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840 Chapter 19 | Electric Potential and Electric Field
Figure 19.6 A system consisting of two point charges initially has the smaller charge moving toward the larger charge
Note that the internal energy of this two-charge system will not change, due to an absence of external forces acting on the system. Initially, the internal energy is equal to the kinetic energy of the smaller charge, and the potential energy is effectively zero due to the enormous distance between the two objects. Conservation of energy tells us that the internal energy of this system will not change. Hence the distance of closest approach will be when the internal energy is equal to the potential energy between the two charges, and there is no kinetic energy in this system.
The initial kinetic energy may be calculated as 50.0 J. Applying Equations (19.38) and (19.2), we find a distance of 9.00 cm. After this, the mutual repulsion will send the lighter object off to infinity again. Note that we did not include potential energy due to gravity, as the masses concerned are so small compared to the charges that the result will never come close to showing up in significant digits. Furthermore, the first object is much more massive than the second. As a result, any motion induced in it will also be too small to show up in the significant digits.
19.2 Electric Potential in a Uniform Electric Field
Learning Objectives
By the end of this section, you will be able to:
• Describe the relationship between voltage and electric field.
• Derive an expression for the electric potential and electric field.
• Calculate electric field strength given distance and voltage.
The information presented in this section supports the following AP® learning objectives and science practices:
• 2.C.5.2 The student is able to calculate the magnitude and determine the direction of the electric field between two electrically charged parallel plates, given the charge of each plate, or the electric potential difference and plate separation. (S.P. 2.2)
• 2.C.5.3 The student is able to represent the motion of an electrically charged particle in the uniform field between two oppositely charged plates and express the connection of this motion to projectile motion of an object with mass in the Earth’s gravitational field. (S.P. 1.1, 2.2, 7.1)
• 2.E.3.1 The student is able to apply mathematical routines to calculate the average value of the magnitude of the electric field in a region from a description of the electric potential in that region using the displacement along the line on which the difference in potential is evaluated. (S.P. 2.2)
• 2.E.3.2 The student is able to apply the concept of the isoline representation of electric potential for a given electric charge distribution to predict the average value of the electric field in the region. (S.P. 1.4, 6.4)
In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship
between voltage and electric field. For example, a uniform electric field is produced by placing a potential difference (or
voltage) across two parallel metal plates, labeled A and B. (See Figure 19.7.) Examining this will tell us what voltage is
needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential
and electric field. From a physicist’s point of view, either or can be used to describe any charge distribution. is
most closely tied to energy, whereas is most closely related to force. is a scalar quantity and has no direction, while
is a vector quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity,
is represented by below.) The relationship between and is revealed by calculating the work done by the force in
moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.
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