Electric Forces, Fields, Energy, and Potential

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Unit V Electric and Magnetic Forces and Fields

Contents

A. Coulomb’s Law
B. Electric Fields
1. Electric Field About a Point Source
2. Electric Field Between Two Large Charged Plates
C. Electrical Potential Energy
D. Potential Difference
1. Potential About a Point Source
2. Potential Between Charged Plates
E. Comparison of Gravitational and Electrical Concepts
F. Magnetic Fields
1. Field Lines
2. Electromotive Force
3. Movement of Charges in Electric and Magnetic Fields
4. Formulas that Apply to Magnetic Fields

This unit presents the basic concepts dealing with the electric and magnetic forces. You have a
couple of options. One is to learn and memorize the formulas and rules that apply to electricity.
The other option is to get an intuitive grasp of the general concepts of force, field, energy, and
potential as they would apply to any broad category: electricity, magnetism, gravitation. Both
options are included here. The first few pages deal with electricity itself. Then a general
comparison of electricity and gravity is presented, so that by seeing the parallels, you may
understand everything better. Finally, magnetic fields are mentioned.

A. Coulomb’s Law
The electric force between two charges q1, q2 (in coulombs) separated by a distance d (in metres)
kq q
is F  12 2 .
d

B. Electric fields
Although it takes two objects, (q1, and q2) the make a force, it is convenient to think of each
object as possessing a field, a zone of influence. When charge q2 is brought into charge q1’s zone
of influence, the charge q2 feels a force. By dividing F by q2, we can get a formula for the
electric field around charge q1:
F kq
   21
q2 d

Now the formula for electric field  about charge q1 only has variables that relate to q1. This
means we can think of a field as a property of a single object.

Units for an electric field: newtons/coulomb. (Compare to gravitational field: N/kg)
V-2
Electric field is a vector that point from positive to negative, in the direction that a positive test
charge would move. When two different electric fields are present, the total field is the vector
sum of the individual fields.

1. Electric field about a point source

The electric field around a point charge is spherical, falling off with the square of the distance.

2. Electric field between two large charged plates

Between a negative and a positive charged plate, the field lines are parallel, going from positive
to negative. The field strength is the same at all points between the plates. That’s because the
force on the test charge is the same no matter how far the charge is from each plate.

C. Electrical Potential energy

Gravitational potential energy is negative. That’s because stored energy increases with the
separation of the object until it reaches zero when the objects are infinitely far apart. This is
consistent with potential energy’s definition as the work done bringing an object from infinity
to a distance d close to another mass. It takes no energy to bring the masses in from infinity and
stop them at a separation d. (The masses when less than infinitely far apart come in by
themselves due to their mutual attraction.) Positive-negative attraction works the same. So the
potential energy in a setup of two charges a distance d apart is
kq q
E 1 2
d
There is no need for a negative sign (as in the gravity formula) because for electrostatic
attraction, one of the q’s itself must be negative.

To find how much energy is required to move a charge from one location to another in an
electric field, you just subtract the potential energies at each point.

D. Potential difference

Just as a field is the force per charge, potential difference (voltage) is the amount of energy per
charge required to move the charge in an electric field (around another charge). Since this energy
is the difference in potential energies at the two locations, we call the energy per unit charge,
potential difference.
V-3
1. Potential near a point source

For the potential difference between two points at different distances from a point source we
have:
E
V 
q2
kq1 kq1
      
d2   d1
 1     1
 kq1 d  d    
 2      1 

It is possible to state the electrical potential at a specific distance d from a point source (by
comparing to the zero potential at infinity) by saying
E kq
V   1
q      d

2. Potential between two charged plates

If a charge q2 is moved a distance d between charged plates, the work done is easy to calculate
because F is constant (because the field is uniform everywhere between the plates).
E  Fd  q 2d
Combining with the formula for potential difference gives us
E
V 
q2
q 2 d

q2
 d
So we have a formula for the field between two plates charged by being attached to a power
source of voltage ΔV:
V
      , where d is the spacing between the plates
d
V-4

