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Reading gravitational field and potential graphs

Key idea: Gravitational field is the negative potential gradient. Add potentials as scalars and fields as vectors; zero field can occur where the potential is still negative.

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  • GCE A-Level H2 Physics 2027
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Field strength and potential describe the same field in two ways: one as a force per unit mass, the other as energy per unit mass. If you know how the potential changes with distance, you can find the field: its strength is the negative of the potential gradient, which you can read from the slope of a potential graph, even between two masses.

Field strength is the negative potential gradient

Take outwards as positive and use g for the signed radial component of field strength. The gravitational force component is mg, so the external force needed for a slow outward move is -mg. Its work over a small distance Δ r is -mg Δ r = mΔφ. Divide by mass and distance, then take the limit:

g = -Δφ/(Δ r) or, at a point, g = -dφ/dr

The gravitational field strength at a point is equal to the negative potential gradient at that point. On a graph of φ against r, the gradient is the slope of the tangent at the point, not the value of φ. Its unit, J kg⁻¹ m⁻¹, is the same as N kg⁻¹.

The minus sign tells you the field points towards lower potential. Around a planet the potential rises with r, so the field points inwards.

Potential and its tangent outside Earth

Negative gravitational potential rises towards zero with decreasing slope. A short dashed tangent touches the curve at a centre distance of ten million metres.

Scroll across the figure to read all labels.

