IB Physics HLTopic 4 — Force FieldsPaper 1 & 2ΔV = 0 ⇒ W = 0~15 min read
Electric Equipotentials
Since W = qΔV, there must be routes through an electric field that cost you nothing at all. Move a charge so that the potential never changes, and ΔV = 0, and the work done is exactly zero. Join up all the points at the same potential and you have drawn a contour on the electric landscape. Field lines are the paths of steepest descent. Equipotentials are the paths that stay level — and the two must cross at right angles.
📘 What you need to know
Equipotential surfaces are lines (or surfaces) of equal electric potential
They are always perpendicular to the electric field lines
Moving a charge along an equipotential does no work: ΔV = 0, so W = qΔV = 0
The charge’s potential energy does not change along an equipotential
In a radial field they are concentric circles, drawn at equal steps of V, getting further apart with distance
In a uniform field they are straight, parallel and equally spaced lines
Closer together means a steeper potential gradient, and so a stronger field
Between two opposite charges there is a central line at 0 V, where the potentials cancel
Between two like charges there is an empty region around the neutral point, and a contour that encircles both
Equipotential lines have no arrows — they are not vectors and have no direction
The idea in one equation
Why no work is doneW = q ΔV and along an equipotential ΔV = 0W = 0so the charge’s potential energy is unchanged, however far you slide it
A contour line on a hiking map joins points of equal height. Walk along one and you neither climb nor descend, and gravity takes nothing from you and gives nothing back. An equipotential is the identical idea for the electric landscape. And notice what follows for free: the steepest way down a hillside is always at right angles to the contour you are standing on. That is why field lines and equipotentials must be perpendicular.
Radial fields: circles that spread out
Around a point charge, V = kQ/r, so every point at the same distance r has the same potential. The equipotentials are concentric circles.
Now draw them at equal steps of potential — every 20 V, say. Because V ∝ 1/r, the radii are not equally spaced. They spread out dramatically.
Because V ∝ 1/r, the 60 V and 40 V circles sit close together while the 20 V circle is far out. Crowded equipotentials mean a strong field.
Read that diagram twice. Near the charge the circles are bunched, so you cross a lot of potential in a short distance — a steep gradient, and by E = −ΔV/Δr, a strong field. Far away they are sparse, the gradient is gentle, the field is weak. Equipotential spacing tells you field strength just as reliably as field-line spacing does — but the other way round.
A charged conducting sphere gives exactly the same picture outside itself, since it behaves as a point charge at its centre. And inside it, where V is constant, the whole volume is a single equipotential.
Uniform fields: parallel and evenly spaced
Between parallel plates the field is the same everywhere, so the potential falls at a steady rate. Equal steps of V now land at equal intervals.
The plates themselves are equipotentials — they are conductors, so every point on one is at the same potential. That is why field lines leave them at 90°.
Two charges
Draw the equipotentials by tracing curves that cut every field line at right angles. The picture below was computed, not sketched: the field lines are faint grey, the equipotentials dashed teal.
Both sets of curves came out of the same field calculation, and they cross at 89.8° on average. Perpendicularity is not a drawing convention — it is forced by the physics.
Opposite charges
Equipotentials are closed loops, positive around the + charge and negative around the −
Straight down the middle runs a line at exactly 0 V, where kQ/r and −kQ/rcancel
The field on that line is not zero. Only the potential is
Like charges
Close in, each charge has its own loops
Further out, a single contour encircles both — from a distance they look like one big charge
Between them lies the neutral point, a quiet region where the field is zero
The potential there is not zero. Both charges contribute a positive potential, and positives cannot cancel
Those last two bullets are the exam question, dressed up. On the 0 V line of a dipole: V = 0 but E ≠ 0. At the neutral point of two like charges: E = 0 but V ≠ 0. Vectors can cancel by pointing opposite ways. Scalars can only cancel by having opposite signs.
Same potential everywhere on it
so
ΔV = 0
and W = qΔV
no work done
Field lines
Equipotentials
What they show
Direction of the force
Points of equal potential
Arrows?
Yes — always label them
No — they are not vectors
Radial field
Straight, radiating
Concentric circles
Uniform field
Parallel, equally spaced
Parallel, equally spaced
Crowded together means
Strong field
Strong field
They meet each other at
90°, always
🗺️ Drawing and reading equipotentials
Cross every field line at 90°. That is the defining property, not a stylistic choice.
No arrows. Ever. They have no direction.
Radial field? Concentric circles, further apart as you move out.
Uniform field? Straight, parallel, equally spaced. The plates themselves are equipotentials.
Two charges? Loops near each one, and further out a contour encircling both.
Work done? Along a line, zero. Between lines, W = qΔV.
