Three charges are placed as shown in Fig. P21.105. The...

Three charges are placed as shown in Fig. P21.105. The magnitude of $q_{1}$ is 2.00$\mu C$ , but its sign and the value of the charge $q_{2}$ are not $2.00 \mu C,$ but its sign and the value of the charge $q_{2}$ are not known. Charge $q_{3}$ is $+4.00 \mu C$ , and the net force $\vec{F}$ on $q_{3}$ is entirely in the neaative $x$ -direction. (a) Considering the different possible signs of $q_{1}$ , there are four possible force diagrams representing the forces $\vec{F}_{1}$ and $\vec{F}_{2}$ that $q_{1}$ and $q_{2}$ exert on $q_{3} .$ Sketch these four possible force configurations. (b) Using the sketches from part (a) and the direction of $\vec{\boldsymbol{F}},$ deduce the signs of the charges $q_{1}$ and $q_{2},$ (c) Calculate the magnitude of $q_{2} .(\mathrm{d})$ Determine $F,$ the magnitude of the net force on $q_{3}$

Three charges are placed as shown in Fig. P21.105. The magnitude of $q_{1}$ is 2.00$\mu C$ , but its sign and the value of the charge $q_{2}$ are not $2.00 \mu C,$ but its sign and the value of the charge $q_{2}$ are not known. Charge $q_{3}$ is $+4.00 \mu C$ , and the net force $\vec{F}$ on $q_{3}$ is entirely in the neaative $x$ -direction. (a) Considering the different possible signs of $q_{1}$ , there are four possible force diagrams representing the forces $\vec{F}_{1}$ and $\vec{F}_{2}$ that $q_{1}$ and
$q_{2}$ exert on $q_{3} .$ Sketch these four possible force configurations. (b) Using the sketches from part (a) and the direction of $\vec{\boldsymbol{F}},$ deduce the signs of the charges $q_{1}$ and $q_{2},$ (c) Calculate the magnitude of $q_{2} .(\mathrm{d})$ Determine $F,$ the magnitude of the net force on $q_{3}$

Key Concepts

Coulomb's Law

Coulomb's Law provides the quantitative basis for calculating the electric force between two point charges. It states that the force is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The law also indicates that the force is attractive if the charges are of opposite signs and repulsive if the charges are of the same sign.

Vector Addition of Forces

The principle of vector addition is essential in the analysis of multiple force interactions, where each force acting on a charge is represented as a vector with both magnitude and direction. By adding these vectors, one can determine the net force acting on a charge, which is crucial in problems where forces are not aligned along a single axis.

Electrostatic Charge Interactions

Understanding how charges interact is key in electrostatics. Like charges repel and opposite charges attract, and this behavior dictates the directions of the forces between charges. Analyzing these interactions helps in deducing unknown charges’ signs based on the resultant force directions observed in the system.

Superposition Principle in Electrostatics

The superposition principle states that the net effect on a charge due to multiple other charges is the sum of the individual effects produced by each of those charges separately. This concept simplifies complex problems by allowing one to consider the force from each interaction independently before combining them vectorially to find the overall force.

Transcript

00:01 For this problem, on the topic of electric charge and electric field, we are given three charges that are placed as shown in the figure.

00:08 We are given specific information about these charges, including their magnitude and sign.

00:15 Now, we are also told that the net force that acts on q3 is entirely in the negative x direction.

00:21 We need to consider the different possible signs of q1 and the four possible force diagrams representing the forces f1 and f2 are three.

00:32 To be sketched.

00:34 Next, we use these sketches from part a and the direction of f to reduce the signs on charges q1 and q2.

00:41 Then we want to calculate the actual magnitude of q2 and determine f the magnitude of the net force on q3.

00:49 So the four possible force diagrams are sketched as follows.

00:55 So the first one is f1 up and to the right, f2 up and to the left.

01:08 Here we have q1 is positive and q2 is positive.

01:17 Next we have the situation where f1 is up and to the right and f2 is down and to the right.

01:27 And in this scenario, q1 is positive, q2, is negative.

01:37 For the third scenario we have f1 down and to the left and f2 down and to the right.

01:47 And here q1 is negative and q2 is also negative.

01:55 And lastly we have the scenario where f2 is up and to the left and f1 is down and to the left.

02:06 And this is when q1 is negative.

02:07 And this is when q1 is negative.

02:08 And q2 is positive.

02:14 Now only the last picture here can result in a net force in the minus x direction.

02:22 So if there's a force in the minus x direction, we have, if we resolve these vectors f2 and f1, we can break these into the vertical and horizontal components.

02:48 So if we call the angle between f2 and the horizontal theta 2 and f1 and the horizontal to be theta 1, then we have the following...

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