Negative gravitational potential rises towards zero with decreasing slope. A short dashed tangent touches the curve at a centre distance of ten million metres.Negative gravitational potential rises towards zero with decreasing slope. A short dashed tangent touches the curve at a centre distance of ten million metres.
Calculated with GME = 3.99 × 10¹⁴ m³ s⁻² and zero potential at infinity, outside radius 6.37 × 10⁶ m. The dashed line is the tangent at r = 1.00 × 10⁷ m.
Open full-size graph
What to noticePotential increases towards zero while its magnitude decreases. The tangent slope at r = 1.00 × 10⁷ m is + 3.99 N kg⁻¹; the radial field component is its negative.
View figure data
Figure data: Potential and its tangent outside Earth
SeriesDistance from Earth's centre, r (10⁷ m)Potential, φ (10⁷ J kg⁻¹)
Potential0.6370-6.2637
Potential0.6491-6.1472
Potential0.6611-6.0350
Potential0.6732-5.9268
Potential0.6853-5.8224
Potential0.6974-5.7216
Potential0.7094-5.6242
Potential0.7215-5.5301
Potential0.7336-5.4391
Potential0.7456-5.3511
Potential0.7577-5.2658
Potential0.7698-5.1832
Potential0.7819-5.1032
Potential0.7939-5.0256
Potential0.8000-4.9875
Potential0.8060-4.9503
Potential0.8181-4.8773
Potential0.8302-4.8064
Potential0.8422-4.7375
Potential0.8543-4.6705
Potential0.8664-4.6054
Potential0.8784-4.5422
Potential0.8905-4.4806
Potential0.9000-4.4333
Potential0.9026-4.4207
Potential0.9147-4.3623
Potential0.9267-4.3055
Potential0.9388-4.2501
Potential0.9509-4.1962
Potential0.9629-4.1436
Potential0.9750-4.0923
Potential0.9871-4.0422
Potential0.9992-3.9934
Potential1.0000-3.9900
Potential1.0112-3.9457
Potential1.0233-3.8991
Potential1.0354-3.8537
Potential1.0474-3.8093
Potential1.0595-3.7659
Potential1.0716-3.7234
Potential1.0837-3.6820
Potential1.0957-3.6414
Potential1.1000-3.6273
Potential1.1078-3.6017
Potential1.1199-3.5629
Potential1.1319-3.5249
Potential1.1440-3.4877
Potential1.1561-3.4513
Potential1.1682-3.4156
Potential1.1802-3.3807
Potential1.1923-3.3465
Potential1.2000-3.3250
Potential1.2044-3.3129
Potential1.2165-3.2800
Potential1.2285-3.2478
Potential1.2406-3.2162
Potential1.2527-3.1852
Potential1.2647-3.1548
Potential1.2768-3.1250
Potential1.2889-3.0957
Potential1.3010-3.0670
Potential1.3130-3.0388
Potential1.3251-3.0111
Potential1.3372-2.9839
Potential1.3492-2.9572
Potential1.3613-2.9310
Potential1.3734-2.9052
Potential1.3855-2.8799
Potential1.3975-2.8550
Potential1.4096-2.8306
Potential1.4217-2.8066
Potential1.4337-2.7829
Potential1.4458-2.7597
Potential1.4579-2.7368
Potential1.4700-2.7144
Potential1.4820-2.6923
Potential1.4941-2.6705
Potential1.5062-2.6491
Potential1.5182-2.6280
Potential1.5303-2.6073
Potential1.5424-2.5869
Potential1.5545-2.5668
Potential1.5665-2.5470
Potential1.5786-2.5275
Potential1.5907-2.5084
Potential1.6027-2.4895
Potential1.6148-2.4709
Potential1.6269-2.4525
Potential1.6390-2.4345
Potential1.6510-2.4167
Potential1.6631-2.3991
Potential1.6752-2.3818
Potential1.6873-2.3648
Potential1.6993-2.3480
Potential1.7114-2.3314
Potential1.7235-2.3151
Potential1.7355-2.2990
Potential1.7476-2.2831
Potential1.7597-2.2675
Potential1.7718-2.2520
Potential1.7838-2.2368
Potential1.7959-2.2217
Potential1.8080-2.2069
Potential1.8200-2.1923
Potential1.8321-2.1778
Potential1.8442-2.1636
Potential1.8563-2.1495
Potential1.8683-2.1356
Potential1.8804-2.1219
Potential1.8925-2.1084
Potential1.9045-2.0950
Potential1.9166-2.0818
Potential1.9287-2.0688
Potential1.9408-2.0559
Potential1.9528-2.0432
Potential1.9649-2.0306
Potential1.9770-2.0182
Potential1.9890-2.0060
Potential2.0000-1.9950
Potential2.0011-1.9939
Potential2.0132-1.9819
Potential2.0253-1.9701
Potential2.0373-1.9584
Potential2.0494-1.9469
Potential2.0615-1.9355
Potential2.0736-1.9242
Potential2.0856-1.9131
Potential2.0977-1.9021
Potential2.1098-1.8912
Potential2.1218-1.8804
Potential2.1339-1.8698
Potential2.1460-1.8593
Potential2.1581-1.8489
Potential2.1701-1.8386
Potential2.1822-1.8284
Potential2.1943-1.8184
Potential2.2063-1.8084
Potential2.2184-1.7986
Potential2.2305-1.7888
Potential2.2426-1.7792
Potential2.2546-1.7697
Potential2.2667-1.7603
Potential2.2788-1.7509
Potential2.2908-1.7417
Potential2.3029-1.7326
Potential2.3150-1.7235
Potential2.3271-1.7146
Potential2.3391-1.7058
Potential2.3512-1.6970
Potential2.3633-1.6883