E. A Comparison of Gravitational and Electrical Concepts
Gravity                                     Electricity
Force             - between masses                            - between charges
Gm1 m2                                      kq q
F                                          F  12 2
d2                                         d
- always attractive                         - attractive if charges are opposite
- falls off with square of the distance     - falls off with square of the distance
Field             - surrounds a mass                          - surrounds a charge
- points in the direction an imaginary      - points in the direction an imaginary
test mass would move (i.e. toward           tiny positive charge would move (i.e.
the other mass)                             toward a negative charge, away from
a positive charge)
- field = force/mass (m2, the test          - field = force/charge (q2, the test
mass)                                       charge)
Gm                                          kq
  21                                        21
d                                           d
- units: newtons/kilogram (= m/s2)          - units: newtons/coulomb ( m/s2)

Field inside a    - gravitational field inside a              - electric field inside a charged
spherical shell     spherical shell is zero                     spherical shell is zero

Field between     - no parallel to electrical field exists,   - electric field between two oppositely-
two plates          because mass is only positive               charge flat plates is constant, directly
(capacitor)                                                     proportional to the charge on the
plates, inversely proportional to the
separation of the plates
Potential               Gm1 m2                                     kq q
Energy            E                                          E 1 2
d                                          d
Note: negative sign means that there        Note: this equation has a positive sign,
is negative potential energy in a            because if charges attract, one will be
configuration where two masses               negative automatically, giving
are separated by a distance d                negative potential energy for a
configuration of opposite charges.
Energy Unit       - joules                                    - joule:
E = QV, so a joule = coulomb.volt
small energy unit: electron volt = the
energy of an electron moving through
a potential difference of one volt.
Thus Q = qe, therefore
1 eV = 1 J x charge on the electron
(1 eV = 1.6 x 10-19 J)
Potential         - change in potential energy per            - change in potential energy per unit
Difference          mass when the mass moves in a               charge (q2) when that charge is
(voltage)           gravitational field                         moved in an electrical field
- little use made of this terminology               E
when applied to gravitation               V 
q2
V-5

F. Magnetic Fields

1. Field Lines

Remember that a field is a region of influence around an object that exerts a force. Gravitational
fields surround masses, electric fields surround charges, and magnetic fields surround magnets.
The direction of the field is the direction a tiny test particle would move. The test particle for
gravitational field is a tiny mass, for an electric field is a proton, and for a magnetic field is an
imaginary “north”. Because magnetic monopoles (north alone) do not exist, in practice the
magnetic field points in the direction a tiny compass would point. By definition, then, magnetic
field lines flow from north to south.

From earlier physics courses you should recall the shapes of magnetic fields around
bar magnets, and current-carrying wires, loops, and coils.

You should also recall the rules that govern the link between current direction and magnetic field
lines. You will have used the left hand rule if your teacher talked about the magnetic fields
around flowing electrons, or the right hand rule if conventional (positive) current was used.
(Mathematically, negative to the left = positive to the right, but in 3-D mirror image situations
result, hence the left/right rules.)
V-6

2. Electromotive force

Finally, you should recall that when charged particles cut across magnetic field lines, they
receive a force at right angles to both their velocity and the magnetic field line. You learned
another left (or right) hand rule for that.

The electromotive force was the basis for the construction of motors. If a loop was free to pivot,
if one end were made to go up and the other down, the loop would spin. (In the diagram below it
would not continue to rotate. A split ring commutator would have to be used so that when the
loop flipped over, the battery stayed where it was, so that the direction of the electromotive force
would stay the same.)

In the setup to the right, the loop
would tend to rotate counter-clockwise
as viewed from above the battery.
V-7

3. Movement of charges in electric and magnetic fields

i. Electron through plates.

Recall that the electric field between two plates is constant, the same value at all distances from
the plates. In the configuration below, an electron flying in from the right receives a constant
downward force. This should remind you of gravity near the Earth’s surface. Both the electron
and a ball should have a parabolic path.

