CP A small sphere with mass m carries a positive charge q and is attached to one end of a silk f

CP A small sphere with mass m carries a positive charge q...

Key Concepts

Electric Field of a Charged Surface

This concept deals with the electric field produced by a uniformly charged surface. Using Gauss’s law, one can derive that an infinite planar sheet with uniform surface charge density produces a constant electric field, independent of distance, given by E = ?/(2??). This field direction is perpendicular to the surface, and its magnitude plays a key role in evaluating the force experienced by other charges in its vicinity.

Static Equilibrium and Force Balance

In static equilibrium, the net force acting on a body must be zero. This principle requires that all forces, including gravitational, electric, and tension forces, balance each other. Understanding how these forces interact and cancel out is crucial in solving problems where an object is at rest under the influence of multiple forces.

Force Decomposition into Components

Force decomposition involves breaking down a force into its horizontal and vertical components based on the geometry of the problem. This is essential when analyzing equilibrium situations, as it allows for the individual balancing of forces in each direction. The use of trigonometric functions to relate these components is a fundamental tool in mechanics and electrostatics.

Trigonometric Relationships in Equilibrium Analysis

Trigonometry provides the link between the geometric setup of the system and the magnitudes of forces acting in different directions. In this context, the tangent of the angle made by a fiber or string with the vertical is expressed as the ratio of the horizontal force (such as the electrical repulsive force) to the vertical force (like gravitational weight). This relationship is key to deriving the equilibrium angle in problems involving inclined supports.

Transcript

00:01 In this exercise, we have a particle here in green that has mass m and positive charge q.

00:07 And this particle is hung by a massless and unconstructive wire that is attached to an infinite sheet of surface charge density sigma.

00:19 And when the mass is in equilibrium, the wire makes an angle theta with the sheet.

00:26 And we must find what is the expression? 4 -theta.

00:35 Okay, so first let's understand what equilibrium means in the system.

00:41 So equilibrium means that the force in the y direction, so the sum of all forces, fy, must be zero, and the sum of the forces acting on the x direction on the green particle, all some must be zero.

01:01 So let's see what forces are acting in.

01:05 In our small particle.

01:07 So first, we know that the particle has a mass, so we can infer that there is a gravitational force acting downwards in the wide direction, which i'll call it fg.

01:23 And also, we know that this particle is subjected to the electric field generated by this surface charge of the sheet.

01:33 So this electric field, this uniform electric field, will also generate a force on the x direction, an electric force, which i'll call fe.

01:49 And furthermore, we have one last force, which is the tension on the wire, which i'll draw here in red, so it's easier for us to see.

02:03 I'll call it t.

02:06 Okay, so we know that the electric force is on the x direction and the gravitational force acts on the y direction.

02:16 We can see that the tension has x and y components.

02:22 So if we do a projection of this force, we have that on the y direction, we have what i will call t y, the y projection of the tension, and in the x direction we have what what i'll call tx, the x projection of the tension.

02:47 And from some trigonometric relations, we can notice that this angle here, from this small triangle, is theta.

02:56 So i'll draw it bigger so we can see better what is going on.

03:01 So here again, we have our particle, we have the gravitational force, acting downwards, fg, we have the electric force acting on the horizontal axis of e we have the tension along the wire and we have the projection of the tension on the x -axis on the y -ex sorry um t y here i made a mistake this is t y and t x instead of the opposite it.

03:50 And here is the projection on the x -axis, tx.

03:57 And those angles here are theta.

04:04 And this is the overall tension vector.

04:09 Okay, so we can apply the equilibrium condition.

04:13 So let's look first at the y components...

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