Electric potential and energy — OCR A-Level Physics
Test yourself on Electric potential and energy with OCR A-Level practice questions.
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Electric potential and energy explained
Electric potential at a point is defined as the work done per unit positive charge in bringing a small test charge from infinity to that point.
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The zero of potential is taken at infinity, meaning that at an infinite distance from the charge creating the field, the potential is exactly zero. The work done W in moving a charge q between two points with potential difference ΔV is W = qΔV. For example, if a charge of +2.0 μC is moved through a potential difference of 500 V, the work done is W = (2.0 × 10⁻⁶) × 500 = 1.0 × 10⁻³ J. You must be able to define electric potential and understand its zero reference at infinity.
(b) electric potential V = Q/(4πε₀r) at a distance r from a point charge; changes in electric potential
Electric potential at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point. For a point charge Q, V = Q/(4πε₀r), where r is the distance from the centre of Q and ε₀ = 8.85 × 10⁻¹² F m⁻¹. Potential is a scalar, so potentials from several charges add algebraically, keeping signs. A positive Q gives positive V; a negative Q gives negative V. Moving between two points changes potential by ΔV = V_final − V_initial, and the work done on a charge q is qΔV. Example: for Q = +2.0 × 10⁻⁹ C at r = 0.30 m, V = (2.0 × 10⁻⁹)/(4π × 8.85 × 10⁻¹² × 0.30) ≈ 60 V.
(c) capacitance C = 4πε₀R for an isolated sphere
An isolated conducting sphere of radius R stores charge on its surface, and its potential is V = Q/(4πε₀R). Since capacitance is defined by C = Q/V, substituting gives C = 4πε₀R. The capacitance depends only on the radius of the sphere and the permittivity of free space, not on the charge stored. For R = 0.10 m, C = 4π × 8.85 × 10⁻¹² × 0.10 ≈ 1.1 × 10⁻¹¹ F, a very small value. This shows why isolated spheres are poor capacitors compared with parallel plates. The equation applies to a sphere in free space, with R measured from the centre to the outer surface.
(d) force–distance graph for a point or spherical charge; work done is area under graph
The electrostatic force between two point charges follows an inverse-square law, F = Qq/(4πε₀r²), so a force–distance graph is a curve that falls steeply as r increases and never crosses the axis for like charges. The work done moving a charge between two separations equals the area under the force–distance graph between those limits. Because the curve is not linear, the area must be found by integration or by counting squares, not by treating the graph as a triangle. For example, moving a charge from r = 0.10 m to r = 0.20 m requires the area between those values, which equals the change in electric potential energy.
(e) electric potential energy = Vq = Qq/(4πε₀r) at a distance r from a point charge Q.
Electric potential energy of a charge q at distance r from a point charge Q is the work done bringing q from infinity to that point. Since V = Q/(4πε₀r), the energy is E_p = Vq = Qq/(4πε₀r). The sign matters: like charges give positive potential energy that decreases as r increases, while unlike charges give negative potential energy that becomes more negative as r decreases. For Q = +3.0 × 10⁻⁶ C, q = +2.0 × 10⁻⁹ C and r = 0.50 m, E_p = (3.0 × 10⁻⁶ × 2.0 × 10⁻⁹)/(4π × 8.85 × 10⁻¹² × 0.50) ≈ 1.1 × 10⁻⁴ J. Energy is a scalar, so contributions from several charges add algebraically.
Your focus
- Define electric potential as work done per unit positive charge.
- State the zero reference for electric potential is at infinity.
- Calculate work done when a charge moves through a potential difference.
Show all 15 objectives
- Define electric potential at a point in terms of work done per unit positive charge from infinity.
- Apply V = Q/(4πε₀r) to calculate potential at a given distance from a point charge.
- Determine changes in electric potential between two points and relate them to work done per unit charge.
- Derive C = 4πε₀R by combining C = Q/V with the potential of an isolated sphere.
- Calculate the capacitance of an isolated sphere of given radius.
- Explain why the capacitance of an isolated sphere is independent of the charge stored.
- Sketch and interpret the force–distance graph for two point or spherical charges.
- Determine work done as the area under the force–distance graph between two separations.
- Relate the area under the graph to the change in electric potential energy.
- Define electric potential energy in terms of work done bringing a charge from infinity.
- Apply E_p = Vq = Qq/(4πε₀r) to calculate potential energy at a given separation.
- Interpret the sign of the potential energy for like and unlike charges.
Electric potential and energy exam tips
Marking Points
- Define electric potential as work done per unit positive charge from infinity to the point.
- State that electric potential is zero at infinity.
- Calculate work done using W = qΔV.
- Recognise that potential is a scalar quantity and can be positive or negative depending on the sign of the source charge.
- States that electric potential is work done per unit positive charge brought from infinity to the point.
