A charge q1=+5.00 nC is placed at the origin of an x y-x -axis at x=4.00 cm . (a) If a third ch

A charge q1=+5.00 nC is placed at the origin of an x y-x...

Key Concepts

Superposition Principle

The superposition principle in electrostatics states that the net force acting on a charge is the vector sum of the individual forces exerted on it by all other charges. This means that forces due to each charge are calculated independently and then combined, which is critical when analyzing systems with multiple charges.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, with the direction of the force along the line connecting the charges. This concept is essential for calculating the force one charge exerts on another in any configuration.

Vector Decomposition

Vector decomposition involves breaking down a vector into its horizontal (x-axis) and vertical (y-axis) components. This is useful in physics when forces are applied at an angle, as it allows the analysis and summation of the effects in each independent coordinate direction, facilitating the determination of the overall force’s magnitude and direction.

Pythagorean Theorem and Trigonometry

When determining distances between points or resolving vectors into components, the Pythagorean Theorem is used to compute the magnitude of the displacement between two coordinates, while trigonometric functions like sine, cosine, and tangent are employed to find the direction and the individual components of a vector. These mathematical tools are fundamental for transitioning from a geometric description of the problem to quantitative analysis.

Transcript

0:00 Hello everyone.

00:02 In this problem, we're asked to find the total force that two charges exert on a third charge in a coordinate system.

00:10 So we have the following scenario.

00:12 We have a charge of five nanoculumps sitting at the origin.

00:17 And then at x equals four and y equals zero centimeters in the coordinate system, we have a charge of minus two nanoculums sitting there.

00:26 And we're placing a third charge of six, positive six, xanaculums and the coordinates of 4 x equals 4 centimeters and y equals 3 centimeters.

00:36 And the question is, the first part of the question is asking find the x and y components of the resulting total force.

00:43 And part b is asking for finding the total magnitude or the magnitude of total force and the direction of the total force.

00:50 So first of all, if we observe the diagram, then we have that the distance between charge 1 and charge 3 is the episode value of the r13 vector.

01:01 This we can already work out, given that this is a right -neigh the triangle, but we will do that in a little bit in a little bit when we get to that force.

01:10 And also, this force is going to be making an angle of theta -1 -3 with the horizontal.

01:15 At the same time, the distance between q2 and q3 is the absolute value for the r -2 -3 vector.

01:24 And that's just going to be pointing straight down.

01:26 So like i said, we're going to have to find the x and y components of the total force.

01:32 And the force is going to be given, since these are, you know, electric charges interacting, is going to be given by the coolant force, where r is the separation between the two charges, big q, capital q and small q are the two charge, magnitudes, k is coolum's constant.

01:48 And r hat is the radial direction vector pointing from one charge to the other.

01:54 So let's start with the force of two exertion three.

01:59 This is easier.

02:00 Because 2 is right below church 3.

02:03 And so the component or the direction of the resulting colon force is going to be pointing straight down in negative y head directions.

02:10 That's why i have this over here.

02:13 So that also tells us that the x component of the force that 2 exerts of 3 is 0.

02:20 And then to find the y component, we just put in the values.

02:24 So we know that q3 is sitting at y equals 3, whereas q2 is sitting at y equals 0.

02:30 So the total distance, so the magnitude of the distance between them, is just three centimeters.

02:36 So using that and the value of 8 .99 times 10309 for a coolum's constant and playing in all the values, we find that the y component of the force that charge 2 exerts and charge 3 is equal to 1 .20 times 1030 minus 4 neutrons acting in the negative y head direction.

02:55 So that's the total charge, or that's the total force that 2 exerts in 3.

03:00 Now, how about the force that 1 exerts on 3? so, like i said, one is a distance of a pythagorean distance of 5 centimeters away from church 3.

03:13 And the way we know that is we know that the x separation between those two points is 4 centimeters, whereas the wide separation between the two points is 3 centimeters, and then we just use pythagoras ' theorem to figure out that the hypotenuse of this right -headed triangle is 5 centimeters.

03:27 And then we also know that this force is going to be making an angle with the horizontal.

03:33 And that angle is going to come from also this triangle.

03:37 So we have to find the angle that the triangle makes with the horizontal order with the x -axis.

03:43 And so the way we do that is we just take the arc 10 of the y and x coordinates.

03:48 So we get that 10 minus 1 over y of y over over x is equal to 36 .9 degrees...

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