OCR Specification focus:
‘Use V = (1/(4πɛ0)) · (Q / r) and apply changes in electric potential with distance.’
Electric potential around a point charge describes how electric energy varies with position; understanding this relationship allows predictions of charge behaviour, field interactions, and potential changes in electric systems.
The Concept of Electric Potential for a Point Charge
Electric potential is central to understanding how charges interact in electric fields. For a point charge, potential varies predictably with distance, enabling analytical descriptions of energy changes in electrostatic systems.
Electric potential: Work done per unit positive charge to bring a small test charge from infinity to a point in an electric field.
This definition emphasises that potential is a positional property, not dependent on the path taken. At infinite separation, potential is defined as zero, forming the reference point used throughout electrostatics.
Electric Potential Produced by a Single Point Charge
A point charge creates an electric field that extends radially outward (for a positive charge) or inward (for a negative charge). Alongside this, the charge also establishes an electric potential in the surrounding space. Unlike the vector nature of electric field strength, potential is a scalar quantity, simplifying its use in energy considerations.
EQUATION
—-----------------------------------------------------------------
Electric Potential of a Point Charge (V) = (1 / (4πɛ₀)) · (Q / r)
V = Electric potential at distance r, measured in volts (V)
Q = Source charge, measured in coulombs (C)
r = Distance from the charge to the point of interest, measured in metres (m)
ɛ₀ = Permittivity of free space, with units F m⁻¹
—-----------------------------------------------------------------
This equation illustrates that potential decreases inversely with distance from the charge. Moving closer increases potential in the case of a positive charge and decreases it for a negative charge. The scalar nature of potential makes it especially useful when dealing with systems containing multiple charges, as potentials simply add.
Variation of Electric Potential with Distance
The inverse proportionality of potential to distance means that the rate of change is most significant close to the charge. Students should recognise several important implications of this relationship:
Potential is highest near a positive point charge.
Potential approaches zero as r tends to infinity.
For negative charges, potential values are negative, reflecting energy gained rather than expended when bringing a positive test charge closer.
Between these extremes, potential changes smoothly, allowing reliable interpretation of physical behaviour in electric fields.
One important feature of potential is its link to electric potential energy. A charged particle in the vicinity of another charge possesses potential energy proportional to the electric potential at that point. This connection underpins later topics such as energy conservation and charge motion.
Equipotential surfaces for a point charge are concentric spheres (circles in a 2D sketch) centred on the charge.
Equipotential lines (green) form concentric circles around a point charge. Electric field lines (blue) are perpendicular to these surfaces, showing that no work is done moving along them. Source.
Interpreting the Formula in Physical Terms
The term 1/(4πɛ₀) acts as a constant in electrostatics, simplifying repeated calculations. Recognising that potential is directly proportional to the source charge gives insight into how strongly different charges influence surrounding space. Likewise, the inverse dependence on distance clarifies why fields weaken as separation increases.
Furthermore, the sign of the charge determines whether the potential is positive or negative:
Positive Q → Positive potential, meaning work must be done to bring a positive test charge closer.
Negative Q → Negative potential, meaning work is released when a positive test charge moves inward.
These principles align neatly with earlier physical intuition about attraction and repulsion.
For a negative point charge, the potential is negative and electric field lines point radially inward.
A negative point charge produces electric field lines directed inward. Equipotential lines remain concentric, but the corresponding potential values are negative. Source.
Key Features of Point-Charge Potential
To ensure clarity when analysing electric potential in examinations and practical contexts, it is essential to identify several consistent features:
Scalar quantity: Potential has magnitude but no direction.
Zero reference point: Defined as zero at infinite separation.
Independence from path: Work done depends only on the initial and final positions.
Direct link to field strength: Electric field strength is the negative gradient of potential with respect to distance.
Superposition principle: In multi-charge systems, total potential equals the algebraic sum of individual potentials.
These characteristics support efficient problem-solving, particularly when studying energy transformations and charge distributions in more complex arrangements.