(Remember, the parabolic path of a projectile is an approximation. In reality, the gravitational
field is greater nearer Earth’s surface, and points to Earth’s centre, so the gravitational field lines
are not exactly parallel.)

Application: In traditional television tubes (cathode ray tubes), electron from an electron gun
enter regions of horizontal and vertical plates. The beam is directed in turn to each region of the
screen. As it sweeps across the stream, the beam hits little dots of chemical (pixels) that glow.
The varying intensity of the beam produces regions of dark and light which become the picture
we see.

ii. Electron through magnetic field

If an electron enters a magnetic field at right angles to the field lines, it will experience a force at
right angles to its velocity. This should not change the electron’s speed, only its direction of
travel. Recall centripetal force: when a force is always perpendicular to an object’s motion, the
object undergoes uniform circular motion.

So if an electron enters a region containing a constant magnetic field, it should travel in a circle.

Application: magnetic bottles. You may have heard that term applied to devices in particle
accelerators which trap fast-moving charged particles.

If the electron runs into some resistance so its speed decreases, the radius of the circle should
decrease.

Application: paths of charged particles in cloud chambers. You may have seen the spiral patterns
that indicate the trajectories of charged particles emitted by radioactive nuclei.

If the magnetic field lines are not constant in strength or converge, the electrons can be focused.
V-8
Application: charged particles from the sun spiral around Earth’s magnetic field lines,
concentrating near the poles, producing northern and southern lights, aurora borealis and aurora
australis.

4. Formulas that apply to magnetic fields

Magnetic fields are measured in tesla, T. Symbol for magnetic field is B. Recall that a force is
exerted on a current-carrying wire in a magnetic field. A tesla is defined as the magnetic field
required to produce a 1 N force on a 1 m long wire carrying a current of 1 A. Therefore a tesla is
a newton/ampere.metre.

i. Force on a charged particle moving at speed v in magnetic field

Think: more charge = greater force; faster charge = greater force (after all, if charge is not
moving, there is no force); stronger field = more force. Put them together:

F = qvB

DON’T MEMORIZE THIS FORMULA, BECAUSE IT IS TOO SPECIFIC.

Think: left/right hand rule. If velocity is parallel to magnetic field line, no force. If velocity is at
90 to magnet field, maximum force. This should remind you of the sine function: sin(0) = 0 but
sin(90) = 1 (the maximum.) Combine with the above expression to get the general formula for
the force on a moving charge in a magnetic field:

F = qvBsin(θ), where θ = the direction between v and the magnetic field lines.

ii. Force on a wire in a magnetic field.

Examine the units of qv in the above formula: units are coulombs metres/second. Coulombs per
second is current in amperes (i.e. 1 A = 1 C/s). So qv works out to current times length. We have
this formula for the force on a wire of length l oriented at angle θ with respect to magnetic field
B, carrying current I. The force is

F = IlBsin(θ)

iii. Magnetic field at a distance from a long, straight wire

The above formulas give the formulas for forces that magnetic fields of a given strength B exert.
Here is the formula for determining the strength of the magnetic field itself, generated by a long,
straight wire carrying current I. Your textbook will derive it, usually in a calculus-like manner.
Look up the derivation if you think your teacher requires you to reproduce it on a test. What you
need to know is the formula itself. The magnetic field at distance r from a long straight conductor
carrying current I is
V-9
 I 
B  o       , where μo is the permeability of free space, whose value is 4 x 10 Tm/A.
-7

 2r 

Don’t let μo scare you. It’s just a proportionality constant to move between current in amperes,
and distance in metres to magnetic field strength in tesla. It does have one characteristic of
significance, though. You can imagine that the magnetic field strength surrounding a wire might
depend on the medium in which the wire was immersed. For example, the magnetic permeability
of iron is about 200 times that of free space. That’s why the magnetic field near the iron core of a
coil is greater than the field at the same location if the iron core were removed.

Alternate units you might see for μo are N/A2 and Wb/m.A. A webber per metre is the same as a
tesla-metre.

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