A small 12.3 g plastic ball is tied to a very light 28.6 cm string that is attached to the ver

A small 12.3 g plastic ball is tied to a very light 28.6...

A small $12.3 \mathrm{~g}$ plastic ball is tied to a very light $28.6 \mathrm{~cm}$ string that is attached to the vertical wall of a room (Fig. $\mathbf{P} 2 \mathbf{1 . 6 3}$ ). A uniform horizontal electric field exists in this room. When the ball has been given an excess charge of $-1.11 \mu \mathrm{C},$ you observe that it remains suspended, with the string making an angle of $17.4^{\circ}$ with the wall. Find the magnitude and direction of the electric field in the room.

Key Concepts

Vector Decomposition

Vector decomposition involves breaking a force into its horizontal and vertical components. This technique is crucial in problems where forces act at angles, allowing each component to be analyzed independently in accordance with equilibrium conditions.

Uniform Electric Field

A uniform electric field is a region where the electric field has the same magnitude and direction at every point. It exerts a constant force on any charged particle placed within it, with the force given by the product of the particle's charge and the field strength.

Electric Force on Charges

The force experienced by a charged object in an electric field is determined by F = qE, where q is the charge and E is the electric field. This concept is fundamental in understanding how charges interact with electric fields, with the direction of the force depending on the sign of the charge.

Static Equilibrium

Static equilibrium refers to the state in which an object is at rest, meaning that the net forces acting on it in all directions are zero. This principle is essential in analyzing systems where multiple forces, such as gravitational, tension, and electric forces, balance each other.

Free-body Diagram

A free-body diagram is a schematic representation that isolates an object and shows all the forces acting upon it. It is a key tool in solving mechanics problems as it helps systematically apply Newton’s laws by visualizing the balance of forces in different directions.

Transcript

00:01 Hello everyone.

00:02 In this problem, we're asked to find the electric field, magnitude, and direction that exists in a room, such that when we charge a plastic bowl of 12 .3 grams to a charge of minus 1 .11 times that minus 6 kulams, it makes an angle of 17 .4 degrees and remains suspended in the air.

00:21 So what we're going to do is we're going to just draw this schematic basically here.

00:26 This kind of already tells you what the direction of the electric field has to be.

00:29 It has to act to the left because the electric field lies tell you in which direction of positive charge we'd want to move.

00:36 So if the positive charge would want to move to the left, then the negative charge will want the moves that are right.

00:41 So that means that the electric field has to point to the left since your bowl is negatively charged and is remaining suspended in the air.

00:50 So then what we do is we just analyze the free video diagram and we see that there is a tension coming from the rope or the string with which the bowl is attached to the wall.

00:59 There is a cool on force, which is equivalent to the charge times the electric field.

01:05 And then there's, of course, the weight of the ball pulling it down.

01:08 So we'll look at the components of these forces.

01:12 And so we look at all the forces acting in the x direction first.

01:15 So in the x direction, we have that the tension has a component acting in this direction, which is t -sign theta.

01:22 And the other force that's acting in the x direction is the coolon force.

01:27 And the thing to notice is that this has to sum to zero.

01:30 And the reason it has to sum to zero is that this is a static situation...