Potential2.3754-1.6798
Potential2.3874-1.6713
Potential2.3995-1.6629
Potential2.4116-1.6545
Potential2.4236-1.6463
Potential2.4357-1.6381
Potential2.4478-1.6300
Potential2.4599-1.6220
Potential2.4719-1.6141
Potential2.4840-1.6063
Potential2.4961-1.5985
Potential2.5081-1.5908
Potential2.5202-1.5832
Potential2.5323-1.5757
Potential2.5444-1.5682
Potential2.5564-1.5608
Potential2.5685-1.5534
Potential2.5806-1.5462
Potential2.5926-1.5390
Potential2.6047-1.5318
Potential2.6168-1.5248
Potential2.6289-1.5178
Potential2.6409-1.5108
Potential2.6530-1.5040
Potential2.6651-1.4971
Potential2.6771-1.4904
Potential2.6892-1.4837
Potential2.7013-1.4771
Potential2.7134-1.4705
Potential2.7254-1.4640
Potential2.7375-1.4575
Potential2.7496-1.4511
Potential2.7616-1.4448
Potential2.7737-1.4385
Potential2.7858-1.4323
Potential2.7979-1.4261
Potential2.8099-1.4200
Potential2.8220-1.4139
Potential2.8341-1.4079
Potential2.8462-1.4019
Potential2.8582-1.3960
Potential2.8703-1.3901
Potential2.8824-1.3843
Potential2.8944-1.3785
Potential2.9065-1.3728
Potential2.9186-1.3671
Potential2.9307-1.3615
Potential2.9427-1.3559
Potential2.9548-1.3503
Potential2.9669-1.3449
Potential2.9789-1.3394
Potential2.9910-1.3340
Potential3.0031-1.3286
Potential3.0152-1.3233
Potential3.0272-1.3180
Potential3.0393-1.3128
Potential3.0514-1.3076
Potential3.0634-1.3025
Potential3.0755-1.2973
Potential3.0876-1.2923
Potential3.0997-1.2872
Potential3.1117-1.2822
Potential3.1238-1.2773
Potential3.1359-1.2724
Potential3.1479-1.2675
Potential3.1600-1.2626
Potential3.1721-1.2578
Potential3.1842-1.2531
Potential3.1962-1.2483
Potential3.2083-1.2436
Potential3.2204-1.2390
Potential3.2325-1.2344
Potential3.2445-1.2298
Potential3.2566-1.2252
Potential3.2687-1.2207
Potential3.2807-1.2162
Potential3.2928-1.2117
Potential3.3049-1.2073
Potential3.3170-1.2029
Potential3.3290-1.1985
Potential3.3411-1.1942
Potential3.3532-1.1899
Potential3.3652-1.1856
Potential3.3773-1.1814
Potential3.3894-1.1772
Potential3.4015-1.1730
Potential3.4135-1.1689
Potential3.4256-1.1648
Potential3.4377-1.1607
Potential3.4497-1.1566
Potential3.4618-1.1526
Potential3.4739-1.1486
Potential3.4860-1.1446
Potential3.4980-1.1406
Potential3.5101-1.1367
Potential3.5222-1.1328
Potential3.5343-1.1290
Potential3.5463-1.1251
Potential3.5584-1.1213
Potential3.5705-1.1175
Potential3.5825-1.1137
Potential3.5946-1.1100
Potential3.6067-1.1063
Potential3.6188-1.1026
Potential3.6308-1.0989
Potential3.6429-1.0953
Potential3.6550-1.0917
Potential3.6670-1.0881
Potential3.6791-1.0845
Potential3.6912-1.0810
Potential3.7033-1.0774
Potential3.7153-1.0739
Potential3.7274-1.0705
Potential3.7395-1.0670
Potential3.7515-1.0636
Potential3.7636-1.0602
Potential3.7757-1.0568
Potential3.7878-1.0534
Potential3.7998-1.0500
Potential3.8119-1.0467
Potential3.8240-1.0434
Potential3.8360-1.0401
Potential3.8481-1.0369
Potential3.8602-1.0336
Potential3.8723-1.0304
Potential3.8843-1.0272
Potential3.8964-1.0240
Potential3.9085-1.0209
Potential3.9205-1.0177
Potential3.9326-1.0146
Potential3.9447-1.0115
Potential3.9568-1.0084
Potential3.9688-1.0053
Potential3.9809-1.0023
Potential3.9930-0.9993
Potential4.0051-0.9962
Potential4.0171-0.9932
Potential4.0292-0.9903
Potential4.0413-0.9873
Potential4.0533-0.9844
Potential4.0654-0.9815
Potential4.0775-0.9785
Potential4.0896-0.9757
Potential4.1016-0.9728
Potential4.1137-0.9699
Potential4.1258-0.9671
Potential4.1378-0.9643
Potential4.1499-0.9615
Potential4.1620-0.9587
Potential4.1741-0.9559
Potential4.1861-0.9531
Potential4.1982-0.9504
Potential4.2103-0.9477
Potential4.2200-0.9455
Potential4.2223-0.9450
Potential4.2344-0.9423
Potential4.2465-0.9396
Potential4.2586-0.9369
Potential4.2706-0.9343
Potential4.2827-0.9317
Potential4.2948-0.9290
Potential4.3068-0.9264
Potential4.3189-0.9238
Potential4.3310-0.9213
Potential4.3431-0.9187
Potential4.3551-0.9162
Potential4.3672-0.9136
Potential4.3793-0.9111
Potential4.3914-0.9086
Potential4.4034-0.9061
Potential4.4155-0.9036
Potential4.4276-0.9012
Potential4.4396-0.8987
Potential4.4517-0.8963
Potential4.4638-0.8939
Potential4.4759-0.8914
Potential4.4879-0.8891
Potential4.5000-0.8867
Tangent at r = 10⁷ m0.8000-4.7880
Tangent at r = 10⁷ m0.9000-4.3890
Tangent at r = 10⁷ m1.0000-3.9900
Tangent at r = 10⁷ m1.1000-3.5910
Tangent at r = 10⁷ m1.2000-3.1920

