Negative charge -Q is distributed uniformly around a quarter-circle of radius a that lies in the

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

Key Concepts

Electric Field Due to Continuous Charge Distribution

This concept involves calculating the net electric field produced by a spread-out charge rather than a single point charge. It is done by dividing the charge distribution into small elements, evaluating the electric field contribution from each using Coulomb's law, and then integrating over the entire distribution.

Uniform Charge Distribution

A uniform charge distribution means that the charge is spread evenly over a given region, resulting in a constant charge density. This uniformity simplifies the calculation of the field because the infinitesimal charge elements are all proportional to the same density.

Coulomb's Law

Coulomb's law provides the expression for the electric field due to a point charge, stating that the field magnitude is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge. For continuous distributions, the law is applied to each infinitesimal element of charge, and the contributions are summed.

Integration Techniques in Electrostatics

Integration is a key mathematical tool in electrostatics, used to sum the contributions of infinitesimally small elements of charge. By appropriately parameterizing the geometry of the charge distribution, integrals can be set up to calculate the total electric field or potential at a point.

Vector Decomposition of Electric Fields

Since electric fields are vector quantities, they have both magnitude and direction. Decomposing the field into its components (for example, along the x- and y-axes) is essential for solving problems in which the geometry of the charge distribution lacks complete symmetry, enabling separate integration of each component.

Transcript

00:01 We have a following situation charge minus q is distributed on the quarter circle surface here.

00:09 So this is 1/4 circle exit.

00:13 And, uh, this is our present alexis x, and this is a war.

00:18 Little by, um, we are asked to find what is the e x and eve? i, um the electric field x component.

00:33 And why component it region.

00:36 So let's call this origin.

00:37 Oh, in order to find the e x and y compliment, um, we start writing.

00:44 Ah, the charge distribution on the the surface.

00:50 Um, we call deke your amount of charge, mr spiritually, let's say here it leant over d s.

00:59 Um, so we can write a dick.

01:00 You will be charged density lambda.

01:03 We call it times d s.

01:05 Is this more land here? so this is a d actually.

01:09 Looks like linda.

01:11 Yes, um, then at the distance off from here to here.

01:16 Recall this distance are, um, we'll call letter r, but initially will call this land.

01:24 Eh? um and here we have a that, uh and this is ah, deed that, um then we can write our d s.

01:39 This moreland d s will be simply eight times a d theta.

01:46 So the small d theta times this radius length will give us a d s then we can write the electric field at the origin will be by the definition off electric feel we know d e electric field will be coolum constant k times there due to the cq charge at the distance off a so called this pay square here.

02:16 So the distance from the surface here the surface so the origin is a so take these a square then we can write our dick you in terms off d s and linda which will give us a t e to be okay times.

02:34 Linda, um times, eh d theta divided by a square.

02:43 When will cancel out so we are left with electric field that is a k ll end up d that divided by hey, so we'll result d e so the eu will be in this direction.

02:57 We can resolve that in its component.

03:01 So, um d will be so resentful and work all component.

03:07 So leave.

03:07 We call this as a he x and this d why we can write that using a trick lamented ratios of than our d x d e x will be d e go state uh, d eve...

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