Gravitational & Electric FieldsJessica Wade (jess.wade@kcl.ac.uk)

www.makingphysicsfun.comDepartment of Physics & Centre for Plastic Electronics, Imperial College LondonFaculty of Natural & Mathematical Sciences, King’s College London

• Newton: there is an attractive force between all objects with mass

• Forces always occur in pairs: object is pulled by earth, earth is pulled by object

• Uniform gravitational field:Gravitational force (FG) = 𝑚𝑎𝑠𝑠  ×  𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛  𝑜𝑓  𝑓𝑟𝑒𝑒  𝑓𝑎𝑙𝑙

𝐹0 = 𝑚  ×  𝑔

• Gravitational field strength = 345= 56

5= g

• What is weight?• Weight of an object is the gravitational force exerted on

that object by the mass of the Earth

Gravitational Field

• Newton proposed that the strength of the Earth’s gravitational field varies inversely with the square from its centre

𝐹 = −𝐺𝑚:𝑚;𝑟;

𝑚:𝑔 = 𝐺𝑚:𝑀𝑟; → 𝑔 =

𝐺𝑀𝑟;

G = Gravitational Constant = 6.67 x 10-11 Nm2kg-1

Non-uniform gravitational fields

• Work is done when a force moves somethingWork done = Force x Distance moved in the direction of the force

• A system has energy if it is capable of doing work

• Gravitational Potential Energy = W = mgh• The total energy of a system is conserved

• Change in GPE = ∆𝐸 = 𝑚𝑔∆ℎ

Gravitational Fields

• Why is gravitational potential energy negative?• Object of mass m in empty space, r∞ away from any other

massive body • Force = GMm/r∞2, but as it is at infinity, there are 0 forces

acting upon it• Cannot fall toward anything à no potential energy à

cannot do any work (GPE = 0)• Mass ‘m’ now sits on Earth• To move away, give it energy. Gets to infinity = 0 GPE• Only way to ‘balance’ is to say it has negative GPE on Earth

Gravitation Potential in a Radial Field

𝐹 = −𝐺𝑀𝑚𝑟;

𝐸A = −𝐺𝑀𝑚B1𝑟;

DE

DFdr = GMm

1𝑟;−1𝑟:

• Gravitational potential is the change in potential energy for a unit mass that moves from infinity to a point at less than infinity (m = 1)

V =−𝐺𝑀𝑟

GPE in Radial Fields

Variations of G with r

V =−𝐺𝑀𝑟

𝑔 = −∆𝑉∆𝑟

Variations of g with r

• Kinetic energy of an asteroid falling to earth

• Loss of gravitational potential = −0NDO

per unit mass

• Gain in KE = loss in GPE • GPE = GP x mass of asteroid

• 𝐸P =:;m𝑣; = −0NO5

DO• Can also calculate escape velocty from massive body:

𝑣RST = 2𝐺𝑀V

𝑟V

Energy of an asteroid falling to Earth

• A satellite moves in a circular orbit with an inward gravitational acceleration g and speed v:

𝑔 =𝑣;

𝑟• Speed of a satellite, 𝑣 = 𝑔𝑟• Geostationary/ Geosynchronous satellites stay still

relative to Earth

Circumference = 2𝜋𝑟S = 𝑣𝑡Where 𝑟S = orbital radius, t = 24 hours = 24 x 60 x 60 seconds

Satellites in Orbit

𝑣 =2𝜋𝑟S𝑡

𝑣 = 𝑔𝑟S𝑔 =

𝐺𝑀𝑟S;

𝑣 =2𝜋𝑟S𝑡 =

𝐺𝑀𝑟S;

𝑟S =𝐺𝑀𝑟S

Geostationary Satellites

• Charged bodies exert a force on each other:• Any charged body in the space around another charged

body is acted on by an electric field• The field between two parallel charged plates is uniform• What is the definition of Electric Field Strength?• Electric field strength = Force [N] on each coulomb of

charge

Electric Fields

• Work done by a force of ‘F’ moving through plates of separation ‘d’𝑊𝑜𝑟𝑘  𝐷𝑜𝑛𝑒 = 𝑄  ×  𝑉 = 𝐹  ×  𝑑

