Topic 10: Fields
Topic 10: Fields
© Kari Eloranta
2015
Jyväskylän Lyseon lukio
International Baccalaureate
September 8, 2015
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Introduction to Fields
A spherical object creates a radial gravitational field around it (Tsokos 396).
A point charge creates a radial electric field around it.
Gravitational and electric fields are conservative fields.
Conservative Field
In a conservative field, the work done by the field on an object, as the object
moves from point A to point B in the field, does not depend on the path taken.
An important property of conservative fields is that the concepts of potential
and potential energy are well defined.
In principle, gravitational and electric fields fill the whole universe.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
General Definition of Potential Energy in Conservative Field
The energy stored in a conservative field is called potential energy.
Only differences in potential energy have physical meaning.
Potential Energy
As an object moves from point A to point B in a conservative field, the work done
by the field on the object W equals the negative of the change in potential energy
∆E pot, that is,
W = −∆E pot.
(1)
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Renormalising Potential Energy
In Topic 10 we redefine potential energy by changing the zero level from the
surface of the Earth to infinitely far away from the Earth.
As a result, the potential energy becomes negative.
The total (mechanical) energy is the sum of kinetic and potential energies. The
negative total energy has a precise physical meaning.
Negative Total Energy
When the total energy of an object in a conservative field is negative, the object is
bound to the field.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Introduction to Gravitational Potential Energy
Consider an object of mass m initially infinitely far away from the Earth (mass
M). When the object moves from infinity to distance r from the centre of the
Earth, the gravitational force does work on the object.
The gravitational force on the object (mass m ) at distance r from the centre of
the Earth is
Mm
(2)
F =G 2
r
where M is the mass of the Earth and G is the gravitational constant.
Because the gravitational force increases as the object approaches the Earth, the
gravitational force is not a constant, but a function of distance (F = F (r )).
Because the force is not constant, we cannot calculate the work done by
ordinary means.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Work Done by the Gravitational Force
By using calculus, we can show (Tsokos page 397) that the work done by the
gravitational force on the object, as the object moves from infinity to distance r
from the centre of the Earth along any path, is
Mm
.
W =G
r
(3)
By definition, the work done by the field on the object equals the negative of the
change in gravitational potential energy ∆E pot
W = −∆E pot
© Kari Eloranta 2015
Topic 10: Fields
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Topic 10: Fields
Gravitational Potential Energy
∞
= 0), we get
If we let the gravitational potential energy at infinity be zero (E pot
a new expression for the gravitational potential energy from the work done by
the gravitational force
W
Mm
initial
final
= −∆E pot = −(E pot − E pot ) = −(G
r
Mm
− 0) = −G
.
r
(5)
Gravitational Potential Energy
The gravitational potential energy of the object at distance r from the centre of
the Earth is
Mm
E pot = −G
.
(6)
r
where M is the mass of the Earth, m is the mass of the object, G is the
gravitational constant, and r is the distance from the centre of the Earth to the
object.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Gravitational Potential
Gravitational potential is the gravitational potential energy per unit mass.
Gravitational Potential
The gravitational potential at distance r from the centre of the Earth is
M
Vg = −G ,
(7)
r
where M is the mass of the Earth, G is the gravitational constant, and r is the
distance from the centre of the Earth to the object.
The SI-unit of gravitational potential is
[E ]
[V ] =
= J kg−1.
[m]
The gravitational potential is a scalar quantity.
© Kari Eloranta 2015
Topic 10: Fields
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Topic 10: Fields
Gravitational Potential and Field Strength
Gravitational Potential
The gravitational potential at point P in a gravitational field is the work done per
unit mass in bringing a small point mass from infinity to point P.
Field Strength g
The average gravitational field strength g between points A and B in the field is
the negative of the gradient of the potential
∆V
g =−
∆r
(9)
where ∆V is the change in the gravitational potential in moving from point A to
point B, and ∆r is the distance across the equipotential lines in the field between
A and B.
If the gravitational potential as a function of distance is known (Tsokos page
409), the gravitational field strength is the slope of the line in a (r,V )
coordinate system.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
10.1 Electric Field
Electric charge creates an electric field around a charged object.
The simplest type of a charged object is a point charge, which is a very small
electrically charged object.
In a field line representation, the density of the lines represents the field
strength, and the arrowheads represent the direction.
