Two tiny spheres of mass 6.80 mg carry charges of equal magnitude, 72.0 nC, but opposite sign. T

Two tiny spheres of mass 6.80 mg carry charges of equal...

Two tiny spheres of mass $6.80 \mathrm{mg}$ carry charges of equal magnitude, $72.0 \mathrm{nC}$, but opposite sign. They are tied to the same ceiling hook by light strings of length $0.530 \mathrm{~m}$. When a horizontal uniform electric field $E$ that is directed to the left is turned on, the spheres hang at rest with the angle $\theta$ between the strings equal to $58.0^{\circ}$ (Fig. P21.74). (a) Which ball (the one on the right or the one on the left) has positive charge? (b) What is the magnitude $E$ of the field?

Key Concepts

Static Equilibrium

Static equilibrium refers to the state in which an object remains at rest because the net force acting on it is zero. This concept is essential in mechanics as it allows us to set up equations by balancing forces in all directions, which can then be used to determine unknown forces or angles in a system where multiple forces like gravitational, tension, and electric forces are present.

Vector Decomposition

Vector decomposition involves breaking forces into their perpendicular components, typically horizontal and vertical, so that the complex interaction of forces at an angle can be analyzed separately along each axis. This technique simplifies the solving of equilibrium conditions by enabling the separate analysis of the forces acting in each direction.

Electric Field and Electric Force

An electric field is a region surrounding a charged object where other charged objects experience a force. The force experienced by a test charge in an electric field is proportional to the magnitude of the charge and the magnitude of the field, given by F = qE. This principle is crucial for understanding how charged particles respond to external electric fields in various configurations.

Coulomb's Law

Coulomb's law describes the magnitude of the electrostatic force between two point charges. The law states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. This law is fundamental in determining the interactions between multiple charged bodies in a system.

Transcript

00:02 All right, so in this problem, we have the set up, uh, we have to threat, connected with to pose.

00:12 And the mess of the bulls are and for both and the charge, uh, they have the same.

00:19 They have the same amount of magnet magnitude of the charge, but artists are so let's suppose this is q isn't that cute? and the advance of the threat is l.

00:30 And this angle is data.

00:34 So we have electric field toward the left direction, and we know ah, yeah, this is an electra feud.

00:43 So for porn, a want to determine the sign of the charge proposed balls.

00:49 So, uh, it's pretty easy because this boy's ah is tends to get tends to get close to the electric food.

00:58 But this one tends to get further away from the electric feud.

01:01 So apparently ah, this wise negative sign and this one's positive.

01:06 Something positive sign.

01:08 So we could just say that cure is positive.

01:12 And for part b ah, we won't to determine the the strange something like to feud.

01:21 So, given the condition theater equal to 58 degrees and they'll is, ah, 0.53 meters.

01:30 Q.

01:31 Is, uh, 72 non locals and is 6.8 milligrams.

01:40 So, uh, let's look at this one.

01:46 So the free body diagram for this fall, we have, ah force allowing the threat.

01:54 And we have the gravity, and we have the electric feud induced by the it's external field.

02:03 Ah, so we just pull this direction.

02:07 Yeah.

02:07 And we also have the, uh, electric force induced a pie baseball.

02:13 Right.

02:15 So this is towards this direction, so just gave them the names.

02:19 This is ft.

02:21 This is f g.

02:24 This is e.

02:26 This is f cube.

02:28 Okay, so we have full forces acting on this full, and this boy is incredible.

02:36 So, uh, we know that after e and f q, your, ah, their peril.

02:41 But they're adjusted toward the opposite direction.

02:44 So, uh, the the expression for f is ah e times cute, but toward the right direction and the full f q...

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