Radial and Uniform Gravitational Fields

Gravitational fields describe how a mass would experience gravitational force at different locations in space. Two important models are the radial gravitational field and the uniform gravitational field.

A radial field is the more complete model for the gravitational field around an isolated spherical object such as a planet or star.

A uniform field is a useful approximation when the field strength and direction change so little across a region that we can treat them as constant.

Understanding the difference between these models allows us to interpret field diagrams and decide which model is appropriate for a particular situation.

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Radial Gravitational Fields

A radial gravitational field is a field whose direction is along lines directed toward a central mass.

Around an isolated spherical planet, gravitational field lines:

• point toward the centre of the planet
• are radial
• become more widely spaced with increasing distance
• form a symmetric pattern around the planet

The arrows point inward because gravity between ordinary masses is attractive.

At every point in the field, the arrow shows the direction in which a small test mass would initially accelerate.

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Why a Planet Produces a Radial Field

For a spherically symmetric planet, gravitational attraction is directed toward its centre.

Imagine placing small test masses at several positions around Earth.

A test mass above the North Pole would accelerate toward Earth's centre.

A test mass above the equator would also accelerate toward Earth's centre.

The same is true at every location around Earth.

Therefore, the gravitational field vectors point inward from all directions.

This produces the characteristic radial pattern.

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Direction in a Radial Field

Gravitational field strength is a vector quantity.

This means it has:

• magnitude
• direction

At any point around a spherical planet, the direction of the gravitational field is:

toward the centre of the planet.

The field direction at a point is tangent to the field line passing through that point.

For a radial field, this tangent points directly toward the central mass.

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Radial Field Lines Spread Out

Near the planet, gravitational field lines are represented as relatively close together.

Farther from the planet, they are represented as more widely spaced.

This indicates that:

the gravitational field becomes weaker with increasing distance.

The change is not linear.

Instead, outside a spherical mass the field follows an inverse-square relationship.

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Field Strength in a Radial Field

Outside a spherical mass:

g = GM/r²

where:

• g = gravitational field strength in N/kg
• G = gravitational constant
• M = mass producing the field in kg
• r = distance from the centre of the mass in m

This equation shows that:

g ∝ 1/r²

if M remains constant.

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The Inverse-Square Relationship

Suppose the gravitational field strength at distance r is:

g

At distance 2r:

field strength = g/4

At distance 3r:

field strength = g/9

At distance 4r:

field strength = g/16

This occurs because distance is squared in:

g = GM/r²

The field becomes weaker rapidly as distance increases.

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Worked Example: Doubling the Distance

At distance r from a planet's centre:

g = 20 N/kg

A spacecraft moves to distance:

2r

Because:

g ∝ 1/r²

the new field strength is:

g₂ = 20/2²

= 20/4

= 5 N/kg

Doubling the distance does not halve the field strength.

It reduces it to one-quarter.

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Worked Example: Tripling the Distance

Suppose:

g = 45 N/kg

at distance r.

At distance 3r:

g₂ = 45/3²

= 45/9

= 5 N/kg

So increasing the distance by a factor of 3 reduces the field strength by a factor of 9.

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Comparing Field Strength at Two Distances

Sometimes we do not need G or the mass of the planet.

Since:

g ∝ 1/r²

we can write:

g₁r₁² = g₂r₂²

or:

g₂/g₁ = r₁²/r₂²

Example

A gravitational field has strength:

g₁ = 12 N/kg

at:

r₁ = 4.0 × 10⁶ m

Find the field strength at:

r₂ = 8.0 × 10⁶ m

Since the distance has doubled:

g₂ = 12 × (4.0 × 10⁶ / 8.0 × 10⁶)²

= 12 × (1/2)²

= 12 × 1/4

= 3 N/kg

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Distance Is Measured from the Centre

In a radial field:

r means distance from the centre of the spherical mass.

This is important.

Suppose a spacecraft is 2,000 km above a planet whose radius is 6,000 km.

The spacecraft's distance from the planet's centre is:

r = 6,000 + 2,000

= 8,000 km

not 2,000 km.

This centre-to-centre distance must be used in gravitational field calculations.

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Radial Fields Extend Through Space

A planet's gravitational field does not suddenly end.

In the Newtonian model, as distance increases:

g becomes smaller and smaller

but does not reach exactly zero at any finite distance from an isolated mass.

This is why Earth can exert gravitational effects on objects far beyond its atmosphere.

The Moon, satellites, and spacecraft can all be affected by Earth's gravitational field.

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Uniform Gravitational Fields

A uniform gravitational field is one in which gravitational field strength is approximately constant in both:

• magnitude
• direction

A uniform field is represented by:

parallel, equally spaced field lines.

