Grade by Grade
D. Gravitational, Electric, and Magnetic Fields · Gravitational, Electric and Magnetic Fields

Gravitational Fields: Near Earth and Far Away

D2.1 D2.2 D3.3About 70 minutes
Learning goals
  • I can explain what a field is and define gravitational field strength as g = F/m.
  • I can derive and use g = GM/r² to find the field at any distance from a planet, moon or star.
  • I can draw and interpret field diagrams for a planet seen from far away and for the region near its surface.
  • I can add gravitational fields from two sources as vectors.

Words to know

field
A region of space in which an object feels a force because of a property it has (mass, charge, or moving charge). The field exists at every point, whether or not an object is there to feel it.
gravitational field strength (g)
The gravitational force per unit mass at a point: g = F/m. Units N/kg, which equal m/s². Its direction is the direction of the force on a mass.
test mass
A small mass used (in imagination) to probe a field. It is small enough that it does not change the field it is measuring.
field line
A line drawn so that its direction at every point shows the direction of the field. Closer lines mean a stronger field.
uniform field
A field with the same size and direction everywhere in a region; shown by parallel, evenly spaced lines.
radial field
A field whose lines point straight toward (or away from) a central point, like the field around a planet seen from far away.

Action at a distance

The Sun pulls on Earth across 150 million kilometres of nearly empty space. How does Earth "know" the Sun is there? Newton himself found this puzzling. The modern answer is the field: a mass changes the space around it, and any other mass placed in that space feels a force from the field right where it is.

The idea of fields was developed by Michael Faraday in the 1800s for electricity and magnetism. It turned out to be so useful that physicists now describe all forces with fields. In this unit you will meet three: gravitational, electric and magnetic. Learning the gravitational field first makes the other two much easier.

Gravitational field strength

To measure a field, imagine putting a small test mass m at a point P and measuring the force F on it. The gravitational field strength at P is the force per kilogram:

g = F / m (units: N/kg)

The field is a vector. Its direction is the direction of the force on the test mass: toward the mass creating the field.

Now use Newton's law. If the field is made by a planet of mass M, the force on m at distance r from the planet's centre is F = GMm/r². Divide by m:

g = GM / r²

The test mass has cancelled out. The field depends only on the source (M) and the position (r). That is what makes a field useful: it describes the space, not the object you put in it.

Since 1 N = 1 kg·m/s², the unit N/kg is the same as m/s². So g is also the acceleration of a freely falling object at that point. Near Earth's surface, both readings say 9.8.

mass Mtest mass m at PFrg = F / m= GM / r²(m cancels)
The field at P is the force on a test mass divided by that mass. Since F = GMm/r², the test mass m cancels and g = GM/r².
Worked example

Example 1: Where does 9.8 N/kg come from?

Use g = GM/r² to find the gravitational field strength at Earth's surface (M_E = 5.97 × 10²⁴ kg, R_E = 6.37 × 10⁶ m).

  1. g = (6.67 × 10⁻¹¹)(5.97 × 10²⁴) / (6.37 × 10⁶)².
  2. Numerator: 3.982 × 10¹⁴. Denominator: 4.058 × 10¹³.
  3. g = 9.81 N/kg.

9.81 N/kg, toward Earth's centre. This matches the 9.8 N/kg you used in SPH3U. (The measured value varies slightly with latitude and local geology, from about 9.78 to 9.83 N/kg.)

Worked example

Example 2: The field at the International Space Station

The ISS orbits about 420 km above Earth's surface. Find g there and the gravitational force on a 70 kg astronaut.

  1. r = R_E + h = 6.37 × 10⁶ m + 0.42 × 10⁶ m = 6.79 × 10⁶ m.
  2. g = (6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(6.79 × 10⁶)² = 8.64 N/kg.
  3. F = mg = (70)(8.64) = 605 N.

g ≈ 8.64 N/kg and F ≈ 605 N, which is about 88% of the astronaut's weight on the ground. Astronauts on the ISS are not in a zero-gravity zone; they float because they are in free fall, as you will see in the next lesson.