For a point mass, φ = -GM/r has gradient dφ/dr = +GM/r², so g = -GM/r²: a field of size GM/r² pointing inwards, as in the previous lesson. Near the Earth’s surface, using g₀ for the positive field magnitude, the same idea gives Δφ ≈ g₀Δ h, which gives Δ UG ≈ mg₀Δ h when the field magnitude is nearly constant.

Gravitational field magnitude outside Earth

A positive field-magnitude curve decreases as inverse distance squared outside the surface. The field itself points inward.

Scroll across the figure to read all labels.

A positive field-magnitude curve decreases as inverse distance squared outside the surface. The field itself points inward.A positive field-magnitude curve decreases as inverse distance squared outside the surface. The field itself points inward.
Calculated for an ideal spherical Earth with GME = 3.99 × 10¹⁴ m³ s⁻² and radius 6.37 × 10⁶ m. Only exterior points are shown. Doubling centre distance quarters the field magnitude.
Open full-size graph
What to noticeCompare r = 1.00 × 10⁷ m and 2.00 × 10⁷ m: the field magnitude falls from 3.99 to 0.9975 N kg⁻¹.
View figure data
Figure data: Gravitational field magnitude outside Earth
Distance from Earth's centre, r (10⁷ m)Field magnitude
0.63709.8332
0.64919.4708
0.66119.1281
0.67328.8037
0.68538.4962
0.69748.2046
0.70947.9278
0.72157.6647
0.73367.4145
0.74567.1764
0.75776.9496
0.76986.7333
0.78196.5270
0.79396.3300
0.80006.2344
0.80606.1418
0.81815.9619
0.83025.7897
0.84225.6250
0.85435.4671
0.86645.3158
0.87845.1707
0.89055.0315
0.90004.9259
0.90264.8978
0.91474.7694
0.92674.6459
0.93884.5272
0.95094.4130
0.96294.3030
0.97504.1971
0.98714.0951
0.99923.9967
1.00003.9900
1.01123.9019
1.02333.8104
1.03543.7220
1.04743.6367
1.05953.5543
1.07163.4747
1.08373.3977
1.09573.3233
1.10003.2975
1.10783.2512
1.11993.1815
1.13193.1140
1.14403.0486
1.15612.9853
1.16822.9239
1.18022.8644
1.19232.8067
1.20002.7708
1.20442.7507
1.21652.6964
1.22852.6437
1.24062.5925
1.25272.5427
1.26472.4944
1.27682.4475
1.28892.4019
1.30102.3575
1.31302.3143
1.32512.2724
1.33722.2315
1.34922.1918
1.36132.1531
1.37342.1154
1.38552.0787
1.39752.0429
1.40962.0081
1.42171.9741
1.43371.9410
1.44581.9087
1.45791.8773
1.47001.8466
1.48201.8166
1.49411.7874
1.50621.7588
1.51821.7310
1.53031.7038
1.54241.6772
1.55451.6512
1.56651.6259
1.57861.6011
1.59071.5769
1.60271.5532
1.61481.5301
1.62691.5075
1.63901.4854
1.65101.4637
1.66311.4426
1.67521.4218
1.68731.4016
1.69931.3817
1.71141.3623
1.72351.3433
1.73551.3247
1.74761.3064
1.75971.2886
1.77181.2711
1.78381.2539
1.79591.2371
1.80801.2206
1.82001.2045
1.83211.1887
1.84421.1732
1.85631.1580
1.86831.1430
1.88041.1284
1.89251.1141
1.90451.1000
1.91661.0862
1.92871.0726
1.94081.0593
1.95281.0463
1.96491.0334
1.97701.0209
1.98901.0085
2.00000.9975
2.00110.9964
2.01320.9845
2.02530.9728
2.03730.9613
2.04940.9500
2.06150.9389
2.07360.9280
2.08560.9173
2.09770.9067
2.10980.8964
2.12180.8862
2.13390.8762
2.14600.8664
2.15810.8567
2.17010.8472
2.18220.8379
2.19430.8287
2.20630.8196
2.21840.8108
2.23050.8020
2.24260.7934
2.25460.7849
2.26670.7766
2.27880.7684
2.29080.7603
2.30290.7523
2.31500.7445
2.32710.7368
2.33910.7292
2.35120.7218
2.36330.7144
2.37540.7072
2.38740.7000
2.39950.6930
2.41160.6861
2.42360.6793
2.43570.6725
2.44780.6659
2.45990.6594
2.47190.6530
2.48400.6467
2.49610.6404
2.50810.6343
2.52020.6282
2.53230.6222
2.54440.6163
2.55640.6105
2.56850.6048
2.58060.5992
2.59260.5936
2.60470.5881
2.61680.5827
2.62890.5773
2.64090.5721
2.65300.5669
2.66510.5618
2.67710.5567
2.68920.5517
2.70130.5468
2.71340.5419
2.72540.5372
2.73750.5324
2.74960.5278
2.76160.5232
2.77370.5186
2.78580.5141
2.79790.5097
2.80990.5053
2.82200.5010
2.83410.4968
2.84620.4926
2.85820.4884
2.87030.4843
2.88240.4803
2.89440.4763
2.90650.4723
2.91860.4684
2.93070.4646
2.94270.4608
2.95480.4570
2.96690.4533
2.97890.4496
2.99100.4460
3.00310.4424
3.01520.4389
3.02720.4354
3.03930.4319
3.05140.4285
3.06340.4252
3.07550.4218
3.08760.4185
3.09970.4153
3.11170.4121
3.12380.4089
3.13590.4057
3.14790.4026
3.16000.3996
3.17210.3965
3.18420.3935
3.19620.3906
3.20830.3876
3.22040.3847
3.23250.3819
3.24450.3790
3.25660.3762
3.26870.3734
3.28070.3707
3.29280.3680
3.30490.3653
3.31700.3627
3.32900.3600
3.34110.3574
3.35320.3549
3.36520.3523
3.37730.3498
3.38940.3473
3.40150.3449
3.41350.3424
3.42560.3400
3.43770.3376
3.44970.3353
3.46180.3329
3.47390.3306
3.48600.3283
3.49800.3261
3.51010.3238
3.52220.3216
3.53430.3194
3.54630.3173
3.55840.3151
3.57050.3130
3.58250.3109
3.59460.3088
3.60670.3067
3.61880.3047
3.63080.3027
3.64290.3007
3.65500.2987
3.66700.2967
3.67910.2948
3.69120.2928
3.70330.2909
3.71530.2891
3.72740.2872
3.73950.2853
3.75150.2835
3.76360.2817
3.77570.2799
3.78780.2781
3.79980.2763
3.81190.2746
3.82400.2729
3.83600.2711
3.84810.2694
3.86020.2678
3.87230.2661
3.88430.2644
3.89640.2628
3.90850.2612
3.92050.2596
3.93260.2580
3.94470.2564
3.95680.2549
3.96880.2533
3.98090.2518
3.99300.2503
4.00510.2487
4.01710.2473
4.02920.2458
4.04130.2443
4.05330.2429
4.06540.2414
4.07750.2400
4.08960.2386
4.10160.2372
4.11370.2358
4.12580.2344
4.13780.2330
4.14990.2317
4.16200.2303
4.17410.2290
4.18610.2277
4.19820.2264
4.21030.2251
4.22000.2241
4.22230.2238
4.23440.2225
4.24650.2213
4.25860.2200
4.27060.2188
4.28270.2175
4.29480.2163
4.30680.2151
4.31890.2139
4.33100.2127
4.34310.2115
4.35510.2104
4.36720.2092
4.37930.2080
4.39140.2069
4.40340.2058
4.41550.2047
4.42760.2035
4.43960.2024
4.45170.2013
4.46380.2002
4.47590.1992
4.48790.1981
4.50000.1970