The parallel lines show that the direction does not change.

The equal spacing indicates that the field strength is constant.

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Earth's Surface as an Approximately Uniform Field

Earth's actual gravitational field is radial.

However, over a small region close to Earth's surface, the radial field lines are almost parallel.

Why?

Earth's radius is enormous compared with everyday distances.

Earth's mean radius is about:

6.37 × 10⁶ m

or:

6,370 km

A classroom only a few metres across is tiny compared with this distance.

Across such a small region:

• the direction toward Earth's centre changes by a negligible amount
• the distance from Earth's centre changes very little
• gravitational field strength changes very little

Therefore, we can approximate the field as uniform.

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Local Versus Global Models

This gives us two useful ways to represent Earth's gravitational field.

Global View

When considering Earth as a whole:

use a radial field.

Field lines converge toward Earth's centre.

Local View

When considering a small region near Earth's surface:

use an approximately uniform field.

Field lines can be treated as parallel and equally spaced.

Both models describe the same gravitational field at different scales.

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Why "Uniform" Is an Approximation

Near Earth's surface:

g ≈ 9.8 N/kg

But g is not exactly identical everywhere.

It changes slightly with:

• altitude
• latitude
• Earth's rotation
• Earth's shape
• local geology

For many introductory physics problems, these differences are small enough to ignore.

We therefore use:

g ≈ 9.8 N/kg

throughout the region being studied.

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Uniform Field-Line Diagram

A uniform gravitational field might be drawn as:

↓ ↓ ↓ ↓ ↓

↓ ↓ ↓ ↓ ↓

↓ ↓ ↓ ↓ ↓

The arrows are:

parallel

because the direction is constant.

They are:

equally spaced

because the magnitude of the field is constant.

They point:

downward

when representing gravity near Earth's surface.

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Radial Field-Line Diagram

A simplified radial field around a planet has arrows directed inward from every direction.

The field lines:

• converge toward the centre
• are not parallel
• become more widely spaced farther away
• indicate changing direction
• indicate decreasing field strength with distance

This is fundamentally different from the uniform-field model.

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Comparing Radial and Uniform Fields

This comparison is especially useful when interpreting field diagrams.

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Reading Field-Line Direction

The arrows on a gravitational field diagram show the direction in which a small positive test mass would accelerate.

For gravity, this means the arrows point:

toward the mass producing the field.

Around a planet:

arrows point inward.

Near Earth's surface:

arrows point downward toward Earth.

Field lines should not point outward from an isolated ordinary mass.

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Reading Field-Line Spacing

In conventional field diagrams, field-line spacing represents relative field strength.

Closer field lines → stronger field

More widely spaced field lines → weaker field

In a radial field, lines spread farther apart as distance increases.

Therefore:

field strength decreases with distance.

In a uniform field, the lines remain equally spaced.

Therefore:

field strength remains approximately constant.

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Field Lines Are Not Physical Objects

Field lines are a visual model.

There are not actual strings or lines extending through space.

A diagram could be drawn using:

8 field lines

or:

16 field lines

or:

32 field lines.

The physical field has not changed.

What matters is the pattern:

• direction
• symmetry
• relative spacing

Field lines allow us to visualize a continuous vector field.

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Field Vectors

Another way to represent gravitational fields is with arrows at selected points.

Each arrow represents the gravitational field vector at that position.

The arrow's:

direction shows the direction of g.

Its:

length can be used to represent the magnitude of g.

Near a planet:

• arrows point toward the centre
• arrows become shorter farther away

In a uniform field:

• arrows have the same direction
• arrows have approximately the same length

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Why Radial Field Lines Spread Out

Imagine a spherical surface surrounding a planet.

As the radius of this imaginary sphere increases, its surface area increases according to:

A = 4πr²

The same gravitational influence is spread across increasingly large spherical surfaces.

Because area increases with r², field strength decreases with:

1/r²

This provides a useful geometric way to understand the inverse-square relationship.

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Surface Area and the Inverse-Square Law

Suppose the distance doubles.

Original spherical area:

A₁ = 4πr²

At distance 2r:

A₂ = 4π(2r)²

= 16πr²

Therefore:

A₂ = 4A₁

The corresponding field strength becomes:

1/4 as large.

If distance triples:

area increases by:

3² = 9

so field strength becomes:

1/9 as large.

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Field Strength and Altitude

As an object moves upward from Earth's surface, its distance from Earth's centre increases.

Therefore:

g decreases.

However, small changes in altitude produce only small changes in g because Earth's radius is so large.

This is why a person standing:

• on the ground
• on the second floor
• on the roof of a building

experiences almost the same gravitational field strength.

For everyday situations, the uniform-field approximation is excellent.