0246810g (N/kg)9.81Surface0 km8.64ISS420 km0.564GPS20 200 km0.224Geostationary35 800 km0.0027Moon's orbit378 000 km
Gravitational field strength of Earth at different heights above the surface. The ISS is still in a strong field; even at the Moon's orbit the field is not zero.

Field diagrams: far away and up close

Field lines show the field at a glance.

  • Far away (distant field): around a planet or star, the lines point straight in toward the centre from every direction. This is a radial field. The lines get farther apart as you move out, showing the field getting weaker (as 1/r²).
  • Near the surface (near-Earth field): zoom in on a small region, a few kilometres across. Earth's surface looks flat, and the lines look parallel and evenly spaced, all pointing down. This is a uniform field: g is the same size and direction everywhere in the region. That is why you could use a constant g = 9.8 N/kg for projectiles in SPH3U.

Rules for gravitational field lines:

  1. Lines point in the direction of the force on a mass (always toward masses, because gravity only attracts).
  2. Lines never cross (a point can only have one field direction).
  3. The closer together the lines, the stronger the field.
  4. Lines end on masses. They never start on masses, because there is no such thing as negative mass.
EarthFar view: radial fieldlines spread out: field weakens as 1/r²ground (Earth's surface)Close-up: near-Earth fieldparallel, evenly spaced: uniform gzoom in
Left: seen from far away, Earth's field is radial and weakens with distance. Right: in a small region near the surface, the same field looks uniform: parallel, evenly spaced lines pointing down.
Worked example

Example 3: Superposition of fields

Find the net gravitational field halfway between Earth and the Moon (d = 3.84 × 10⁸ m between centres). M_E = 5.97 × 10²⁴ kg, M_M = 7.35 × 10²² kg.

  1. Distance to each centre: r = 1.92 × 10⁸ m.
  2. Earth's field there: g_E = (6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(1.92 × 10⁸)² = 1.080 × 10⁻² N/kg, toward Earth.
  3. Moon's field there: g_M = (6.67 × 10⁻¹¹)(7.35 × 10²²)/(1.92 × 10⁸)² = 1.33 × 10⁻⁴ N/kg, toward the Moon.
  4. The fields point in opposite directions, so subtract: 1.080 × 10⁻² − 0.0133 × 10⁻² = 1.07 × 10⁻² N/kg.

1.07 × 10⁻² N/kg toward Earth. Fields add as vectors, exactly like forces.

Worked example

Example 4: Weighing a planet with a pendulum

A probe on a distant rocky planet measures g = 5.0 N/kg at the surface. Radar shows the planet's radius is 3.0 × 10⁶ m. Find the planet's mass.

  1. Rearrange g = GM/r² to M = gr²/G.
  2. M = (5.0)(3.0 × 10⁶)²/(6.67 × 10⁻¹¹) = (5.0)(9.0 × 10¹²)/(6.67 × 10⁻¹¹).
  3. M = 4.5 × 10¹³/6.67 × 10⁻¹¹ = 6.7 × 10²³ kg.

M ≈ 6.7 × 10²³ kg, a little more than Mars. Measuring g and r is a real way to find the masses of planets and moons.

Field strength and acceleration

Two different things are called g. The gravitational field strength g = GM/r² is about the place. The free-fall acceleration is about the object. They are equal only when gravity is the only force acting. Keep using g = 9.8 N/kg near Earth's surface unless a question asks for more precision.

Just for fun

Physicists love this daydream: dig a tunnel straight through Earth, remove the air (and the molten iron), and jump in. In a simple model where Earth has the same density all the way through, you would reach the far side in about 42 minutes, stop for an instant, and fall all the way back. Please do not try this at home.

