Resource Lesson
Electric Field Strength vs Electric Potential
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When two charged objects are brought into proximity they either attract or repel each other with an electric force described by
Coulomb's Law
. Since this force is
conservative
; that is, path-independent, it can be expressed as the negative derivative of its associated
potential energy function
.
As in gravitation where an object's gravitational potential energy is proportional to its mass,
m
,
a charged object's electrical potential energy is proportional to the magnitude of its charge. The greater the charge placed on an object in a given position in an electric field, the larger its electric potential energy.
The ratio of
electric potential energy per unit charge
is therefore a property of the electric field and is called the field's electric potential, or
voltage (volt = joule/coulomb)
. If you were to connect together a series of positions having the same voltage you would produce an
equipotential surface
. Electric potential is a scalar quantity.
Electric Field Strength and Potential
We are now going to derive two important relationships between the quantities electric field strength and electric potential.
To derive an expression for the local electric field, E, in terms of its electric potential, V, we begin with the definition of a conservative force and the following two facts:
the force exerted on a charge by an electric field equals the product of the charge times the field strength (F = qE)
a volt is defined as electric potential energy per unit charge,
This expression gives rise to an alternate unit for measuring electric fields,
volt/meter
.
The electric field is the
negative gradient of the potential
; that is, field lines point from positions of high potential to points of low potential. The more rapidly the voltage changes the stronger the electric field in that region.
Note that knowing the potential of one position in a field is insufficient to allow you to calculate the electric field strength at that position. You must know the equation for the potential over a region to take its derivative (rate of change with respect to position) and calculate E.
Also recall that the negative sign in this formula was originally introduced since a conservative electric force reduces a charged object's electric potential energy as it accelerates the object to positions of lower potential: Δ KE
_{gained}
+ Δ U
_{lost}
= 0.
Now, instead of solving for the electric field strength, let's solve for the change in the potential between two positions,
a
and
b
, in an electric field.
Note in this expression that we are starting at
a
and going to
b
. The integral notation is read as "the negative integral from
a
to
b
of
E dr
."
Point Charges
Use the formula for the electric field surrounding a point change to calculate an expression that would allow you to evaluate the electric potential (or voltage) at a given distance from the point charge.
How does the graph of
E vs r
for a charged spherical conductor of radius R compare to its graph of
V vs r
?
Refer to the following information for the next five questions.
In a hydrogen atom, an electron (q = -e = -1.6 x 10
^{-19}
C) orbits a proton (q = +e = +1.6 x 10
^{-19}
C) at a radius of 0.53 x 10
^{-10}
meters.
Considering the proton to be the central charge, what is the electric potential at the electron's orbital radius?
How much electric potential energy does the electron have by virtual of its position in the proton's electric field?
How much kinetic energy does the electron have as it orbits the proton?
What is the electron's total energy?
How much additional energy would the electron need to escape from its energy well - that is, to be ionized?
Given an electric field function, determine the potential difference
Refer to the following information for the next five questions.
In a certain region of space the electric field is defined as
E = 1500 i - 800 j
. Use this information to answer the following questions.
Is this a uniform field?
Find the potential difference V
_{B}
- V
_{A}
if A is at the origin and B is at (0, -4).
Find the potential difference V
_{C}
- V
_{A}
if A is at the origin and C is at (-4, 0).
Find the potential difference V
_{C}
- V
_{B}
.
How much work would be required to move an electron from point B to point C?
Refer to the following information for the next five questions.
Outside of a long charged wire, the electric field is pointing radially inward and is defined by the function
E
_{r}
= -1200/r
N/C
for r > R, the radius of the wire.
Is this field uniform?
Is the wire positively or negatively charged?
Find V
_{B}
- V
_{A}
if r
_{B}
= 60 cm and r
_{A}
= 20 cm.
Is A or B at the higher potential?
How much work would be required to move a proton from point A to point B?
Given a potential function, determine the electric field
Refer to the following information for the next six questions.
A charge distribution creates an electric potential that obeys the function
V(x) = 3(x - 1)
^{2}
+ 5
along the x-axis.
What is the electric potential at x = 2.00?
What is the electric potential at x = 2.10?
What is the voltage gradient in this region around x = 2.05?
Differentiate to determine the associated electric field function along the x-axis.
Evaluate your electric field function to determine the x-component of the electric field at x = 2.05?
How closely does the value of the voltage gradient approximate your previous answer for the electric field at x = 2.05?
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