PhysicsLAB Resource Lesson
Electric Potential: Point Charges

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When charged particles are moved from one position in an electric field to another position, a new unit of measurement is needed. A volt represents the amount of work per unit charge required to move a charge between two positions in an electric field. If it takes 1 joule of work to move 1 coulomb of charge between two positions in an electric field, then those positions have a potential difference of 1 volt. Voltage is a scalar property of an electric field, it has no direction, only magnitude. In general,
 
1 volt = 1 joule / 1 coulomb

Rearranging these units (1 joule = 1 coulomb x 1 volt) shows us that the amount of work done on a charge by an external agent as it is moved around an electric field is expressed as
 
Wexternal = qΔV
Wexternal = q(Vfinal - Voriginal)
Wexternal = EPEfinal - EPEoriginal
 
For a point charge the absolute potential of any position in its electric field can be calculated using the equation
 
Vabs = kQ/r
 
EPE represents a charge's electric potential energy which is calculates as the magnitude of the charge times the absolute potential of its position in the electric field produced by the dentral charge.
 
EPE = qVabs = kqQ/r
 
When the charge creating the field is positive, the voltage is positive; when the central charge is negative, the voltage is negative. As the distance from the central charge, r, grows larger and larger; that is, as r approaches infinity, the absolute potential is defined to be zero. You can almost think of the "voltage" as being an indicator of the "elevation of the terrain" surrounding a central point charge. The steeper the terrain, the faster the voltage changes from one location to another. The work done by an external agent can be envisioned as "pushing or pulling" a second charge up or down these changes in elevation. The EPE that the second charge gains or loses can anticipated by the difference in the elevations of its starting and ending positions.
 
voltage "profile" - charge
voltage "profile" + charge
 
Refer to the following information for the next seven questions.

The central charge, Q, has a charge of 10 µC.
 
 What is the potential at surface A where rA = 3 meters?

 What is the potential at surface B where rB = 1 meter?

 Which surface has the higher potential?

 How much work is required by an external agent to move a +1 nC charge from surface A to surface B?

 On which surface, A or B, does the +1 nC charge have more potential energy?

 How much electric potential energy (EPE) does the +1 nC charge have when placed on surface B?

 How much kinetic energy would the +1 nC charge posses as it reaches "infinity" if it is released from rest from surface B?

Surfaces which connect points that are at the same absolute potential, or voltage, are called  equipotential surfaces. In the diagram of the point charge shown in the previous example, two equipotential surfaces were labeled, A and B. Notice that equipotential surfaces meet field lines at right angles. The closer together two equipotential surfaces are to each other, the more rapid the change in voltage. This indicates a stronger electric field which is shown in the second diagram below by the fact that the field lines are grouped more closely together on the left side where the equipotential surfaces are also more closely clustered together.
 
 
Note that the electric field strength, E, can be measured in either the units V/m, or equivalently, in the unit N/C.
 
N/C = V / d
      = (J/C) / m
      = [(Nm)/C] / m
      = N/C
 
Shown below are two more illustrations showing the relationships between field lines and equipotential surfaces. The image on the left shows two positive point charges of equal magnitude; while the image on the right shows two charges of equal magnitude but differing in sign. Note that the field lines (represented by vectors) meet every equipotential surface at right angles.
 

These images were provided by the The Wolfram Demonstrations Project
 
The following two graphs compare the voltage around a positively charged conducting sphere and the electric field for the same positively charged conducting sphere under electrostatic conditions. Note that the electric field strength (E ∝ 1/r2) drops off more rapidly than does the voltage (V ∝ 1/r). Also notice that within a conducting sphere, the voltage remains constant in contrast to the fact that no electric field exists.
 
For a conducting sphere,
V = kQ/r
For a conducting sphere,
E = kQ/r2
 
Remember that the electric field strength, E, is a vector quantity. You are required to state both its magnitude and its direction to completely describe it at any given location. If you are ever asked to calculate the net electric field in 2-dimensions, you should first take the x- and y-components of each field, add the components to determine the net Ex and net Ey, and then calculate the resultant field and its direction. Voltage, on the other hand, is a scalar quantity and can be added directly without considering components or directions.
 
Let's work through the next examples to show you the difference in these two field properties. In each set of diagrams, compare the charge configuration diagram and voltage diagram to determine the requested information for each midpoint. The charges are assumed to be a distance "2r" apart. That os, the midpoint is located a distance "r" from each point charge.
 
Charge Configuration Voltage Diagram
 
 What is the value of the electric field at point A, the midpoint of the line?

 What is the value of the voltage at point A, the midpoint of the line?

Charge Configuration Voltage Diagram
 
 What is the value of the electric field at point B, the midpoint of the line?

 What is the value of the voltage at point B, the midpoint of the line?

Charge Configuration Voltage Diagram
 
 What is the value of the electric field at point C, the midpoint of the line?

 What is the value of the voltage at point C, the midpoint of the line?

Charge Configuration Voltage Diagram
 
 What is the value of the electric field at point D, the midpoint of the line?

 What is the value of the voltage at point D, the midpoint of the line?

Refer to the following information for the next four questions.

Use the following charge configurations to check your knowledge of when the electric field and/or the net voltage equal(s) zero in the center of each square.
In which group is net E ≠ 0 but net V = 0?
 
In which group is net E = 0 but net V ≠ 0?
 
In which group are both net E and net V equal to 0?
 
In which group are neither net E or net V equal to 0?
 
Refer to the following information for the next two questions.

Suppose two conductive spheres are both charged to +6 µC. The smaller sphere has a radius of 0.5 cm while the larger sphere has a radius of 1.5 cm.

 What is the potential of the smaller sphere?

 What is the potential of the larger sphere?

Refer to the following information for the next three questions.

The spheres are now connected with a conducting wire.

 
In which direction will the charges flow:
  1. from the small sphere towards the larger sphere
  2. from the large sphere towards the smaller sphere

 How much charge will be present on the small sphere once equilibrium is reached?

 What will be the common voltage once equilibrium is reached?




 
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