- Uses V = Q/(4πε₀r) with ε₀ = 8.85 × 10⁻¹² F m⁻¹ and r measured from the centre of the point charge.
- Recognises V is a scalar and adds potentials algebraically, retaining the sign of each charge.
- Calculates a change in potential as ΔV = V_final − V_initial and links it to work done per unit charge.
- Keeps the correct sign: positive Q gives positive V, negative Q gives negative V.
- States that capacitance is charge stored per unit potential difference, C = Q/V.
- Combines V = Q/(4πε₀R) with C = Q/V to derive C = 4πε₀R.
- Uses R as the radius of the isolated sphere and ε₀ = 8.85 × 10⁻¹² F m⁻¹.
- Recognises that capacitance depends only on R and ε₀, not on the charge or potential.
- Evaluates C for a given radius and comments on the very small magnitude compared with typical capacitors.
- States that the electrostatic force between point charges obeys F = Qq/(4πε₀r²), an inverse-square relationship.
- Describes the force–distance graph as a curve decreasing with distance, asymptotic to the distance axis.
- States that work done equals the area under the force–distance graph between the two separations.
- Recognises that the area must be found by integration or square-counting because the graph is curved.
- Links the area under the graph to the change in electric potential energy between the two positions.
- States that electric potential energy is the work done bringing charge q from infinity to distance r from Q.
- Uses E_p = Vq and substitutes V = Q/(4πε₀r) to obtain E_p = Qq/(4πε₀r).
- Applies the correct sign: like charges give positive E_p, unlike charges give negative E_p.
- Recognises that E_p is a scalar and that contributions from multiple charges add algebraically.
- Calculates E_p for given values of Q, q and r with consistent units.
Examiner Tips
- 💡Remember that electric potential is a scalar, so no direction is needed.
- 💡When calculating work done, ensure the charge and potential difference have consistent signs.
- 💡Use the definition precisely: work done per unit positive charge from infinity.
- 💡Write the equation, substitute values with units, then evaluate; show the power-of-ten handling clearly.
- 💡Check whether the question asks for potential at a point or a change in potential between two points before calculating.
- 💡Use ε₀ = 8.85 × 10⁻¹² F m⁻¹ and keep at least three significant figures through the working.
- 💡Write C = 4πε₀R, substitute R in metres and ε₀ in F m⁻¹, and check the unit of the result is the farad.
- 💡If asked to compare, calculate both values and state the ratio or order-of-magnitude difference explicitly.
- 💡Keep the full expression until the final step to avoid rounding errors in the 4π factor.
- 💡Sketch the curve first, mark the two limits on the distance axis, and shade the area you need to find.
- 💡If counting squares, state the area of one square in J and multiply by the number of squares counted.
- 💡Check the sign of the work done: moving like charges closer together requires positive work by an external agent.
- 💡Write E_p = Qq/(4πε₀r), substitute with signs included, and check the unit is the joule.
- 💡If the question asks for a change in energy, calculate E_p at both separations and subtract.
- 💡Keep ε₀ = 8.85 × 10⁻¹² F m⁻¹ and handle powers of ten carefully to avoid order-of-magnitude errors.
Common Mistakes
- Confusing electric potential with electric potential energy; correction: potential is work done per unit charge, while potential energy is work done on a specific charge.
- Forgetting the zero reference at infinity; correction: always state that potential is zero at infinity when defining it.
- Mixing up potential difference and potential; correction: potential difference is the difference in potential between two points.
- Treating potential as a vector and adding components: potential is a scalar, so values are added algebraically with their signs.
- Measuring r from the surface of a charged sphere rather than from its centre: for an external point, r is the distance from the centre.
- Confusing electric potential V with electric potential energy: V is energy per unit charge, measured in J C⁻¹ or V.
- Forgetting the sign of Q when substituting: a negative charge gives a negative potential at every finite distance.
- Using the diameter instead of the radius in C = 4πε₀R: R is the radius from the centre to the surface.
- Thinking capacitance increases as more charge is added: for an isolated sphere, C is fixed by its geometry.
- Confusing the sphere equation with the parallel-plate equation C = ε₀A/d, which applies to a different geometry.
- Omitting the factor 4π when substituting, which changes the answer by more than an order of magnitude.
- Treating the curve as a straight line and using a triangle area: the inverse-square curve requires integration or square-counting.
- Using the force at one separation as a constant force across the whole distance: force varies continuously with r.
- Forgetting that work done is the area between two specific limits, not the total area from zero to infinity.
- Mixing up the force–distance graph with the potential–distance graph, which has a different shape and meaning.
- Omitting the sign of the charges: like charges give positive potential energy and unlike charges give negative potential energy.
- Confusing electric potential energy with electric potential: energy is measured in joules, potential in volts or joules per coulomb.
- Using r measured from the surface of a charged sphere instead of from its centre.
- Treating potential energy as a vector and adding components: it is a scalar and adds algebraically.