Question

FIGURE 16–58 Problem 33.Question: (II) Determine the electric field \({\bf{\vec E}}\) at the origin 0 in Fig. 16–58 due to the two charges at A and B.

FIGURE 16–58 Problem 33.

Short answer

The magnitude of the net electric field at the origin due to charges at A and B is \(3.6 \times {10^7}\ N/C, and the direction of the net electric field at origin is (30^\circ \) below the positive x-axis.

Step-by-step solution

Meaning of electric field

The electric field is the area nearby an electrical charge or electrically charged body in which a positive unit charge can experience a force.

Given information

The magnitude of the charges, \(Q = 26\;{\rm{\mu C}}\).

The distance of both charges from the origin, \(l = 8.0\;{\rm{cm}}\).

Evaluation of the magnitude of the electric field at the origin

The diagram for the electric fields at the origin due to the charges at A and B is as follows:

The electric field at origin due to charge at A can be calculated as follows:

The electric field at the origin due to charge at A is toward downwards because the charge at A is positive.

The x-component of the electric field \({E_{\rm{A}}}\) is as follows:

\({E_{{\rm{Ax}}}} = 0\)

The y-component of the electric field \({E_{\rm{A}}}\) is as follows:

E_AY =

The electric field at origin due to charge at B can be calculated as follows:

The x-component of the electric field \({E_{\rm{B}}}\) is as follows:

The y-component of the electric field \({E_{\rm{B}}}\) is as follows:

The net electric field at the origin due to charge at A and B can be calculated as follows:

The magnitude of the net electric field at the origin due to charge at A and B can be calculated as follows:

Thus, the magnitude of the net electric field at the origin due to charge at A and B is

Evaluation of the direction of the electric field at the origin

The direction of the net electric field at the origin with a positive x-axis in the counterclockwise direction can be calculated as follows:

The negative sign indicates the net electric field will be below the positive x-axis.

Thus, the direction of the net electric field at the origin with a positive x-axis in the clockwise direction is \(30^\circ \).