18.6 Electric field lines: multiple charges

Learning objectives

By the end of this section, you will be able to:

• Calculate the total force (magnitude and direction) exerted on a test charge from more than one charge.
• Describe an electric field diagram of a positive point charge and of a negative point charge with twice the magnitude of the positive charge.
• Draw the electric field lines between two points of the same charge and between two points of opposite charge.

The information presented in this section supports the following AP® learning objectives and science practices:

• 2.C.1.2 The student is able to calculate any one of the variables – electric force, electric charge, and electric field – at a point given the values and sign or direction of the other two quantities.
• 2.C.2.1 The student is able to qualitatively and semiquantitatively apply the vector relationship between the electric field and the net electric charge creating that field.
• 2.C.4.1 The student is able to distinguish the characteristics that differ between monopole fields (gravitational field of spherical mass and electrical field due to single point charge) and dipole fields (electric dipole field and magnetic field) and make claims about the spatial behavior of the fields using qualitative or semiquantitative arguments based on vector addition of fields due to each point source, including identifying the locations and signs of sources from a vector diagram of the field. (S.P. 2.2, 6.4, 7.2)
• 2.C.4.2 The student is able to apply mathematical routines to determine the magnitude and direction of the electric field at specified points in the vicinity of a small set (2-4) of point charges, and express the results in terms of magnitude and direction of the field in a visual representation by drawing field vectors of appropriate length and direction at the specified points. (S.P. 1.4, 2.2)
• 3.C.2.3 The student is able to use mathematics to describe the electric force that results from the interaction of several separated point charges (generally 2-4 point charges, though more are permitted in situations of high symmetry). (S.P. 2.2)

Drawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude and direction, it is a vector. Like all vectors , the electric field can be represented by an arrow that has length proportional to its magnitude and that points in the correct direction. (We have used arrows extensively to represent force vectors, for example.)

[link] shows two pictorial representations of the same electric field created by a positive point charge $Q$ . [link] (b) shows the standard representation using continuous lines. [link] (b) shows numerous individual arrows with each arrow representing the force on a test charge $q$ . Field lines are essentially a map of infinitesimal force vectors.