[F]=N, [d]=m, [Q]=C, [V]=V=JC-1

𝐹𝑄=𝑉𝑑

• The magnitude of a uniform electric field:

𝐸 =𝑉𝑑

[E]=V m-1

• Calculating the speed of moving charges from an electron gun• Thermionic Emission: Electrons with enough energy escape the surface of the

wire • Charges accelerate between filament and anode, gaining KE in E:

• 𝐾𝐸 = :;𝑚𝑣

; = 𝑒𝑉

Moving Charges

• Direction of a positive charge (from positive to negative)

• Strength of field = spacing of lines

• Arrows on lines = direction of electric field

• Parallel, evenly spaced lines = uniform electric field strength

Direction of an Electric Field

• The voltage measured in the fieldbetween two plates is the electricpotential

• Electric potential is the potential difference between the 0 V plate and the probe (voltmeter)

• Equipotentials are always at right angles to field lines

• Take care at corners of plates where field no longer uniform𝐹𝑖𝑒𝑙𝑑  𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ = − 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙  𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡

𝐸 = −𝑑𝑉𝑑𝑟

Electric Potential

• Electric field between two parallel plates can store charge (capacitor)

• Charge on plates ∝ potential difference CV• Charge on plates ∝ area plates

𝑄𝐴 ∝

𝑉𝑑

• Medium between plates (dielectric) is an insulator 𝑄𝐴 = 𝜀b

𝑉𝑑

• 𝜀b is the permittivity of free space, [𝜀b] = F m-1

• A 1 farad capacitor charged by a potential difference of 1 volt carries a charge of 1 coloumb

Parallel Plate Capacitor

• Capacitance:

𝐶 =𝑄𝑉 = 𝜀b

𝐴𝑑

𝐶 = 𝜀b𝜀D𝐴𝑑

• 𝜀b is the relative permittivity of the medium

• 𝜀d air = 1, paper = 2– 3, water = 80

Parallel Plate Capacitor

• Coulomb’s Law: Force depends on Q1, Q2 and r:

𝐹 = 𝑘𝑄:𝑄;𝑟;

• Notice any similarities?

𝐹 = −𝐺𝑚:𝑚;𝑟;

• Gravity = always attractive• Electric = attractive/ negative• Electric Field Strength considers force on a

‘test charge’ Q2 at a distance r from Q1

𝐸 =𝑘𝑄:𝑄;𝑟;

1𝑄;

=𝑘𝑄:𝑟;

Non-Uniform Electric Fields

𝑉 =𝑘𝑄:𝑟

• Find k: isolated charged sphere, where radius = r and charge = Q

𝑄𝐴= 𝜀b

𝑉𝑑=𝜀b𝑘𝑄𝑟;

Where 𝐴 = 4𝜋𝑟;𝑄

4𝜋𝑟;=𝜀b𝑘𝑄𝑟;

𝑘 =1

4𝜋𝜀b

Potential in a radial field

• Electric field = –potential gradient = −fgfD

𝑉 =1

4𝜋𝜀b𝑄𝑟

𝐸 =−14𝜋𝜀b

𝑑 :D𝑑𝑟

𝐸 =𝑄

4𝜋𝜀b𝑟;

Potential in a radial field

Comparing Electric and Gravitational Fields

Gravitational Electric

Force 𝐹 = −𝐺𝑚1𝑚2𝑟2 𝐹 = 𝑘

𝑄1𝑄2𝑟2

Field Strength𝐸 = −𝐺

𝑚𝑟2 𝐸 =

𝑄4𝜋𝜀0𝑟2

Potential 𝑉 = −𝐺𝑚𝑟 𝑉 =

14𝜋𝜀0

𝑄𝑟