A long line carrying a uniform linear charge density +50.0 μC / m runs parallel to and 10.0 cm

A long line carrying a uniform linear charge density +50.0...

A long line carrying a uniform linear charge density $+50.0 \mu \mathrm{C} / \mathrm{m}$ runs parallel to and $10.0 \mathrm{~cm}$ from the surface of a large, flat plastic sheet that has a uniform surface charge density of $-100 \mu \mathrm{C} / \mathrm{m}^{2}$ on one side. Find the location of all points where an $\alpha$ particle would feel no force due to this arrangement of charged objects.

Key Concepts

Electric Force on a Charged Particle

The electric force experienced by a charged particle is directly related to the electric field present at its location, as expressed by the equation F = qE, where q is the charge of the particle and E is the electric field strength. Analyzing the conditions under which the net force is zero, such as finding equilibrium points where opposing electric fields cancel each other out, is key to many electrostatics problems.

Principle of Superposition

Superposition is a fundamental principle in physics which states that when multiple sources of electric field are present, the net electric field at any point is the vector sum of the electric fields produced by each individual source. This principle allows us to analyze complex charge configurations by adding their individual contributions together to determine the overall effect.

Electric Field of a Line Charge

This concept involves understanding how a continuous line of charge creates an electric field in the space around it. The field generated by a line charge decreases with distance and its magnitude can be calculated using a formula derived from Gauss's Law. The direction of the field is radially outward or inward depending on the sign of the charge. It is an essential concept in electrostatics for analyzing systems with cylindrical symmetry.

Electric Field of an Infinite Sheet

An infinite sheet of uniform charge produces a constant electric field in space, meaning that the field’s magnitude does not depend on the distance from the sheet. The direction of the field is perpendicular to the surface of the sheet, pointing away for positive charge and toward the sheet for negative charge. This concept simplifies the analysis of problems where planar symmetry is present.

Transcript

00:01 Okay, so for this problem, we have a very large plane and it has some sigma.

00:06 The sigma is negative.

00:09 And above this, at a certain distance away, we have a linear charge density along this line.

00:16 It's called this a distance d.

00:19 And this is positive.

00:24 So we can see that if we draw the electric field vectors due to each contribution, the plane and the line, that it should look something like this.

00:35 If we first begin with the plane, it's negative.

00:38 The line should go towards it, okay, something like this.

00:49 And for the line, things should radiate away, something like this.

01:05 Immediately we can see that if we had an alpha particle, which is a helium nucleus, if this particle were to be in this region right here, that it would never be able to come to rest because the electric fields would never cancel.

01:26 Remember that if we write down the newton's second law, f equals ma for electrical systems, this is the net electric field on that particle times its charge.

01:41 If i want this to equal zero, the sum of electric fields must equal each other.

01:47 Victorily, we can see that in this region between the plane and the line of charge, they are actually only adding together.

01:56 They're helping each other out, making the field stronger...

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