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Example: A Small Change in Height

Suppose an object moves upward by 10 m.

Earth's radius is approximately:

6,370,000 m

The distance changes from approximately:

6,370,000 m

to:

6,370,010 m

This is a tiny fractional change.

Therefore, the corresponding change in gravitational field strength is also tiny.

We can reasonably treat:

g ≈ constant

over this height.

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When the Uniform Approximation Breaks Down

The uniform approximation becomes less appropriate when distances become large compared with the planet's radius.

Examples include:

• high-altitude satellites
• spacecraft travelling far from Earth
• motion between planets
• comparing surface gravity with gravity far above a planet

In these situations, use the radial-field model and:

g = GM/r²

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Example: Choosing the Correct Model

A Ball Falling 2 m in a Classroom

Use:

uniform field approximation

because the change in distance from Earth's centre is negligible.

A Satellite Orbiting Earth

Use:

radial field

because field direction changes around the orbit and field strength depends on distance from Earth's centre.

A Skydiver Falling Several Kilometres

For many introductory calculations:

uniform field approximation

may still be acceptable.

For high-precision calculations:

changes in g could be considered.

A Spacecraft Travelling from Earth Toward the Moon

Use:

radial gravitational fields from both bodies.

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Comparing Motion in the Two Models

In a uniform field, an object moving vertically experiences approximately constant gravitational acceleration:

a ≈ g

Near Earth's surface:

a ≈ 9.8 m/s² downward

In a radial field, acceleration changes as the object's distance from the central mass changes:

a = g = GM/r²

Therefore, a spacecraft moving away from a planet experiences progressively smaller gravitational acceleration.

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Uniform Fields and Potential Energy

Near Earth's surface, because g is approximately constant, changes in gravitational potential energy can often be calculated using:

ΔEₚ = mgΔh

This works well for relatively small changes in height.

For very large changes in distance from a planet, g changes significantly, so the simple near-surface equation becomes less appropriate.

This is another example of how the uniform-field approximation simplifies calculations.

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Radial Fields and Orbits

Satellites move through Earth's radial gravitational field.

At every point in a circular orbit, the gravitational force points toward Earth's centre.

The satellite's velocity is tangent to the orbit, while gravitational acceleration points inward.

Gravity continually changes the direction of the satellite's velocity.

This inward gravitational acceleration allows the satellite to follow a curved path around Earth.

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Field Strength at the Surface of Different Planets

At the surface of a spherical planet:

g = GM/R²

where R is the planet's radius.

This means surface gravity depends on both:

• planetary mass
• planetary radius

A planet with greater mass does not necessarily have greater surface gravity if it also has a much larger radius.

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Worked Comparison of Two Planets

Suppose Planet B has:

4 times the mass of Planet A

but:

2 times the radius.

For Planet A:

gA = GMA/RA²

For Planet B:

gB = G(4MA)/(2RA)²

gB = 4GMA/4RA²

Therefore:

gB = gA

Despite having four times the mass, Planet B has the same surface gravitational field strength because its radius is twice as large.

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Changing Mass but Keeping Radius Constant

Suppose two planets have the same radius.

Planet B has twice the mass of Planet A.

Since:

g = GM/R²

and R is unchanged:

gB = 2gA

At the same distance, doubling the source mass doubles the gravitational field strength.

This is different from changing distance, where the inverse-square relationship applies.

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Multiple Radial Fields

When several astronomical objects are present, each produces its own radial gravitational field.

For example:

• Earth produces a field.
• The Moon produces a field.
• The Sun produces a field.

At any point in space, the total gravitational field is the vector sum of all relevant fields.

This is called superposition.

g_net = g₁ + g₂ + g₃ + ...

Direction must be considered because gravitational field strength is a vector.

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Neutral Points

Between two masses, their gravitational fields may oppose each other.

At some point, their magnitudes may become equal:

g₁ = g₂

If they point in opposite directions:

g_net = 0

This is a neutral point in the gravitational field.

For two equal isolated masses, the neutral point is at the midpoint.

For unequal masses, the neutral point between them is closer to the smaller mass.

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Identifying Fields from Diagrams

When shown an unfamiliar field diagram, use the following clues.

Radial Field

Look for:

• lines converging toward a central object
• changing direction
• changing line spacing
• spherical symmetry

Uniform Field

Look for:

• straight parallel lines
• equal spacing
• identical direction throughout the region

You should be able to identify the type of field without knowing which planet or object created it.

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A Useful Decision Rule

Ask:

Is the region being considered small compared with the radius of the planet?

If yes:

uniform-field approximation may be appropriate.

If no:

use the radial-field model.

Then ask:

Does g change significantly over the distances involved?

If yes, the radial model is required.