Hands-on activity

Measure g with a pendulum

A pendulum's period depends on g. For small swings, T = 2π√(L/g), so g = 4π²L/T². (You will study pendulums and simple harmonic motion in the unit Work, Energy and Simple Harmonic Motion.)

You need: about 1.5 m of thin string, a small heavy object such as a metal nut or a key ring with keys, tape, a metre stick or tape measure, and a phone stopwatch.

  1. Tape the string to the edge of a table or door frame so the weight hangs freely. Measure L from the support point to the centre of the weight. Aim for about 1.00 m.
  2. Pull the weight about 10 cm to the side (a small angle) and let go.
  3. Time 20 complete swings (over and back is one swing). Divide by 20 to get T. Repeat three times and average.
  4. Calculate g = 4π²L/T². Compare with 9.81 N/kg and find the percentage difference.
  5. Change L to 0.50 m and repeat. Does g stay the same?

What to look for: with L = 1.00 m, T should be close to 2.0 s, giving g ≈ 9.8 N/kg. Timing 20 swings instead of 1 reduces the effect of your reaction time. A result within about 2% is good. g should stay the same when L changes, because g describes the place, not the pendulum.

Your turn

1

Define gravitational field strength in words and give its units.

2

Find g at the surface of the Moon (M = 7.35 × 10²² kg, R = 1.74 × 10⁶ m).

3

Find g at the surface of Mars (M = 6.42 × 10²³ kg, R = 3.39 × 10⁶ m).

4

A 70 kg astronaut is 1000 km above Earth's surface. Find g there and the gravitational force on the astronaut.

5

At what height above Earth's surface is g half its surface value?

6

Sketch the gravitational field (a) around the Moon seen from far away and (b) in your classroom. Describe each.

More practice: warm-up

1

What is the unit of g, and what other unit is it equal to?

2

In which direction does Earth's gravitational field point?

3

What do closely spaced field lines tell you?

4

If you move to twice the distance from a planet's centre, g becomes ___.

5

Can two gravitational field lines cross? Why?

More practice: on your own

1

Jupiter has mass 1.90 × 10²⁷ kg and radius 6.99 × 10⁷ m. What would a 60 kg student's weight be at the cloud tops?

2

Find g at geostationary orbit, 4.22 × 10⁷ m from Earth's centre.

3

Use g = 9.8 N/kg and R_E = 6.37 × 10⁶ m to calculate Earth's mass, then its average density.

4

At what height above Earth's surface is g = 9.0 N/kg?

5

Why can SPH3U projectile problems treat g as constant, but satellite problems cannot?

Challenge

1

Planet X has the same density as Earth but twice the radius. Compare its surface g with Earth's.

2

Find the point between Earth and the Moon where the net gravitational field is zero, and explain why a spacecraft there would still not stay put forever.

3

Mount Everest's summit is 8849 m above sea level. By what percentage is g smaller there than at sea level (use the simple formula, ignoring rotation and local mass)?

Check your understanding

Quiz

1.

Gravitational field strength is defined as

2.

The units N/kg are equivalent to

3.

A field diagram has parallel, evenly spaced lines all pointing down. This represents

4.

Planet Y has the same mass as Earth but half the radius. Its surface g is about

5.

Why do gravitational field lines end on masses but never start on them?

6.

On the ISS, 420 km up, Earth's gravitational field strength is closest to

For parents and teachers

Big idea (D2.2, D3.3): a field describes the space around a mass. g = GM/r² gives its size, and field diagrams show its shape: radial when seen from far away, uniform near the surface.

How to help: Ask your teen to explain why the test mass m cancels out, and why astronauts on the ISS are not in "zero gravity" (the field there is 8.6 N/kg). The pendulum activity is safe and quick; you can time the swings while your teen counts.

Common difficulties:

  • Using the altitude instead of the distance from the centre.
  • Thinking g is zero in space. It never reaches zero; it just becomes small far away.
  • Drawing field lines that cross, or that point away from a planet.
  • Mixing up g (field strength, N/kg) and G (universal constant).