Worked example 1

Field strength from a tangent

Problem

On the potential graph above, the tangent drawn at r = 1.00 × 10⁷ m rises by 0.399 × 10⁷ J kg⁻¹ over a distance of 0.100 × 10⁷ m. Find the gravitational field strength at that point.

Show full solution

  1. Gradient of the tangent

    Method

    Divide the rise of the tangent by its run.

    Reason

    The field depends on the slope of the graph at the point, not on the value of φ there.

    Working

    Answer

    dφ/dr = (0.399 × 10⁷)/(0.100 × 10⁷) = +3.99 N kg⁻¹

  2. Field strength

    Working

    Answer

    g = -dφ/dr = -3.99 N kg⁻¹: a field of 3.99 N kg⁻¹ pointing towards the Earth.

Guided practice 2

Average field over a short distance

About 3 min

Problem

Moving outwards by 2.00 × 10⁵ m, the gravitational potential rises from -6.00 × 10⁷ J kg⁻¹ to -5.80 × 10⁷ J kg⁻¹. Find the potential change, the chord gradient and the average field strength over this distance, taking outwards as positive.

Find the potential change, gradient and field

Show the steps Optional working
Unit: J kg⁻¹
Unit: N kg⁻¹
Unit: N kg⁻¹

Hints

Hint 1

gradient, then sign

Divide the change in potential by the distance, then reverse the sign.

Show full solution

  1. Field strength

    Working

    Answer

    Δφ/(Δ r) = (+2.00 × 10⁶)/(2.00 × 10⁵) = +10.0 N kg⁻¹, so g = -10.0 N kg⁻¹: 10.0 N kg⁻¹ inwards.

Try it yourself 3

Field from the potential equation

Minimal support

Problem

For a planet with GM = 3.20 × 10¹⁴ N m² kg⁻¹, use the gradient of φ = -GM/r to find the field strength at r = 8.00 × 10⁶ m, taking outwards as positive. What is its size at twice this distance, to three significant figures?

Find both values

Unit: N kg⁻¹
Unit: N kg⁻¹

Hints

Hint 1

differentiate

Write the potential as a power of r before differentiating.

Show solution step by step

  1. Gradient and field

    Reason

    g = -dφ/dr.

    Working

    Answer

    dφ/dr = GM/r² = (3.20 × 10¹⁴)/((8.00 × 10⁶)²) = 5.00 N kg⁻¹, so g = -5.00 N kg⁻¹

  2. Twice the distance

    Working

    Answer

    The field falls as 1/r², so its size is 5.00/4 = 1.25 N kg⁻¹.

Check your understanding 1 Derive the signed field

Starting from φ = -GM/r, show by differentiation why the radial field is -GM/r² when outwards is positive. Explain why the size of the field at twice the distance is one quarter, while the potential’s magnitude is only halved. You may work on paper.

Show answer

Compare the signs and powers: φ = -GM r⁻¹ gives dφ/dr = +GM r⁻²; g = -dφ/dr = -GM/r² points inwards. Doubling r quarters |g| but halves |φ|. Field is a gradient of potential, not its plotted value.

Between two masses

Put a mass M at x = 0 and a mass 4M at x = 3d. Potentials are scalars, so add them; both are negative. Fields are vectors: between the masses they point in opposite directions and can cancel. Taking right as positive,

φ = -GM/x-4GM/(3d-x) g = -GM/x² + 4GM/((3d-x)²)

The graphs plot x in units of d, φ in units of GM/d and g in units of GM/d².

Potential between two unequal masses

A negative potential curve has a maximum at x divided by d equals one; its value there is minus three.

Scroll across the figure to read all labels.

