Two identical beads each have a mass m and charge q . When...

Two identical beads each have a mass $m$ and charge $q$ . When placed in a hemispherical bowl of radius $R$ with frictionless, nonconducting walls, the beads move, and at equilibrium they are a distance $R$ apart (Fig. P23.68). Determine the charge on each bead.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges, stating that the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance separating them. This law is fundamental in understanding interactions between charged objects and is expressed by the formula F = k * |q1 * q2| / r^2, where k is Coulomb’s constant.

Static Equilibrium

Static equilibrium refers to a state where an object is at rest and the vector sum of all forces acting on it is zero. In such a situation, there is no net force or moment, ensuring that the object does not accelerate. This principle allows one to set up equations balancing gravitational, normal, and other forces to solve for unknown quantities.

Force Decomposition

Force decomposition involves breaking down forces into perpendicular components, typically along directions that simplify the analysis (such as horizontal and vertical axes or parallel and perpendicular to a surface). This technique is essential in solving equilibrium problems, especially when forces are applied at angles or in non-linear configurations.

Geometric Analysis in Equilibrium

Geometric analysis is crucial in problems where the configuration of components (such as distances, angles, or curvatures) influences the forces acting on an object. By understanding the spatial relationships and constraints imposed by shapes like curves or surfaces, one can accurately relate physical quantities, such as distances between charges, to the resulting forces in the system.

Transcript

00:01 In this question we have a hemispherical bowl, which is kind of like a semicircle bowl, and we have two beats being placed in the bowl.

00:15 And these two bits are each charged with the same charge of q and they have the same mass am.

00:24 Now at equilibrium we are given that they are at a distance of our apart.

00:30 So because of the same charge, they are likely to repel each other.

00:36 And we are given that they are at a distance of r apart, where r is the radius of this spherical bowl.

00:49 Now we want to find what is the charge on each of the beat, which is basically q.

00:53 And to do that, we will need to take a look at the forces that are cycling on each of the beats.

00:59 Now if you are to connect the bead to the center of the bow, this, the length over here is the radius, right? so what we have is in fact an equilateral triangle and as such we expect that the angle that it makes must be 60 degrees throughout because equilateral right and we're going to use this property to look at the free body diagram of a b so let us consider this left beat over here it experiences first of all the electric force from the other bit.

01:48 This is a repulsive force because they are the same charge.

01:51 So it's repulsing directly horizontal.

01:55 And then we have the normal, sorry, we have the weight.

02:00 That's the easier ones, which is perpendicular downwards.

02:03 This is the electrical force.

02:06 And then we have the normal force that's coming from the bow.

02:13 And we know that the normal force is acting at perpendicular to the to the bowl, to the surface of the bowl.

02:25 So it must be pointing towards the center of the spherical bowl.

02:30 And as such, we can label this angle as 60 degrees.

02:35 So this is where the 60 degrees angle comes in.

02:39 And because we know that it's an equilibrium, that is the net force acting on it must be equals to zero...