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00:00:00 – 00:08:13
The YouTube video provides a detailed instructional guide on calculating the electric field due to a charged solid disc. The instructor begins by introducing the concept of area charge density, which is the charge per unit area of the disc. To find the electric field at a point a certain distance away, the instructor considers an infinitesimal ringlet of charge on the disc and uses the electric field formula for a ring of charge.
The calculation involves integrating the contributions of these differential ringlets across the entire disc. Constants such as ( K ), ( Sigma ), ( pi ), and ( a ) are factored out to simplify the integral, with a substitution simplifying the integrand. During this process, the instructor ensures terms cancel properly, ultimately expressing ( K ) as ( frac{1}{4 pi epsilon_0} ) for further simplification.
In the final steps, the instructor reviews and concludes the integration, resulting in an expression for the electric field involving the surface charge density ( sigma ), the permittivity of free space ( epsilon_0 ), and the disc's dimensions. The complete derivation illustrates how the electric field is derived, considering only the horizontal components that do not cancel out.
00:00:00
In this segment of the video, the instructor discusses how to find the electric field due to a solid disc with charge. They introduce the concept of area charge density, which is derived by dividing the charge ( Q ) by the area ( pi R^2 ), where ( R ) is the radius of the disc. The goal is to determine the electric field at a point a certain distance away from the disc.
To do this, the instructor considers a small ringlet on the disc with a differential charge ( dQ ). They recall the electric field formula for a ring of charge from a previous video, which involves the charge ( dQ ) and the distance from the disc to the point of interest. For the current problem, ( dQ ) is determined by multiplying the area charge density ( sigma ) by the area of the ringlet ( dA ), which is the circumference ( 2pi x ) (where ( x ) is the radius of the ringlet) times the thickness ( dx ).
The instructor then expresses the differential electric field ( dE ) using the formula involving ( dQ ), and explains that to find the total electric field, they need to integrate this expression across all ringlets from the center to the outer edge of the disc. Only the horizontal components of ( dE ) are considered, as the perpendicular components cancel out when integrated around the ringlet. The segment concludes with the preparation to integrate and sum the contributions of all the ringlets to find the total electric field.
00:03:00
In this part of the video, the speaker explains the process of computing the electric field (E) by integrating over the radius of a disk. The integral is set up with constants ( K ), ( Sigma ), ( pi ), and ( a ) factored out, leaving an integral in terms of ( x ). The integral is from ( x = 0 ) to ( x = R_0 ), with the integrand involving a term in the denominator raised to the ( frac{3}{2} ) power which simplifies after performing a substitution to make the integration straightforward. The result of the integration includes constants that simplify through cancellations, including rewriting ( K ) as ( frac{1}{4 pi epsilon_0} ) to facilitate further simplification, ultimately leading to a simpler expression for the electric field.
00:06:00
In this part of the video, the instructor concludes the evaluation of the electric field generated by a charged disc. The field equation simplifies to an expression involving the surface charge density (σ), permittivity of free space (ε₀), and the disc’s dimensions. By multiplying and rearranging terms, the final result indicates the electric field in vector form pointing in the X direction. Throughout the segment, the instructor reviews the integration process, which involves summing up contributions from infinitesimally small ringlets of charge across the disc’s surface, demonstrating how the electric field at a specific location is derived.