CALC Negative charge -Q is distributed uniformly around a quarter-circle of radius a that lies i

CALC Negative charge -Q is distributed uniformly around a...

Key Concepts

Coulomb's Law

Coulomb's law quantifies the electric field produced by a point charge, stating that the field is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge. It serves as the fundamental starting point for calculating the electric field contributions of differential charge elements in a continuous charge distribution.

Continuous Charge Distributions

A continuous charge distribution treats the charge as spread smoothly over a region, requiring the use of integration to sum the electric field contributions from infinitesimal charge elements. This technique allows the calculation of the net electric field generated by the entire distribution, rather than considering isolated charges.

Superposition Principle

The superposition principle states that the net electric field created by a group of charges is the vector sum of the electric fields produced by each individual charge. This concept is essential for problems involving charge distributions, as it permits the integration of differential fields over the distribution to obtain the total field.

Vector Decomposition

Vector decomposition involves breaking down a vector into its component parts along specified coordinate axes. In electric field calculations, this process is used to resolve each differential electric field contribution into its x and y components, facilitating the integration of these components over the charge distribution.

Integration over a Curved Geometry

When dealing with charge distributions along curved geometries, such as arcs, it is necessary to express differential elements of the curve (using variables like the polar angle) and incorporate the geometry into the integration process. This method allows for the accurate computation of the net electric field by taking into account the changing directions and distances of the differential charge elements relative to the point of interest.

Transcript

00:01 In this exercise, we have a negative charge minus q that is uniformly distributed on a quarter of circle of radius a.

00:09 And in this exercise, what we want to calculate is what is the electric field generated by minus q measured by someone that is placed on the orange of the circle.

00:21 And the exercise asks us explicitly to calculate the x and y components of this field.

00:29 So how do we calculate the electric field generated by this weird wire? well, killam's law only allow us to calculate the electric fields generated by a point charge.

00:46 And here we are not working with a point charge at all, but we can treat this wire of charges as a collection of infinite small, small, small charge.

00:59 Charges the q here as an infinite collection of charges so we can use the infinitesimokum's law.

01:13 Okay, so what is the q? so to calculate the q, we must first notice that q can be expressed as the perimeter of a quarter of circle, which is pi over 2 times a, which is the radius of circle.

01:36 This is a the perimeter of a quarter of circle times the density of charges along the line, which i'll call it lambda.

01:49 Here, pi over 2 times a, our perimeter will be our equivalent on the infinitesimal charge, the q, to something that we call the line element, the s.

02:06 So the q is just yes times times lambda.

02:14 And what is the s in our case? well, we are working with polar coordinates.

02:19 In this case, the s is just this small element here of length.

02:26 And we can notice that this length can be easily measured if we take, if we notice that it comprehends a infinitesimal angle, d theta.

02:40 So as we took that the wire comprehends a pi over two, two degrees over circle, we can take that the s comprehends a theta angle over circle of radius a so that the s can be expressed as a d theta and the q is a d theta times pi.

03:10 So before we move on and substitute to the q in the infinitesimun column law, we must it is important to know to where the electric field is pointing to.

03:25 So we know that the electric field generated by these negative charges, we're pointing in the radial direction because of symmetry.

03:40 And also because those charges are negative, we know that the electric field must be pointing towards the charges.

03:48 So they're pointing, if we talk about polar coordinates, r and theta, it points to the r direction, to the radio direction.

04:09 From polar coordinates, but we have that the x component is x times r, which in our case is a, the radius of the circle times theta, and y is equal to a times sine theta, so that we can rewrite the radial direction r as x times the x direction plus y in the y direction over a, that is the magnitude of the radio vector.

04:58 And in the end, we have that the radial direction in this exercise is just cosine of theta in the x direction plus sine theta in the y direction.

05:15 Okay, now that we have this quantity, we know to where the field is pointing to, and we decompose it into the x and y direction, so we can calculate ex, ey.

05:29 We are ready to apply the infinitesimal columns law.

05:35 So we have that, the infinitesimal columns law, the e, is equal to the q, which we already saw that it is equal to a, the theta times gamma over 4, pi, epsilon zero, a squared.

05:58 So in our case, we can integrate the e so that we have that the vector, the electric field vector that we already know that points to the r direction is the integral over theta...

Please give Ace some feedback