Three charges are at the corners of an isosceles triangle as...

Three charges are at the corners of an isosceles triangle as shown in $\textbf{Fig. E21.57.}$ The $\pm$5.00-$\mu$C charges form a dipole. (a) Find the force (magnitude and direction) the $-$10.00$\mu$C charge exerts on the dipole. (b) For an axis perpendicular to the line connecting the $\pm$5.00-$\mu$C charges at the midpoint of this line, find the torque (magnitude and direction) exerted on the dipole by the $-10.00 \space \mu$C charge.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electric force between charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. This principle is critical for determining the interactions between charged particles and is the foundation for calculating forces in electrostatics.

Electric Dipole

An electric dipole consists of two equal but opposite charges separated by a small distance. The dipole moment, a vector quantity, characterizes its strength and orientation. Understanding dipoles is essential in the study of molecular interactions, polarization, and the behavior of materials in external electric fields.

Force on Charges

The concept of force on charges involves calculating the vector sum of the forces acting on a charge due to other charges, using principles such as Coulomb's Law. The net force takes into account both the magnitude and direction of each individual interaction, and vector addition is used to combine these effects.

Torque on an Electric Dipole

Torque on an electric dipole in an external electric field arises due to the tendency of the dipole to align with the field. This torque is given by the cross product of the dipole moment and the electric field. It is a measure of the rotational effect about a pivot point, which is important for understanding the stability and orientation of dipoles in electric fields.

Superposition Principle

The superposition principle in electrostatics asserts that the net force on any charge due to a system of other charges is the vector sum of the individual forces exerted by each charge independently. This principle allows complex charge interactions to be broken down into simpler parts, making the computation of resultant forces and torques tractable.

Transcript

00:01 In this question, we have this setup or this arrangement of three charges.

00:06 So the 5 microculum, the plus 5 microculum and the negative minus 5 microculum charges, they form a dipole.

00:16 So there are two parts in the question.

00:19 We want to find the next force that the 10 microculum charge exits on the dipole and the top.

00:28 Okay, so first thing, you to find a net force.

00:35 Okay, so this is kulom's law.

00:38 Okay, so using kulom's law, okay, and then we just find the force on the individual charge, like the individual plus and minus five microculum charge by the minus 10, negative 10, microculum charge.

00:58 Okay, so f, the electric force, is k k u1 q2 divide by r square okay so on the five microculum charge this is the electric force and then on the negative five microcholomb charge this is the electric force okay so just to draw it better okay so this is f1 and then this is f2 okay and then uh by symmetry okay, what i meant here is equidistant.

01:43 Okay, so f1 is equal to f2 in magnitude.

01:51 Okay.

01:53 And then, and if you look at the geometry, from the arrangement, the horizontal components of f1 and f2 hands out.

02:25 So we only need the the, i need to sum the vertical components.

02:43 Okay, so this is the angle that we are interested in.

02:47 Okay, and this part will be 1 .5 meters, um, 0 .5 cm, sorry.

02:55 So, um, and, uh, and the, based on the diagram, the vertical component will be cosine, the cosine data, okay.

03:07 So, f net, okay.

03:13 Or the magnitude of the net force, magnitude of the net force.

03:24 Okay, so this is equal to 2, f1, coside, data.

03:35 So we know that they have the same magnitude, and then the horizontal components can set out, so the vertical components they just sum, and then it comes to f1 cosine data, okay? and then f1 would just be k q1 q2 divide by r square cosine theta so q1 is q1 is 5 microculum i'm just going to write micro and then the q2 is 10 microculum and then divide by r in this case is 2 cm so 0 .02 meters and then you square it cosine data from the diagram.

04:21 Okay, so you have 1 .5, you have two as a hypotenous, so it becomes 1 .5, you are 2.

04:29 Okay, and do this you get 1690 newton's, okay, and then the direction of the net force, okay, it is downward.

04:52 So this many tube and direction of the net force.

04:59 So this is part a.

05:01 Next we move on to part b.

05:03 Part b is to find a net torque.

05:05 So we just have the diagram again.

05:09 And the forces are pointing in this way.

05:23 All right...