A negative potential curve has a maximum at x divided by d equals one; its value there is minus three.A negative potential curve has a maximum at x divided by d equals one; its value there is minus three.
Point masses M at x = 0 and 4M at x = 3d. The plotted interval stays away from both sources. Potential is scaled by GM/d and position by d.
Open full-size graph
What to noticeAt x = d, the tangent is horizontal although potential is negative. On either side, the slope changes sign.
View figure data
Figure data: Potential between two unequal masses
Position, x/d (1)Total potential
0.500-3.600
0.505-3.583
0.510-3.567
0.515-3.551
0.520-3.536
0.525-3.521
0.530-3.506
0.535-3.492
0.540-3.478
0.545-3.464
0.550-3.451
0.555-3.438
0.560-3.425
0.565-3.413
0.570-3.400
0.575-3.389
0.580-3.377
0.585-3.366
0.590-3.355
0.595-3.344
0.600-3.333
0.605-3.323
0.610-3.313
0.615-3.303
0.620-3.294
0.625-3.284
0.630-3.275
0.635-3.266
0.640-3.257
0.645-3.249
0.650-3.241
0.655-3.232
0.660-3.225
0.665-3.217
0.670-3.209
0.675-3.202
0.680-3.195
0.685-3.188
0.690-3.181
0.695-3.174
0.700-3.168
0.705-3.161
0.710-3.155
0.715-3.149
0.720-3.143
0.725-3.138
0.730-3.132
0.735-3.127
0.740-3.121
0.745-3.116
0.750-3.111
0.755-3.106
0.760-3.102
0.765-3.097
0.770-3.092
0.775-3.088
0.780-3.084
0.785-3.080
0.790-3.076
0.795-3.072
0.800-3.068
0.805-3.065
0.810-3.061
0.815-3.058
0.820-3.054
0.825-3.051
0.830-3.048
0.835-3.045
0.840-3.042
0.845-3.040
0.850-3.037
0.855-3.034
0.860-3.032
0.865-3.030
0.870-3.027
0.875-3.025
0.880-3.023
0.885-3.021
0.890-3.019
0.895-3.018
0.900-3.016
0.905-3.014
0.910-3.013
0.915-3.011
0.920-3.010
0.925-3.009
0.930-3.008
0.935-3.007
0.940-3.006
0.945-3.005
0.950-3.004
0.955-3.003
0.960-3.002
0.965-3.002
0.970-3.001
0.975-3.001
0.980-3.001
0.985-3.000
0.990-3.000
0.995-3.000
1.000-3.000
1.005-3.000
1.010-3.000
1.015-3.000
1.020-3.001
1.025-3.001
1.030-3.001
1.035-3.002
1.040-3.002
1.045-3.003
1.050-3.004
1.055-3.004
1.060-3.005
1.065-3.006
1.070-3.007
1.075-3.008
1.080-3.009
1.085-3.010
1.090-3.012
1.095-3.013
1.100-3.014
1.105-3.016
1.110-3.017
1.115-3.019
1.120-3.021
1.125-3.022
1.130-3.024
1.135-3.026
1.140-3.028
1.145-3.030
1.150-3.032
1.155-3.034
1.160-3.036
1.165-3.038
1.170-3.040
1.175-3.043
1.180-3.045
1.185-3.048
1.190-3.050
1.195-3.053
1.200-3.056
1.205-3.058
1.210-3.061
1.215-3.064
1.220-3.067
1.225-3.070
1.230-3.073
1.235-3.076
1.240-3.079
1.245-3.082
1.250-3.086
1.255-3.089
1.260-3.093
1.265-3.096
1.270-3.100
1.275-3.103
1.280-3.107
1.285-3.111
1.290-3.114
1.295-3.118
1.300-3.122
1.305-3.126
1.310-3.130
1.315-3.134
1.320-3.139
1.325-3.143
1.330-3.147
1.335-3.151
1.340-3.156
1.345-3.160
1.350-3.165
1.355-3.170
1.360-3.174
1.365-3.179
1.370-3.184
1.375-3.189
1.380-3.194
1.385-3.199
1.390-3.204
1.395-3.209
1.400-3.214
1.405-3.220
1.410-3.225
1.415-3.230
1.420-3.236
1.425-3.241
1.430-3.247
1.435-3.253
1.440-3.259
1.445-3.264
1.450-3.270
1.455-3.276
1.460-3.282
1.465-3.288
1.470-3.295
1.475-3.301
1.480-3.307
1.485-3.314
1.490-3.320
1.495-3.327
1.500-3.333
1.505-3.340
1.510-3.347
1.515-3.354
1.520-3.361
1.525-3.368
1.530-3.375
1.535-3.382
1.540-3.389
1.545-3.396
1.550-3.404
1.555-3.411
1.560-3.419
1.565-3.426
1.570-3.434
1.575-3.442
1.580-3.450
1.585-3.458
1.590-3.466
1.595-3.474
1.600-3.482
1.605-3.490
1.610-3.499
1.615-3.507
1.620-3.516
1.625-3.524
1.630-3.533
1.635-3.542
1.640-3.551
1.645-3.560
1.650-3.569
1.655-3.578
1.660-3.587
1.665-3.597
1.670-3.606
1.675-3.616
1.680-3.626
1.685-3.635
1.690-3.645
1.695-3.655
1.700-3.665
1.705-3.675
1.710-3.686
1.715-3.696
1.720-3.706
1.725-3.717
1.730-3.728
1.735-3.738
1.740-3.749
1.745-3.760
1.750-3.771
1.755-3.783
1.760-3.794
1.765-3.805
1.770-3.817
1.775-3.829
1.780-3.840
1.785-3.852
1.790-3.864
1.795-3.877
1.800-3.889
1.805-3.901
1.810-3.914
1.815-3.926
1.820-3.939
1.825-3.952
1.830-3.965
1.835-3.978
1.840-3.992
1.845-4.005
1.850-4.019
1.855-4.033
1.860-4.046
1.865-4.060
1.870-4.075
1.875-4.089
1.880-4.103
1.885-4.118
1.890-4.133
1.895-4.148
1.900-4.163
1.905-4.178
1.910-4.193
1.915-4.209
1.920-4.225
1.925-4.240
1.930-4.256
1.935-4.273
1.940-4.289
1.945-4.306
1.950-4.322
1.955-4.339
1.960-4.356
1.965-4.374
1.970-4.391
1.975-4.409
1.980-4.427
1.985-4.445
1.990-4.463
1.995-4.481
2.000-4.500