WE 1
A charge of +5.0 nC is moved along an equipotential line at 250 V, travelling a distance of 12 cm. (a) Calculate the work done. (b) It is then moved to an equipotential at 100 V. Calculate the work done on this second journey. (c) Explain why the distance travelled was irrelevant in part (a).
(a) Step 1 — along an equipotential, the potential never changesΔV = 250 − 250 = 0W = qΔV = (5.0 × 10⁻⁹) × 0W = 0 J(b) Step 2 — now the potential does changeΔV = 100 − 250 = −150 VW = (5.0 × 10⁻⁹)(−150)W = −7.5 × 10⁻⁷ J, so the field did the work(c) Step 3 — why 12 cm never appeared
Work depends only on ΔV, not on the path or its length.
Along an equipotential ΔV is zero everywhere.
the distance is a red herringThe 12 cm is there purely to see whether you reach for W = Fd. You don’t. Electric potential energy depends only on where you start and where you finish, never on the route between.
WE 2
Equipotential circles around an isolated point charge are drawn at 60 V, 40 V and 20 V. (a) Explain why the circles are not equally spaced. (b) State where the electric field is strongest, and justify your answer in two different ways.
(a) why the spacing grows
For a point charge V = kQ/r, so r ∝ 1/V.
Equal steps of V therefore need ever larger steps of r.
r₆₀ : r₄₀ : r₂₀ = 1/60 : 1/40 : 1/20 = 1 : 1.5 : 3the circles spread out as you move away(b) where the field is strongestclosest to the charge
Reason 1: the equipotentials are most crowded there,
so ΔV/Δr is steepest, and E = −ΔV/Δr is largest.
Reason 2: E = kQ/r², which grows as r falls.
Both arguments are worth marks, and they are the same argument seen from two ends. Crowded equipotentials are a steep potential gradient, and a steep potential gradient is a strong field.
WE 3
A student is shown the field lines around two point charges. All the lines point outwards, and they are more densely packed around the left-hand charge. (a) State the sign of each charge. (b) Deduce which charge is larger, giving two pieces of evidence. (c) State what happens to the equipotentials far from both charges.
(a) reading the arrows
Field lines point away from positive charges.
both charges are positive(b) which is larger?
Evidence 1: the left charge has a greater density of field lines.
Evidence 2: its equipotentials reach further out — a larger sphere of influence.
Evidence 3: the neutral point lies closer to the right charge.
the left-hand charge is larger(c) far from both charges
From a great distance the pair looks like a single charge of their combined value.
the equipotentials become circles encircling bothPart (b) wants two reasons and there are three available. The neutral point one is the subtlest: you must move nearer the weaker charge before its field can match the stronger one.
💡 Top tips
No arrows on equipotentials. They are lines of equal potential, not vectors.
They must cross field lines at 90° everywhere — including where they meet a conductor.
Along an equipotential, W = 0. Say “because ΔV = 0″, not “because there is no force”.
Radial field: circles that get further apart. Uniform field: lines equally spaced.
Crowded equipotentials = strong field, exactly as for field lines.
Dipole: the middle line is 0 V but the field there is not zero.
Like charges: the neutral point has E = 0 but V≠ 0.
⚠ Common mistakes
Drawing arrows on equipotential lines
Drawing them equally spaced in a radial field. They spread out, because V ∝ 1/r
Letting an equipotential cross a field line at anything other than 90°
Saying no work is done “because the force is perpendicular to the motion” and stopping there. The reason is ΔV = 0
Assuming V = 0 means E = 0 (dipole midline), or that E = 0 means V = 0 (neutral point)
Forgetting that a whole conductor — sphere or plate — is a single equipotential
Using W = Fd for a charge moved between equipotentials. Use W = qΔV
Quick recap:Equipotentials join points of equal potential. They are always perpendicular to field lines, and they carry no arrows. Move a charge along one and ΔV = 0, so W = 0 — the potential energy is unchanged whatever the distance. In a radial field they are concentric circles that spread out with distance (since V ∝ 1/r); in a uniform field they are straight, parallel and equally spaced. Crowded equipotentials mean a steep gradient and a strong field. Between opposite charges lies a 0 V line where E ≠ 0; between like charges lies a neutral point where V ≠ 0.
And that closes the electric half of this topic. Look back at what we built: a force between two charges, then a field that one charge makes on its own, then the energy stored, then the potential per unit charge, and finally a map of that potential. Five ideas, one structure. Now we change the subject entirely — no charges at rest, but charges on the move, and a new kind of field that only appears when they do. Next page: Magnetic Fields.
Equipotentials and field lines getting tangled?
Book a free meeting and we’ll practise sketching them at right angles, and settle the V = 0 versus E = 0 confusion for good.
