Tru Physics

Chapter 5: Electric Potential Energy

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5.1 Introduction to Electric Potential Energy

Electric potential energy is the energy an electrically charged object possesses due to its position within an electric field. This type of energy is a scalar quantity and is an essential concept for understanding the behavior of charged particles in electric fields.

5.2 Definition of Electric Potential Energy

The electric potential energy (U) between two point charges q_1 and q_2 separated by a distance r can be defined as:

U = k \dfrac{|q_1 q_2|}{r}

where

  • k is the electrostatic constant, approximately 8.99 \times 10^9 \text{ N} \text{m}^2 \text{/C}^2 (newton meters squared per coulomb squared),
  • q_1 and q_2 are the magnitudes of the charges, and
  • r is the distance between the charges.

5.3 Work and Electric Potential Energy

Work must be done to move a charged particle within an electric field. The work done (W) is equal to the change in electric potential energy (ΔU) of the particle:

W = \Delta U = U_f - U_i

where

  • W is the work done,
  • U_f is the final electric potential energy, and
  • U_i is the initial electric potential energy.

5.4 Electric Potential

Electric potential (V) is the electric potential energy per unit charge. It is a scalar quantity and can be defined as the work done per unit charge to move a test charge from infinity to a specific point in the electric field:

V = \dfrac{U}{q_0}

where

  • V is the electric potential,
  • U is the electric potential energy, and
  • q_0 is the test charge.

5.5 Relationship between Electric Field and Electric Potential

The electric field (E) and electric potential (V) are related through the gradient operator:

\vec{E} = -\nabla V

The negative sign indicates that the electric field points in the direction of greatest decrease in electric potential.

The gradient operator is simply introduced here. We will discuss it in much greater detail in the next chapter.

5.6 Equipotential Surfaces

Equipotential surfaces are surfaces where the electric potential is constant. The work done to move a charge along an equipotential surface is zero since there is no change in electric potential energy. The electric field is always perpendicular to equipotential surfaces.

The figure above shows a point charge (+q) at the origin. Electric field vectors point radially outward in all directions. The three concentric spheres represent three equipotential surfaces. Thus, all points on any given sphere share the exact same electric potential energy. If you were to move a particle along the surface of one of the spheres, no work would be done by the Coulomb force. A two dimensional visualization of this same situation can be seen below.

Three equipoential surfaces (V1, V2, and V3) are shown. A positive point charge is shown at the center with a test charge located on the V2 surface. As the test charge is moved along V2, no work will be done be the Coulomb force because there is no change in electric potnetial on an equipotential surface.
Three equipoential surfaces (V1, V2, and V3) are shown. A positive point charge is shown at the center with a test charge located on the V2 surface. As the test charge is moved along V2, no work will be done be the Coulomb force because there is no change in electric potnetial on an equipotential surface.

Chapter Summary

In this chapter, we explored electric potential energy, a key concept in electromagnetism. We defined electric potential energy and its relationship with work, introduced electric potential, and discussed the relationship between electric field and electric potential. We also covered equipotential surfaces, which are essential for understanding the behavior of charged particles in electric fields. These concepts are vital for further study in topics such as capacitance, current, and electromotive forces.

Continue to Chapter 6: The Gradient Operator

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