towards a negative source charge. (1 mark)
