A small, rigid object carries positive and negative 3.50 -nC...

A small, rigid object carries positive and negative 3.50 -nC charges. It is oriented so that the positive charge has coordinates $(-1.20 \mathrm{mm}, 1.10 \mathrm{mm} \text { ) and the negative charge is }$ at the point $(1.40 \mathrm{mm},-1.30 \mathrm{mm} \text { ). (a) Find the electric }$ dipole moment of the object. The object is placed in an electric field $\overrightarrow{\mathbf{E}}=(7800 \hat{\mathbf{i}}-4900 \hat{\mathbf{j}}) \mathrm{N} / \mathrm{C}$ . (b) Find the torque acting on the object. (c) Find the potential energy of the object-field system when the object is in this orientation. (d) Assuming the orientation of the object can change, find the difference between the maximum and minimum potential energies of the system.

Key Concepts

Uniform Electric Field

A uniform electric field is one in which the magnitude and direction remain constant at all points in space. This simplification allows for direct calculation of forces, torques, and potential energies acting on charges and dipoles, making it a fundamental concept in electrostatics.

Potential Energy of an Electric Dipole

The potential energy of an electric dipole in an electric field is given by the negative dot product of the dipole moment and the electric field. This relationship reflects how the energy of the dipole depends on its orientation relative to the field, with minimum energy achieved when the dipole aligns with the field.

Torque on an Electric Dipole

The torque on an electric dipole in an external electric field is generated by the tendency of the dipole to align with the field. This torque is computed as the cross product of the dipole moment and the electric field, and it determines the rotational effect experienced by the dipole.

Electric Dipole Moment

An electric dipole moment is a measure of the separation of positive and negative charges, defined as the product of one of the charges and the displacement vector between the charges. It is a vector quantity that influences how the dipole interacts with external electric fields.

Transcript

00:01 We are given an electric dipole with a positive charge oriented at a position r plus, and the negative charge oriented at a position r minus.

00:09 And we know the charge at each end of the dipole is given by q.

00:13 Now we are asked to find the dipole moment p, which we know is p equals qd, where d is the vector from q minus to q plus, essentially describing the difference between these two positions.

00:29 So we can rewrite this, the difference of their position vectors, right? so that would mean d is just r plus minus r, oops, minus.

00:41 So let's go ahead and do that, and you should find that d is going to be equal to negative 1 .2 minus 1 .4i plus 1 .1 plus 1 .3j.

01:01 And if you work this out, you should see that d is equal to negative 2 .6i plus 2 .4j.

01:13 And remember, we're looking for the dipole moment p, which is given by qd.

01:19 And we have d.

01:20 So to get p, we should multiply this by q, which we know is 3 .5 times 10 to the negative 9.

01:29 And we should also divide this by 1 ,000 because we want to be working in meters.

01:35 And our original vectors were given in millimeters.

01:40 So doing all this, you should find that our dipole moment p is equal to negative 9 .1i plus 8 .4 j times 10 to negative 12 koums per meter.

02:00 Now imagine that our dipole, which i have shifted over shown by that's blue vector in our diagram, is now put in an electric field with position and magnitude given by e.

02:09 Now when a dipole is put in an electric field, it wants to align with that electric field, and the force it feels we can describe with torque.

02:16 So what is this torque? well, we know that torque, torque is just equal to p.

02:25 Cross e.

02:30 So knowing how to do a cross product, we end up with this equation.

02:35 And since this is only a two -dimensional cross product, we end up with just a formulation that looks a little bit like this over here.

02:43 So let's do that out.

02:45 So torque tau is equal to negative 9 .1 times 10 to negative 12 times negative 4 ,900, minus 7 ,800 times 8 .4 times 10 to the negative 12.

03:06 Remember, it's all in the k direction.

03:09 Keep in mind how we compute our cross product.

03:11 And we find that torque is just going to be equal to negative 2 .09 times 10 to the negative 8 in the k direction, per meter.

03:33 All right...

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