Field component between two unequal masses

The signed field crosses zero at x divided by d equals one. It points left before that position and right after it.

Scroll across the figure to read all labels.

The signed field crosses zero at x divided by d equals one. It points left before that position and right after it.The signed field crosses zero at x divided by d equals one. It points left before that position and right after it.
Same two sources and position scale as the potential plot. Right is positive; the field component is scaled by GM/d².
Open full-size graph
What to noticeThe smaller left mass dominates near the left source. At x = d the fields cancel; farther right the larger right mass dominates.
View figure data
Figure data: Field component between two unequal masses
Position, x/d (1)Resultant field
0.500-3.36000
0.505-3.27862
0.510-3.19952
0.515-3.12263
0.520-3.04786
0.525-2.97512
0.530-2.90434
0.535-2.83545
0.540-2.76837
0.545-2.70304
0.550-2.63940
0.555-2.57737
0.560-2.51691
0.565-2.45796
0.570-2.40047
0.575-2.34437
0.580-2.28964
0.585-2.23621
0.590-2.18404
0.595-2.13310
0.600-2.08333
0.605-2.03471
0.610-1.98718
0.615-1.94072
0.620-1.89529
0.625-1.85086
0.630-1.80739
0.635-1.76485
0.640-1.72322
0.645-1.68246
0.650-1.64255
0.655-1.60346
0.660-1.56517
0.665-1.52765
0.670-1.49087
0.675-1.45482
0.680-1.41947
0.685-1.38480
0.690-1.35079
0.695-1.31742
0.700-1.28467
0.705-1.25253
0.710-1.22097
0.715-1.18998
0.720-1.15954
0.725-1.12964
0.730-1.10026
0.735-1.07139
0.740-1.04300
0.745-1.01510
0.750-0.98765
0.755-0.96066
0.760-0.93411
0.765-0.90798
0.770-0.88227
0.775-0.85695
0.780-0.83203
0.785-0.80749
0.790-0.78332
0.795-0.75951
0.800-0.73605
0.805-0.71293
0.810-0.69015
0.815-0.66768
0.820-0.64553
0.825-0.62368
0.830-0.60213
0.835-0.58087
0.840-0.55989
0.845-0.53919
0.850-0.51875
0.855-0.49857
0.860-0.47864
0.865-0.45896
0.870-0.43952
0.875-0.42031
0.880-0.40133
0.885-0.38256
0.890-0.36401
0.895-0.34567
0.900-0.32754
0.905-0.30960
0.910-0.29185
0.915-0.27429
0.920-0.25692
0.925-0.23972
0.930-0.22269
0.935-0.20583
0.940-0.18914
0.945-0.17260
0.950-0.15622
0.955-0.13999
0.960-0.12390
0.965-0.10796
0.970-0.09215
0.975-0.07648
0.980-0.06094
0.985-0.04552
0.990-0.03023
0.995-0.01506
1.0000.00000
1.0050.01494
1.0100.02978
1.0150.04451
1.0200.05914
1.0250.07366
1.0300.08809
1.0350.10243
1.0400.11668
1.0450.13084
1.0500.14491
1.0550.15890
1.0600.17282
1.0650.18665
1.0700.20042
1.0750.21411
1.0800.22773
1.0850.24129
1.0900.25478
1.0950.26821
1.1000.28159
1.1050.29490
1.1100.30817
1.1150.32138
1.1200.33454
1.1250.34765
1.1300.36072
1.1350.37375
1.1400.38674
1.1450.39968
1.1500.41259
1.1550.42547
1.1600.43831
1.1650.45112
1.1700.46391
1.1750.47667
1.1800.48940
1.1850.50211
1.1900.51480
1.1950.52747
1.2000.54012
1.2050.55276
1.2100.56539
1.2150.57800
1.2200.59060
1.2250.60320
1.2300.61579
1.2350.62838
1.2400.64096
1.2450.65354
1.2500.66612
1.2550.67871
1.2600.69130
1.2650.70389
1.2700.71650
1.2750.72911
1.2800.74173
1.2850.75437
1.2900.76702
1.2950.77968
1.3000.79237
1.3050.80507
1.3100.81779
1.3150.83054
1.3200.84331
1.3250.85611
1.3300.86893
1.3350.88179
1.3400.89467
1.3450.90759
1.3500.92054
1.3550.93353
1.3600.94655
1.3650.95962
1.3700.97272
1.3750.98587
1.3800.99906
1.3851.01229
1.3901.02558
1.3951.03891
1.4001.05230
1.4051.06573
1.4101.07922
1.4151.09277
1.4201.10637
1.4251.12004
1.4301.13376
1.4351.14755
1.4401.16140
1.4451.17532
1.4501.18931
1.4551.20337
1.4601.21749
1.4651.23170
1.4701.24597
1.4751.26033
1.4801.27476
1.4851.28928
1.4901.30388
1.4951.31856
1.5001.33333
1.5051.34819
1.5101.36314
1.5151.37819
1.5201.39332
1.5251.40856
1.5301.42389
1.5351.43933
1.5401.45487
1.5451.47051
1.5501.48626
1.5551.50213
1.5601.51810
1.5651.53419
1.5701.55039
1.5751.56671
1.5801.58316
1.5851.59972
1.5901.61642
1.5951.63324
1.6001.65019
1.6051.66728
1.6101.68450
1.6151.70186
1.6201.71936
1.6251.73700
1.6301.75480
1.6351.77274
1.6401.79083
1.6451.80907
1.6501.82748
1.6551.84604
1.6601.86477
1.6651.88366
1.6701.90273
1.6751.92196
1.6801.94138
1.6851.96097
1.6901.98074
1.6952.00070
1.7002.02084
1.7052.04118
1.7102.06172
1.7152.08245
1.7202.10339
1.7252.12453
1.7302.14588
1.7352.16745
1.7402.18923
1.7452.21124
1.7502.23347
1.7552.25593
1.7602.27863
1.7652.30156
1.7702.32474
1.7752.34816
1.7802.37183
1.7852.39576
1.7902.41995
1.7952.44441
1.8002.46914
1.8052.49414
1.8102.51942
1.8152.54498
1.8202.57084
1.8252.59699
1.8302.62345
1.8352.65021
1.8402.67728
1.8452.70467
1.8502.73239
1.8552.76044
1.8602.78882
1.8652.81754
1.8702.84662
1.8752.87605
1.8802.90584
1.8852.93600
1.8902.96654
1.8952.99746
1.9003.02878
1.9053.06049
1.9103.09260
1.9153.12514
1.9203.15809
1.9253.19147
1.9303.22529
1.9353.25956
1.9403.29428
1.9453.32947
1.9503.36513
1.9553.40128
1.9603.43792
1.9653.47506
1.9703.51271
1.9753.55089
1.9803.58960
1.9853.62885
1.9903.66867
1.9953.70904
2.0003.75000

