A dipole consisting of charges ±e, 220 nm apart, is placed between two very large (essentially i

step 1 In this question, we have two really large plates. One plate has a positive charge, surface char

A dipole consisting of charges ±e, 220 nm apart, is placed...

Key Concepts

Potential Energy of a Dipole in an Electric Field

The potential energy of a dipole in an external electric field is given by the negative dot product of the dipole moment and the electric field. This energy reaches an extremum when the dipole is aligned in a particular orientation relative to the field. Understanding this relationship is key to analyzing stable and unstable configurations of dipoles in external fields.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. This configuration is fundamental in electrostatics and is used to model many physical situations, from molecular polarity to antenna behavior. The dipole’s behavior in external electric fields provides insight into both the field and the intrinsic properties of the dipole itself.

Dipole Moment and Its Orientation

The dipole moment is a vector quantity defined as the product of the magnitude of one of the charges and the separation distance, directed from the negative to the positive charge. Its orientation relative to an external electric field determines the potential energy and the torque experienced by the dipole. The alignment or misalignment between the dipole moment and the field is crucial in determining the system's behavior and stability.

Electric Field Due to Infinite Sheets

Infinite charged sheets produce a uniform electric field that is constant in magnitude and direction at all points in space, excluding edge effects. This uniformity greatly simplifies calculations as the field does not vary with distance from the sheet. The uniform field is central to many theoretical models and is derived from Gauss's law in electrostatics.

Torque on a Dipole in an Electric Field

The torque experienced by an electric dipole in an external electric field is calculated as the cross product of the dipole moment and the electric field. This torque tends to rotate the dipole so that it aligns with the field, minimizing the system’s potential energy. The magnitude of the torque depends on both the strength of the dipole moment and the electric field, as well as the sine of the angle between them.

Net Force on a Dipole in a Uniform Field

In a uniform electric field, the net force on an electric dipole is zero because the forces on the individual charges are equal in magnitude but opposite in direction. Despite experiencing a torque which tends to align the dipole with the field, the dipole does not translate because the spatially constant field exerts no net force. This principle distinguishes the behavior of dipoles in uniform versus non-uniform fields.

Transcript

00:01 In this question, we have two really large plates.

00:13 One plate has a positive charge, surface charge density, the other one has a negative charge density, and then in between the two plates, there's a dipole, okay, and then they are separated by 220 nanometers apart, and this is plus e, this is negative, this is negative.

00:36 E, okay, in this question, we want to look at the maximum potential energy and the maximum torque and the net's force acting on the dipole, and the dipole is inside the electric field due to the charge plates.

00:55 Okay, and then we are given that the sigma is 125 times 10 to the negative six, uh, columns per meter square okay so okay so first we want to find a maximum potential energy okay the formula will be uh equals to u equals to negative p .e okay so from the diagram okay that if you is pointing to the right okay that if you the dipole moment is pointing from the negative charge to the positive charge so that's how the two vectors look like on the diagram.

01:41 And then this is negative p times e cosine data.

01:50 So umax will be negative p times e cosine 180 degrees.

01:59 All right, so and cosine 180 degrees is negative one.

02:03 So you have p and times e together.

02:05 Okay so the dipole moment is e times d and then the electric field in this case it would be sigma over xll0 okay so what happens is that inside inside the in between the two plates you have the e plus is equal to sigma over to e0 and then you also have the e minus equals to sigma over 2 e0 okay both are pointing to the right in the in between the space of the two plates so the e will be sigma over a epsilon or not you just sum the two vectors okay so now you can substitute the numbers 1 .6 times 10 to the negative 19 d is 220 nanometers okay sigma is 125 times 10 to the negative 6 and epsilon 0 .85 times 10 to the negative crop.

03:11 You calculate this you get 4 .97 times 10 to the negative 19 juice.

03:19 Okay.

03:23 And then what orientation? the orientation for the dipole moment.

03:27 We know that this is 180 degrees, which means that p and e are opposite in direction.

03:41 To get umax okay so the positive charge is facing the positive plate okay yeah so the diagram would look like this okay so you have the p pointing to the left and then the e pointing to the right okay this is negative sigma okay yeah so this is the answer for the orientation okay so yeah this is plus sigma okay yeah in part b i want to find a maximum torque okay will be using the formula for torque which is p cross e okay so the many to our torque will be many 2 of p times 92 of e times sine data okay so talk max would be p times e times e times x 90 degrees this is just p times e this is the same as the previous part so then value the many 2 will be 4 .97 times 10 to the negative 19 newton meters okay so this is a answer for part b and then the orientation...

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