Applying the Concept in A-Level Physics
Understanding how potential varies with distance helps students analyse changes in energy and interpret the behaviour of charges in electric fields. This subsubtopic requires fluent use of the given equation and clear physical reasoning when predicting how potential alters as a charged particle moves. Students should focus on the relationship between algebraic signs, physical direction, and energy changes. Bullet-point reasoning is often effective in structured analysis:
Identify the sign of the source charge.
Relate the sign of potential to the charge.
Compare distances to determine relative potential values.
Interpret whether energy is gained or lost.
For a point charge with V=(1/(4πε0))⋅(Q/r)V = (1/(4\pi\varepsilon_0))\cdot (Q/r)V=(1/(4πε0))⋅(Q/r), the potential is chosen to be zero at infinity and falls off as 1/r1/r1/r with increasing rrr.
This graph illustrates the 1/r1/r1/r decrease of electric potential outside a spherical or point charge. The levelling inside the sphere represents conductor behaviour, which extends beyond this subsubtopic but is consistent with point-charge equivalence externally. Source.
FAQ
Electric potential is based on work done per unit charge, and work depends on energy change rather than direction. Because energy is a scalar, so is potential.
This means potentials from multiple charges combine through simple addition, which greatly simplifies problems involving several point charges.
The corresponding electric field, however, requires vector addition because it involves force, which has both magnitude and direction.
Equipotential surfaces around a point charge form concentric spheres where each sphere corresponds to a single potential value.
Their spacing reveals how rapidly potential changes:
Close to the charge: surfaces are close together, indicating a steep potential gradient.
Far away: surfaces spread out, reflecting slower changes with distance.
This spherical pattern arises because potential depends only on radial distance, not direction.
The equation V = kQ/r means small changes in r near the charge produce large changes in potential.
For example, moving from 0.1 m to 0.2 m halves the potential, while moving from 10.0 m to 10.1 m produces only a tiny change.
This behaviour reflects how electric influence weakens with distance and why the region close to a charge is most sensitive to positional changes.
A positive test charge naturally moves from higher potential to lower potential, losing electric potential energy as it does so.
A negative test charge behaves oppositely, moving towards higher potential because this reduces its potential energy.
These tendencies underpin many electrostatic phenomena, including acceleration of charges in fields and the formation of stable or unstable equilibrium points near isolated charges.
Setting the potential to zero at infinity establishes a universal reference point, meaning all potential values closer to the charge are measured relative to this baseline.
This helps avoid ambiguity when comparing potentials created by isolated charges, where energy changes depend only on differences in potential.
It also ensures that a positive point charge always produces positive potential values at finite distances, consistent with the work required to move a positive test charge inward.
Practice Questions
Question 1 (2 marks)
A point charge produces an electric potential at a distance r from its centre. State the equation for the electric potential due to a point charge and identify the physical meaning of either Q or r.
Mark scheme:
• States correct equation: V = (1 / (4 pi epsilon0)) (Q / r). (1)
• Identifies meaning of Q (source charge) or r (distance from charge to point). (1)
Question 2 (5 marks)
A positively charged sphere can be treated as a point charge when calculating electric potential outside its surface.
A sphere carries a charge of +6.0 microcoulombs.
(a) Explain why the sphere can be modelled as a point charge when calculating the potential at distances much larger than its radius.
(b) The electric potential at a distance of 0.40 m from the centre of the sphere is measured. Using the appropriate equation, describe how the potential changes if the distance is doubled.
(c) Comment on how the sign of the charge determines the sign of the electric potential.
Mark scheme:
(a)
• States that at distances much larger than the radius, the field behaves as if all charge is concentrated at the centre. (1)
• Notes that external potential depends only on total charge and radial distance. (1)
(b)
• States that potential is inversely proportional to distance. (1)
• Correctly explains that doubling the distance halves the potential. (1)
(c)
• Explains that a positive charge produces positive potential; a negative charge produces negative potential. (1)
