Two charges are placed as shown in Fig. P21.96. The magnitude of q1 is 3.00 , but its sign and t

Two charges are placed as shown in Fig. P21.96. The...

Two charges are placed as shown in $\textbf{Fig. P21.96}$. The magnitude of $q_1$ is 3.00 $\mu$C, but its sign and the value of the charge $q_2$ are not known. The direction of the net electric field $\overrightarrow{E}$ at point $P$ is entirely in the negative $y$-direction. (a) Considering the different possible signs of $q_1$ and $q_2$ , four possible diagrams could represent the electric fields $\overrightarrow{E}_1$ and $\overrightarrow{E}_2$ produced by $q_1$ and $q_2$. Sketch the four possible electric-field configurations. (b) Using the sketches from part (a) and the direction of $\overrightarrow{E}$ , deduce the signs of $q_1$ and $q_2$ . (c) Determine the magnitude of $\overrightarrow{E}$.

Key Concepts

Vector Addition of Electric Fields

Electric fields are vector quantities, meaning both magnitude and direction must be considered when combining them. By breaking fields into components and using vector addition, one can determine how different charges’ fields combine to produce a net field in a specific direction, which in turn can provide insights into the relative magnitudes and signs of the contributing charges.

Charge Sign and Field Direction Relationship

The relationship between the sign of a charge and the direction of its electric field is crucial. For positive charges, the field lines point away from the charge, while for negative charges they point toward the charge. This concept is used to deduce the sign of unknown charges by analyzing the direction of the resultant field.

Electric Field of a Point Charge

This concept describes the electric field produced by a point charge, where the field’s magnitude is proportional to the absolute value of the charge and inversely proportional to the square of the distance from the charge. It also includes the important detail that the field radiates outward for a positive charge and inward for a negative charge.

Superposition Principle

The superposition principle is fundamental in electrostatics and states that the total electric field produced by multiple charges is the vector sum of the electric fields produced by each individual charge. This allows for the calculation of the net field at a point by considering the contributions and directions from all nearby charges.

Transcript

00:01 In this problem, we have a setup q1, q2, and their separation is 13 centimeters.

00:09 We have also an interested point over here, point p, and the distance between p and q1 is 5 centimeters, a distance between p and q2 is 12 centimeters.

00:26 So for part a, we have four configurations of different signs of q1 and q2.

00:31 So we want to label the electric field for different configurations.

00:36 So suppose we have positive and positive, then for the point p, this is pointing up, this is pointing to this direction, and for that one is pointing to this direction.

00:48 So the y component has to be pointing up.

00:53 And if we have positive and negative, then for the interested point, because q1 is positive, then it's point to this direction.

01:03 For q2 is negative, so it's pointing to this direction.

01:07 In this case, the x component is not zero, as you can see clearly.

01:13 For third case, negative and positive, we have this point.

01:22 Because q1 is negative, so the electric field is pointing to this direction, and q2 is positive, so it's pointing to this direction.

01:30 In that case, the x component is not zero as well.

01:34 And for the last case, if we have negative and negative.

01:40 So we have two points.

01:42 We have this interesting point over here.

01:44 Because q and q2, they are both negative.

01:47 So the electrify is pointing toward them.

01:50 In that case, it is possible to set the x component being equal to 0, and the y component is pointing down.

01:59 And so for part b, the only possible case is just the negative and negative.

02:03 For part c, we want to find out the value of q2 and the value of the electric field...

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