At x = d, the distances are d and 2d, and the fields cancel: -GM/d² + 4GM/(2d)² = 0. The potential graph has a horizontal tangent there, because g = -dφ/dx = 0. But the potential itself is -GM/d-4GM/2d = -3GM/d, not zero. Zero field does not mean zero potential.

Check your understanding 2 The midpoint

Is the field zero at the midpoint, x = 1.5d, between the masses M and 4M?

Field at the midpoint
Show answer

No. The distances are equal there, but the right-hand mass is four times larger, so the field points towards it. The fields cancel nearer the smaller mass, at x = d.

Try it yourself 4

Midway between two equal masses

Minimal support

Problem

Two equal point masses M are a distance 2d apart. Find the gravitational potential, in units of GM/d, and the field strength, in units of GM/d², at the point midway between them. What does the potential graph look like along the line joining the masses there?

Find the potential and the field

Unit: GM/d
Unit: GM/d²
Potential graph along the line of centres

Hints

Hint 1

scalars and vectors

Add the potentials as numbers; add the fields as vectors, with their directions.

Show solution step by step

  1. Potential

    Reason

    Each mass is a distance d away, and potentials add.

    Working

    Answer

    φ = -GM/d-GM/d = -2GM/d

  2. Field and graph

    Working

    Answer

    The two fields are equal and opposite, so g = 0. The potential graph has a maximum there, with a horizontal tangent, at -2GM/d.

Check your understanding 3 A descending potential graph

Along a line between two masses, take increasing x to the right. At a point, the tangent falls by 6.00 × 10⁶ J kg⁻¹ for a rightward run of 2.00 × 10⁶ m. Calculate the signed field component. If the potential at the point is negative, does that change the field direction?

Unit: N kg⁻¹
Does negative potential change this direction?
Show answer

The tangent gradient is -6.00 × 10⁶/(2.00 × 10⁶) = -3.00 N kg⁻¹, so g = +3.00 N kg⁻¹, to the right. The negative potential value does not alter that direction.

Common mistakes

  • Reading the field strength from the value of φ instead of from the gradient of the graph.
  • Leaving out the minus sign, so the field points towards higher potential.
  • Using a chord across a curved graph when the field at one point is needed; draw the tangent.
  • Thinking zero field means zero potential.

Spotted an error? Report a correction

Syllabus and review details

Last reviewed:

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