IB Physics HLTopic 4 — Force FieldsPaper 1 & 2Radial vs uniform~15 min read
Mapping Gravitational Fields
You can now calculate g at any single point. Do that a thousand times and you have a thousand numbers — and no picture. Physicists solved this beautifully: draw an arrow at every point, then join the arrows up into lines. Suddenly a whole field is visible at a glance, its direction shown by where the lines point and its strength shown by how tightly they crowd together. It is a map, and like any map it has rules.
📘 What you need to know
Field lines (or lines of force) show the direction of the force on a mass placed at that point — and therefore its direction of acceleration
Arrows always point towards the centre of mass of the body making the field, because gravity is attractive only
Closer lines → stronger field. The spacing tells you the size of g
Field lines never cross
Around a point mass or a uniform sphere the field is radial: straight lines pointing inwards, spreading apart as you go out. Radial fields are non-uniform — g changes with r
Close to a surface the field is uniform: parallel, equally spaced lines, and g is the same everywhere
Point mass approximation: a body can be treated as a point mass at its centre when the distances involved are much larger than its size
A uniform sphere produces exactly the same external field as a point mass at its centre
What a field line actually means
Put a small mass down at any point. It gets pulled in some direction. Draw a tiny arrow in that direction. Do it again a millimetre along, and again, and again, and you have traced out a field line.
A field line tells you two thingswhich way the arrow points → the direction of the force on a mass there how crowded the lines are → how strong the field is there
Because F = mg, and force and acceleration point the same way, a field line also shows you the direction a released mass would accelerate. Field lines are the paths gravity would like you to fall along.
The radial field of a planet
Every field line around an isolated sphere runs straight at its centre. Nothing points sideways, nothing loops around. Draw it and you get a hedgehog — only with the spines pointing in.
The same twelve lines pass through a much bigger area when they get far out. That thinning-out is the inverse square law, drawn as a picture.
This is worth pausing on. The twelve lines never disappear — they just spread. Double your distance and those lines are sharing four times the area, so the field is a quarter as strong. The picture and the equation g = GM/r² are saying exactly the same thing in two different languages.
So a radial field is non-uniform: the value of g depends entirely on how far from the centre you are.
Why the field near the ground looks uniform
Here is the puzzle. If Earth’s field is radial, why does every mechanics question you have ever done draw gravity as parallel arrows straight down?
Because you are standing on a very, very big sphere. Take a small patch of that radial field — a classroom, a football pitch — and zoom in. The lines are still converging on the Earth’s centre, 6400 km away, but over ten metres they converge by an amount far too small to draw. They look parallel. They look equally spaced. For all practical purposes they are.
The red box is the whole of your physics lab. Blow it up and the radial field inside it is indistinguishable from a uniform one — the curvature has not gone away, you simply cannot see it over a few metres.
Feature
Radial field
Uniform field
Where
Around a point mass or sphere
Small region close to a surface
Lines
Straight, meeting at the centre
Parallel and equally spaced
Spacing
Grows with distance
Constant
Field strength g
Changes: g = GM/r²
Same at every point
Type
Non-uniform
Uniform
Two rules you must never break
Examiners look for two things in a sketched field, and they are both easy marks.
Unlabelled arrows and crossing lines are the two easiest marks to throw away on a field sketch.
When is a planet a point?
Newton’s law is written for point masses, yet we happily apply it to Jupiter. Two results let us get away with it.
The point mass approximationa body may be treated as a point mass at its centre when the distances involved are much greater than its size
A uniform sphere — one whose mass is spread evenly — produces exactly the same field outside itself as a point mass sitting at its centre. That is not an approximation at all; it is exactly true.
For anything travelling a huge distance compared with its own dimensions — planets, moons, stars — the size simply stops mattering. The Earth is a dot at the scale of its orbit.
Which is why the field lines around a planet are drawn identical to those around a point mass. If you sketch a sphere and then draw its field lines stopping at the surface with arrows aimed at the centre, you have drawn a point mass’s field. Nature does not know the difference, and neither does the mark scheme.
