(III) An 8.00 μ C charge is on the x axis of a coordinate system at x = + 5.00 cm . A - 2

step 1 This is chapter 21 problem number 94. This problem requires plotting a function as a graph. So I

(III) An 8.00 μ C charge is on the x axis of a coordinate...

Key Concepts

Coulomb's Law

Coulomb's law is the fundamental principle used to calculate the magnitude and direction of the electric field produced by a point charge. It states that the electric field is proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point of interest and the charge. This law is crucial when analyzing forces between charges and determining field strengths in various configurations.

Electric Field

The electric field is a vector field that represents the force per unit positive charge at any given point in space. It is created by electric charges and is defined mathematically by Coulomb's law, which states that the field due to a point charge decreases with the square of the distance from the charge. Understanding the electric field is essential for explaining how charges interact in space without direct contact.

Superposition Principle

The superposition principle is a key concept in electrostatics stating that the total electric field created by multiple charges is the vector sum of the individual fields produced by each charge separately. This principle allows for the analysis of complex charge distributions by breaking them down into simpler point charge contributions and then summing the effects, which is especially useful when dealing with multiple charges in different spatial arrangements.

Vector Decomposition

Vector decomposition involves breaking a vector, such as the electric field, into its component parts along defined axes (e.g., x and y directions). This process is critical when analyzing fields in a coordinate system because it allows for separate analysis of the effects in each direction. Understanding vector decomposition is essential for plotting and interpreting the direction and magnitude of fields, particularly in situations with symmetry or when assessing fields along specific lines or axes.

Electric Field Mapping and Visualization

Mapping the electric field involves plotting its components over a region of space to visually represent how the field varies with position. This concept is important for understanding spatial variations in field strength and direction, and it aids in the interpretation of physical phenomena such as force distributions and regions of equilibrium. Visualization through plots helps in comprehending the influence of multiple charges and the effects resulting from their positions.

Transcript

00:01 This is chapter 21 problem number 94.

00:05 This problem requires plotting a function as a graph.

00:11 So i suggest that you either have mathematica or matlab or any other software that you would prefer.

00:19 And ready for this one.

00:21 So what we're given basically is this.

00:25 We're giving two point charges, equidistant as far as the origin is concerned.

00:34 Okay, q1, let's call this q1, it's a positive one, 8 microcoloms, and q2 is a negative 1, negative charge, negative 2 microcoloms.

00:45 Now we're asked in part a, plot the x component of the electric field, the x, for points on x -axis from, for points on, on x axis from x equals negative 30 centimeters to x equals 30 centimeters.

01:10 So this is our range, right? so we are actually trying to plot the electric field in the x direction as a function of where we are in the extraction.

01:22 This is the x -axis, right? so we might be either here in this region.

01:30 So there are several regions is what i'm, i guess, trying to say here.

01:34 Region 1, where we have the x less than the negative d, negative d being the distance of the second charge with respect to the origin.

01:47 And this will also be, so the location of q1 is positive d, location of k2 is negative d.

01:52 So if we're in this region, region one, let's call it, then what would be the direction of the net electric field due to these two charges? so pick an arbitrary point here, call it that, well, let's assume that it's x away from the origin, okay? so the electric field due to a negative charge at this point, but let's call that e2.

02:18 E2 would be towards the charge, right? this would be the direction of e2 and q1, the electric field generated by q1 would be the opposite direction because it's a positive charge.

02:29 Always the electric field emanating from it is radially outward.

02:32 So the net electric field would be e2 minus e1, right? so we're writing this as a function of x.

02:38 So then it's going to be k times q2, the magnitude of it minus magnitude of x minus d squared, minus e1bb k21 over again x plus d squared, right? so then that's going to be equal to kk2 over negative x negative d squared minus kq1 over negative x plus d squared.

03:06 Okay, this is the region one.

03:08 Now, what if our x, what if we're calculating the electric field in region two, meaning that net field anywhere between the origin and where q2 is located.

03:23 So then if that is the case, then our negative d, negative d is less than x less than zero, right? this is the region 2.

03:32 So the net electric field, let's see what happens.

03:35 Pick an arbitrary point here.

03:38 Due to the electric field due to q2 would be toward q2 because it's a negative charge.

03:45 So it's the direction of e2.

03:48 And q1 is going to generate the electric grid that is also in the same direction radially outward.

03:54 Then the net field will be negative e2, negative e1, right? so e would be negative e2, negative e1.

04:03 Both are negative, right? that's better.

04:09 Then what is the magnitude of negative e2, negative k, q2.

04:14 This is the magnitude.

04:18 Over d minus x squared minus b1 is kk1 over x plus d squared right so then it's going to be equals negative q2 kk2 over d plus x squared minus kk1 over negative x plus d squared now what happens if you're in the third region so we're looking at the electric fields from the origin to where where q1 is located.

04:50 A arbitrary point on picking, q1 would be away, right? this would be e1, the electric field due to q1.

04:58 Also, the electric field due to q2 would be in the same direction because it's a negative charge.

05:04 So this is pretty much the same region, same thing as region two, right? so 0x, d, e, again, equals negative e2 minus e1.

05:17 So we're getting the same result from here.

05:26 So the function is going to look pretty much similar within these regions.

05:35 Now we have the last region, right? it could be between q1 and positive x anywhere.

05:44 So let's pick the point here.

05:45 Q1 is going to be to the right, right? this would be e1.

05:49 E2 would be to the left.

05:51 E2.

05:52 So for the region where we have x greater than d, we have e equals e1 minus e2, right? e1 being in the positive restriction, then kq1 over d plus x, pardon me, d minus x, minus x, right? not plus it anymore.

06:16 Minus x squared minus k q2 over x plus d squared all right now we have all these functions right what you need to do is to plot them in a software that you know how to use so this function this function where they're pretty much the same so we have four regions when you plot the these functions you're going to see it looks something like this.

06:51 Let's say this is the strength of the electric field.

06:54 This is the x -axis...

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