Unit 10 Fields
Let’s start with a reminder about electric fields from unit 5
Electric Force and Electric field
The force between two charges was investigated by Charles Augustin Coulomb in 1785
Electric Force and Electric field
Coulomb found that the force between two point charges is proportional to the product of the two charges
and inversely proportional to the square of the
distance (r) between the charges
Coulomb’s law
It follows that
Coulomb’s law
The constant k is sometimes written as
where εo is called the permittivity of free space.
Coulomb’s law review
This force can be attractive or repulsive. The magnitude of the force can be calculated, and the direction should be obvious from the signs of the interacting charges. (Actually, if you include the signs of the charges in the equation, then whenever you get a negative answer for the force, there is an attraction, whereas a positive answer indicates repulsion).
In all realistic cases, the electric force between 2 charges objects absolutely dwarfs the gravitational force between them, as the first of the worked examples will show.
Coulomb’s law
This applies to POINT charges. In the case of a charged sphere, we imagine the charge is concentrated at the centre of the sphere (although as we shall see, the field INSIDE the sphere is zero.
Gravity? (Unit 6)
Mathematically (almost) identical
Gravitational Force and Field
The force between masses was formulated (discovered?) by Isaac Newton in 1687
Gravitational Force and Field
Newton found that the force between two masses is proportional to the product of the two masses
and inversely proportional to the square of the distance (r) between the masses
Newton’s law of universal gravitation
It follows that
Newton’s law of universal gravitation
The constant G is known as “Big G” and is equal to 6.667 x 10-11 N m2 kg-2
Electric field
An area or region where a charge feels a force is called an electric field.
The electric field strength at any point in space is defined as the force per unit charge (on a small positive test charge) at that point.
E = F/q (in N C-1)
Gravitational field
An area or region where a mass feels a gravitational force is called a gravitational field.
The gravitational field strength at any point in space is defined as the force per unit mass (on a small test mass) at that point.
g = F/m (in N kg-1)
Summary so far:
Electric field
Electric field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one charge) must use vector addition.
Field here due to both charges?
q1
q2
Electric field
Electric field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one charge) must use vector addition.
Field here due to both charges?
Field due to q1
q1
q2
Electric field
Electric field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one charge) must use vector addition.
q1
q2
Resultant field
Field due to q1
Field due to q2
Electric field patterns
An electric field can be represented by lines and arrows on a diagram, in a similar ways to magnetic field lines.
Electric field patterns
An electric field can be represented by lines and arrows on a diagram , in a similar ways to magnetic field lines.
The arrows show the direction of force that would be felt by a positive charge in the field
Electric field patterns
An electric field can be represented by lines and arrows on a diagram , in a similar ways to magnetic field lines.
The closer the lines are together, the stronger the force felt.
This is an example of a radial field
Point charge and solid sphere?
E = 0 inside the sphere
Is this positive or negative?
Field between charged parallel plates
Uniform field E = V/d
V
d
“Edge effects”
NOT in data book
+ + + + + + + + + + + +
- - - - - - - - - - - -
Gravitational field
Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition.
m1
m2
Field here due to both masses?
Gravitational field
Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition.
m1
m2
Field due to m1
Field here due to both masses?
Gravitational field
Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition.
m1
m2
Field due to m1
Field due to m2
Gravitational field
Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition.
m1
m2
Field due to m1
Field due to m2
Resultant Field
Gravitational field patterns
A gravitational field can be represented by lines and arrows on a diagram, in a similar ways to magnetic field lines.
Gravitational field patterns
A gravitational field can be represented by lines and arrows on a diagram, in a similar ways to magnetic field lines.
Note, gravity is ALWAYS attractive
The closer the lines are together, the stronger the force felt.
Field around a uniform spherical mass
Field close to the earth’s surface
Uniform
Summary so far:
Gravitational Potential Energy
GPE = mgh?
How much GPE?
GPE = mgh?
How much GPE?
GPE = mgh?
How much GPE?
GPE = mgh?
We do know that the GPE must be decreasing. But where is the GPE zero?
Gravitational potential energy
Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point.