Force on a test mass
joined up into a continuous line
Field line direction of g
how tightly they pack
Strength of the field
✏️ Sketching a field for full marks
Draw the body first. A circle for a planet, a straight line for a surface.
Straight lines only. For a radial field, every line must aim at the centre, not at the edge.
Put arrowheads on. All pointing inwards. An unlabelled line scores nothing.
Space them evenly around the sphere, and evenly apart in a uniform field.
Never let two lines touch or cross. Check before you put the pen down.
WE 1
Describe the gravitational field lines around an isolated spherical planet, stating three features that a correct diagram must show.
Feature 1 — shape and direction
The lines are straight and radial, with arrowheads pointing inwards, towards the centre of the planet.
Feature 2 — spacing
They are evenly spaced around the sphere, but get further apart as distance increases — so the field is non-uniform.
Feature 3 — the rule
The lines never cross.
radial, inwards, spreading, never crossing“Inwards” is the mark that gets forgotten. Gravity is attractive only, so there is no other option.
WE 2
The Earth’s gravitational field is radial. Explain why, in a laboratory, the field is drawn as parallel, equally spaced lines instead.
Step 1 — what is really happening
The lines still converge on the Earth’s centre, about 6400 km below the floor.
Step 2 — the scale of the lab
Over a few metres the lines converge by a negligible amount, and r barely changes.
Step 3 — the consequence
So the direction of g is the same everywhere in the room, and its magnitude is too.
the field is uniform over that small regionThe word examiners want is small region close to the surface. A uniform field is always a local approximation to a radial one.
WE 3
At the surface of a planet of radius R, the field strength is 6.0 N kg⁻¹. A student looks at a field line diagram and says: “The lines are twice as far apart at 2R, so the field there must be 3.0 N kg⁻¹.” Identify the error and calculate the correct value.
Step 1 — spot the error
The student is thinking about spacing in one direction. Field strength depends on lines per unit area, and area grows as r².
Step 2 — the correct reasoning
Double r, and the same lines are shared over 4× the area.
g ∝ 1/r² → g′ = g/4Step 3 — substituteg′ = 6.0 / 4g′ = 1.5 N kg⁻¹A useful reality check on any field diagram: the lines are spreading in two directions at once, not one. That second direction is where the square comes from.
💡 Top tips
Always draw the arrowheads, and always point them at the centre. Unlabelled lines score zero.
Use a ruler. Field lines in a radial or uniform field are straight, and examiners notice.
Say “closer lines mean a stronger field” if asked how a diagram shows the strength.
A radial field is non-uniform; a uniform field only exists over a small region near a surface. Use both phrases.
Field lines show the direction of the force and hence the acceleration of a mass placed there.
Outside a uniform sphere, the field is exactly that of a point mass at its centre — you may use g = GM/r² without apology.
⚠ Common mistakes
Drawing field lines with no arrows, or with arrows pointing outwards — gravity never repels
Letting the lines cross, or bend, or stop in mid-air
Aiming the lines at the edge of the sphere instead of its centre
Drawing a radial field with equally spaced lines all the way out — the spacing must grow
Calling the Earth’s field uniform. It is radial; it merely looks uniform close to the ground
Thinking that doubling the line spacing halves the field. Lines spread in two dimensions, so g drops to a quarter
Forgetting that the point mass approximation needs distance much greater than size
Quick recap:Field lines point along the force a test mass would feel, so they always run inwards, towards the centre of the body making the field, and they never cross. Their crowding shows the strength of g. Around a sphere the field is radial and therefore non-uniform; over a small patch near a surface the same lines look parallel and equally spaced, giving a uniform field. And because a uniform sphere behaves exactly like a point mass at its centre, the two pictures are really one.
We have mapped the direction gravity would push you. But maps have contour lines as well as arrows — heights, not just slopes. Ask instead “how much energy would it cost to be here?” and you get a completely different, and in many ways more powerful, description of the same field. That quantity is called gravitational potential, and it is where we go next.
Field sketches costing you marks?
Book a free meeting and we’ll practise radial and uniform field diagrams, and the questions examiners build on them.