2
2
1
1
0
0
+
-1
-1
-2
-2
-3
-3
Figure : A positive point charge creates an
outward radial electric field.
© Kari Eloranta 2015
−
Figure : A negative point charge creates an
inward radial electric field.
Topic 10: Fields
Topic 10: Fields
Electric Field
#»
10.2 Electric Field Strength E
When we place a small positive test charge q to a point in an electric field, the
field exerts an electric force on the charge in the direction of the field.
The electric field strength is a quantity that gives the magnitude and direction of
the electric field at a certain point in the field.
Electric Field Strength
The electric field strength is the electric force per small positive unit charge.
Electric Field Strength
The electric field strength is
#»
F
E=
q
#»
#»
(10)
where F is the electric force on a small positive test charge q at a point in the
electric field.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
#»
10.2 Electric Field Strength E (cont.)
The SI derived unit of the electric field strength is N C−1 (newton per coulomb)
(or V m−1 (volt per metre), as we study later).
Electric field strength is a vector quantity.
Direction of Electric Field
The direction of the electric field at a point is the direction of the electric force on
a small positive test charge at that point.
Charge q should be as small as possible, so that it exerts essentially no force on
the charges which created the field. That way the test charge does not change
the original charge distribution that created the field.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
10.2 Electric Field Strength of Point Charge
Electric Field Strength of Point Charge
The magnitude of the electric field strength at a distance r from a point charge q
is
q qt
F k r2
q
E= =
=k 2
qt
qt
r
(11)
where k = 8.99 × 109 N m2 C−2 is the Coulomb constant, and qt a charge of a small
positive test charge.
A point charge creates a conservative radial field around itself.
In principle, the field extends to infinity, but decreases rapidly as distance to the
charge increases.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
10.1 Electric Potential
Electric Potential
The electric potential at point P in an electric field is the work done per unit
charge in bringing a small point charge from infinity to point P.
4
4
3
3
2
2
1
1
0
0
+
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
Figure : The equipotential lines are at right
angles to the field lines.
© Kari Eloranta 2015
−
Figure : The density of the equipotential lines
increases as the field strength increases.
Topic 10: Fields
Topic 10: Fields
Electric Field
10.2 Electric Potential
The electric potential at point P in an electric field is the electric potential
energy per unit charge at that point.
Electric Potential
The electric potential at distance r from the a point charge
q
Ve = −k ,
(12)
r
where q is the charge of the point charge, k is the electric constant, and r is the
distance from the point charge to point P.
The SI-unit of electric potential is
[E ]
[V ] =
= J C−1.
[q]
The electric potential is a scalar quantity.
© Kari Eloranta 2015
Topic 10: Fields
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Topic 10: Fields
Electric Field
10.2 Electric Potential Difference
In a radial field, the equipotential lines are concentric circles around the charge.
The density of the field lines decreases in moving in radial direction away from
the charge.
Only differences in electric potential have physical meaning, not the value of a
potential on a point which may be artificially normalised.
Electric Potential Difference V
When charge q travels from point A to point B in an electric field, the electric
potential difference between the points is
W
∆Ve = VA − VB =
q
(14)
where W is the work done by the electric field on the charge.
Electric potential is a scalar quantity.
The unit of electric potential is [E ]/[Q] = J C−1 = 1V (joule/coulomb=volt),
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
10.1 Electric Field of Two Long Plates
Long, oppositely charged metallic plates produce a constant electric field
between them. Notice that the field is not constant near the edges. This is
known as the edge effect.
+ + + + + + + + + + + +
−- −- −- −- −- −- −- −- −- −- −- −By definition, the direction of the field is away from the positive charge.
© Kari Eloranta 2015
Topic 10: Fields
Topic 10: Fields
Electric Field
10.2 Accelerating with Electric Fields
Consider a small positive test charge in a homogeneous electric field.
As the particle moves in the field, the work done by the field on the charge is
W = q∆Ve. By the work energy theorem, this is equal to the change in the
kinetic energy of the particle ∆E K.
Kinetic Energy in Homogeneous Field
When an electric field is used to accelerate a charged particle, the kinetic energy of
the particle changes. By the work energy theorem, the work done by the field on
the particle is
W = q∆Ve = ∆E K
(15)
where q is the charge of the particle, ∆Ve is the change in the electric potential
between the starting and final points, and ∆E K is the change in the kinetic energy
of the particle.
© Kari Eloranta 2015
Topic 10: Fields