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

Two charges are placed as shown in Fig. P21.96. The magnitude of $q_{1}$ is $3.00 \mu \mathrm{C},$ but its sign and the value of the charge $q_{2}$ are not known. The direction of the net electric field $\overrightarrow{\boldsymbol{E}}$ 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 $\vec{E}_{1}$ and $\vec{E}_{2}$ produced by $q_{1}$ and $q_{2} .$ Sketch the four possible electricfield configurations. (b) Using the sketches from part (a) and the direction of $\overrightarrow{\boldsymbol{E}}$, deduce the signs of $q_{1}$ and $q_{2}$. (c) Determine the magnitude of $\overrightarrow{\boldsymbol{E}}$.

Key Concepts

Vector Decomposition and Analysis

Vector decomposition involves breaking down vectors into their components (typically along the x, y, and z axes). In the context of electric fields, analyzing the components of each field allows one to determine the net field direction and magnitude by adding the corresponding components, which is essential when the overall field points entirely in a specific direction.

Charge Sign Effect on Field

The sign of a charge determines the direction of the electric field it produces: the field radiates outward from positive charges and inward toward negative charges. This concept is crucial when analyzing and deducing the behavior of combined electric fields in a system.

Electric Field

The electric field is a vector field that represents the force per unit charge exerted on a small positive test charge placed in the vicinity of other charges. The field's magnitude and direction are determined by the source charge's magnitude, sign, and position relative to the point of interest.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the magnitude of the force is proportional to the product of the charges and inversely proportional to the square of the distance between them. This law underpins the calculation of the electric field produced by point charges.

Superposition Principle

The superposition principle in electrostatics states that the net electric field created by multiple charges is the vector sum of the individual fields produced by each charge. This principle is key when determining the overall field at a given point due to several sources.

Transcript

0:00 Hello, everyone.

00:01 In this problem, we're asked to find out worries things about a charge configuration that you can see in a diagram that i've drawn over here.

00:10 So, you know, this is just a sketch, of course, but basically what happens here is you have this triangular setup, such that you have two charges q1 and q2, separated along the horizontal distance of 13 centimeters or 0 .13 meters.

00:24 And there is a point p of interest that is at right angles that makes a right hand.

00:30 Hand a triangle with these two points and is a distance of 0 .05 meters away from q1 and a distance of 0 .12 meters away from q2.

00:40 Now we know that q1 has a magnitude of three microculems.

00:45 So that's what we are given.

00:48 And we're also told that e points straight down.

00:53 So e is parallel to the negative y head direction, right? if we choose up to b plus y.

01:02 So these are two pieces of information that we're told.

01:05 And then we're kind of left to fend for ourselves.

01:07 So we have to figure out essentially or eventually what we want to find is the magnitude of the electric field.

01:15 So what is the size of e? that's what we really want to find.

01:22 So we kind of let on to the solution here.

01:26 But basically what happens, you're first asked to consider all the various, configurations for the two electric fields produced by q1 and q2 and see which of these, like based on their size and which of these makes sense for the problem.

01:41 So if q1 and q2 are both positive, then this is the electric field coming from e1 or from q1, and something like this is the electric field coming from q2.

01:57 So these are the two electric fields produced by, q1 and q2.

02:04 If q1 is positive, then this is the field produced by q1.

02:12 And if q2 is negative, then the field produced by q2 is going to point towards q2.

02:20 So it's going to do something like this.

02:24 So these are two possibilities.

02:25 Now what if q1 is negative and q2 is positive? if q1 is negative, then e1 points towards q1.

02:38 And if q2 is positive, then the electric field of q2 points away from q2.

02:45 So it points in that direction.

02:49 So that's e2.

02:50 If they're both negative, then the electric field from p points towards, like e1 points towards e1, and e2 points towards e2.

03:04 Or q2.

03:11 So these are the four possibilities that we are faced with.

03:15 Now in part b we're as to find based on these diagrams which configuration is probably true.

03:21 So which, like what are the, what is the sign of the on the charge of q1 and q2? so we notice that if e is pointing straight down, so this we were told that e is going to be this and this is equal to the sum of e1 plus e2 as vectors, then for this to happen or to be able to happen, we have to have that a, so part a's fourth solution is the correct one.

03:54 So this diagram that we drew here, that is the correct one.

03:58 And so both q1 and q2 are negative charges.

04:09 And so in part c, we are actually just have to, we have to find what the magnitude of e is.

04:15 And so the way we do this is the following.

04:17 So since we don't know what q2 is and we don't know its electric field, what we first have to do is either find its electric field or find its, find the charge's magnitude, knowing that it's negative, right? so basically, since we know that e is going to point straight down, we know that the x components of the two electric fields are going to cancel out, right? so the total x component of the net electric field is going to be zero.

04:47 Now, there is some trigidometry that you have to do here to figure out what the angles are.

04:52 So e1 makes an angle off.

04:55 You can worry this out.

04:56 So if theta is the angle that e1 makes with the horizontal, then theta is the angle also at the charge q1 of the triangle.

05:08 So this is q1 over here...

Please give Ace some feedback