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Common Mistakes

Thinking Earth's field is actually uniform

Earth's gravitational field is radial. Uniform gravity is an approximation used over relatively small regions near the surface.

Drawing radial field arrows outward

Gravitational field arrows point toward the mass.

Drawing uniform field lines that converge

Uniform field lines should be parallel.

Drawing uniform field lines with changing spacing

Changing spacing suggests changing field strength.

Thinking gravitational field strength decreases linearly with distance

Outside a spherical mass:

g ∝ 1/r²

not 1/r.

Thinking twice the distance means half the gravity

Twice the distance means:

1/4 the field strength.

Using altitude instead of distance from the centre

For radial-field calculations:

r = planetary radius + altitude

when the object is above the surface.

Assuming field lines physically exist

Field lines are a visual model used to represent a continuous gravitational field.

Thinking the field suddenly ends

A gravitational field becomes progressively weaker but does not abruptly stop.

Using the radial model unnecessarily

For small distances near Earth's surface, treating g as constant is often much simpler and sufficiently accurate.

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Key Terms

Gravitational field — a region in which a mass experiences gravitational force.

Gravitational field strength (g) — gravitational force per unit mass at a location.

Radial field — a field whose direction is along radial lines toward a central mass.

Uniform field — a field whose magnitude and direction are approximately constant throughout the region considered.

Field line — an imaginary line representing the direction of a field.

Field vector — an arrow representing the magnitude and direction of gravitational field strength at a particular point.

Inverse-square law — a relationship in which a quantity is proportional to 1/r².

Distance from centre (r) — the radial distance between a point and the centre of the spherical mass producing the field.

Altitude — height above the surface of a planet or other astronomical body.

Uniform-field approximation — treating a slightly changing gravitational field as constant over a sufficiently small region.

Radial symmetry — a pattern that is equivalent in every direction from a central point for an isolated spherical source.

Superposition — combining individual gravitational field vectors to determine the net field.

Neutral point — a point where opposing gravitational fields cancel to give zero net field.

Gravitational constant (G) — the constant in Newton's law of gravitation, approximately 6.67 × 10⁻¹¹ N·m²/kg².

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Key Takeaways

• The gravitational field around an isolated spherical planet is a radial field.
• Radial gravitational field lines point inward toward the centre because gravity is attractive.
• Field direction changes from place to place in a radial field.
• Gravitational field strength decreases as distance from the central mass increases.
• Outside a spherical mass, g = GM/r².
• Therefore, gravitational field strength follows an inverse-square relationship with distance.
• Doubling distance from the centre reduces field strength to one-quarter.
• Tripling distance reduces field strength to one-ninth.
• The distance r is measured from the centre of the spherical mass.
• A uniform field has approximately constant magnitude and direction.
• Uniform fields are represented by parallel, equally spaced field lines.
• Earth's gravitational field can be approximated as uniform over small regions near its surface.
• Earth's complete gravitational field is radial, not uniform.
• Radial and uniform models describe the same gravitational interaction at different scales.
• Field-line spacing can represent relative field strength.
• The appropriate model depends on the size of the region being considered compared with the size of the planet.

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Check Your Understanding

1. Describe the shape and direction of the gravitational field around an isolated spherical planet.
2. Explain why gravitational field lines around a planet point inward rather than outward.
3. Describe how the spacing of field lines changes as distance from a planet increases and explain what this indicates.
4. Define a uniform gravitational field and describe how it is represented in a field-line diagram.
5. Explain why Earth's gravitational field can be approximated as uniform inside a classroom even though Earth's actual field is radial.
6. A planet produces a gravitational field strength of 24 N/kg at distance r. Calculate the field strength at 2r.
7. The gravitational field strength at distance r is 90 N/kg. Calculate the field strength at 3r.
8. A spacecraft moves from distance r to distance 5r from a planet's centre. By what factor does the gravitational field strength change?
9. A spacecraft is 1,500 km above the surface of a planet with radius 5,000 km. Determine the value of r that should be used in a radial-field calculation.
10. Explain why doubling the distance from a planet does not simply halve the gravitational field strength.
11. Compare a radial gravitational field and a uniform gravitational field in terms of field-line direction, field-line spacing, and field strength.
12. A student draws Earth's gravitational field using parallel lines from Earth all the way to the Moon. Explain why this model is inappropriate.
13. Planet B has four times the mass and twice the radius of Planet A. Compare their surface gravitational field strengths.
14. Explain why a ball falling a few metres can usually be analysed using a uniform gravitational field, while an orbiting satellite should be analysed using a radial field.
15. Two identical planets are separated in space. Describe the gravitational fields at the midpoint between them and explain why the net gravitational field strength can be zero even though both planets produce a gravitational field there.