M
m
R
Work done = force x distance
The force however is changing as the mass gets closer
I’ve come from infinity!
Gravitational potential energy
M
m
R
I’ve come from infinity!
W =
R
Fdr
R
GMmdr
r2
=
=
[
]
GMm
r
R
=
GMm
R
-
-
Gravitational potential energy
Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point.
Ep is always negative
Gravitational Potential
It follows that the Gravitational potential at a point is the work done per unit mass on a small point mass moving from infinity to that point. It is given by
Note the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)
Moving masses in potentials
If a mass is moved from a position with potential V1 to a position with potential V2, work = m(V2 – V1) = mΔV
V1
V2
(independent of path)
Equipotential surfaces/lines
Field and equipotentials
Equipotentials are always perpendicular to field lines. Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour maps give us information about geographical heights.
Field strength = potential gradient
In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field)
Also note when Δr tends to zero...
Hold on!
Isn’t electricity similar?
Gravitational Potential
It follows that the Gravitational potential at a point is the work done per unit mass on a small point mass moving from infinity to that point. It is given by
Note the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)
Electrical Potential
The Electrical potential at a point is the work done per unit charge on a small positive test charge moving from infinity to that point. It is given by
Note the difference between electrical potential energy (J) and Electrical potential (J.C-1)
Gravitational potential energy
Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point.
Ep is always negative
Electrical potential energy
Electrical potential energy at a point is defined as the work done to move a positive charge from infinity to that point.
Equipotential surfaces/lines
Field and equipotentials
Equipotentials are always perpendicular to field lines. Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour maps give us information about geographical heights.
Field strength = potential gradient
In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field)
Summary so far:
Escape speed
infinity
Escape speed
infinity
Escape speed
infinity
Conservation of energy
Total energy = 0 J
Calculate v minimum
Escape speed
Object | Radius/m | Mass/kg |
The Moon | 1.74 x 106 | 7.35 x 1022 |
Jupiter | 6.99 x 107 | 1.90 x 1027 |
Neutron Star | 1.2 x 104 | 4 x 1030 |
Extension: At the event horizon of a black hole the escape velocity is the speed of light (3 x 108 m.s-1). For a black hole with the same mass as the earth (6 x 1024 kg) what would be the radius of the event horizon? What about for a black hole of 5 solar masses (one solar mass is 2 x 1030 kg)?
Calculate the escape velocities at the surface of the following:
Animal lovers - do not shout at me please (I borrowed the following presentation)
Even bigger cannon
Gravity
Gravity
Gravity
VERY big cannon?
VERY big cannon
Gravity
Humongous cannon?
Dog in orbit!
The dog is now in orbit! (assuming no air resistance of course)
Dog in orbit!
The dog is falling towards the earth, but never gets there!
Dogs in orbit!
The force that keeps an object moving in a circle is called the centripetal force (here provided by gravitational force)
Gravity
Calculations
A geostationary satellite must orbit the earth along a path of radius 4.23 x 107 m. Find:
Calculations
A geostationary satellite must orbit the earth along a path of radius 4.23 x 107 m. Find:
Calculations
A geostationary satellite must orbit the earth along a path of radius 4.23 x 107 m. Find:
Calculations
A geostationary satellite must orbit the earth along a path of radius 4.23 x 107 m. Find:
Challenge
Use Kepler’s Law to explain why low orbit polar satellites orbit very quickly.
Energy of a satellite
A satellite has kinetic energy and gravitational potential energy.
Total energy = Ek + Ep = ½mv2 - GMm/r
Energy of a satellite
A satellite has kinetic energy and gravitational potential energy.
Total energy = Ek + Ep = ½mv2 - GMm/r
Energy of a satellite
A satellite has kinetic energy and gravitational potential energy.
Total energy = Ek + Ep = ½mv2 - GMm/r
so
Total energy
Total energy = -½mv2
Energy of a satellite
Total energy of satellite = -GMm/2r
Kinetic energy = GMm/2r
Potential energy = - GMm/r
Energy of a satellite
Ep
Ek
ET
energy
distance
Questions - Tsokos pg 152
Q 11, 12, 13, 15, 29